Wikipedia - User contributions [en]

Calreticulin

2025-07-30T04:28:01Z

174.138.212.166: wl

{{cs1 config|name-list-style=vanc}}
{{short description|Soluble protein}}
{{distinguish|calretinin}}
{{Infobox_gene}}

'''Calreticulin''' also known as '''calregulin''', '''CRP55''', '''CaBP3''', '''calsequestrin-like protein''', and '''endoplasmic reticulum resident protein 60''' ('''ERp60''') is a [[protein]] that in humans is encoded by the ''CALR'' [[gene]].<ref name="pmid2365822">{{cite journal | vauthors = McCauliffe DP, Zappi E, Lieu TS, Michalak M, Sontheimer RD, Capra JD | title = A human Ro/SS-A autoantigen is the homologue of calreticulin and is highly homologous with onchocercal RAL-1 antigen and an aplysia "memory molecule" | journal = The Journal of Clinical Investigation | volume = 86 | issue = 1 | pages = 332–5 | date = Jul 1990 | pmid = 2365822 | pmc = 296725 | doi = 10.1172/JCI114704 }}</ref><ref name="entrez">{{cite web | title = Entrez Gene: calreticulin| url = https://www.ncbi.nlm.nih.gov/gene?Db=gene&Cmd=ShowDetailView&TermToSearch=811}}</ref>

Calreticulin is a multifunctional soluble [[protein]] that binds [[calcium|Ca2+]] [[ion]]s (a [[second messenger]] in [[signal transduction]]), rendering them inactive. The Ca2+ is bound with low [[chemical affinity|affinity]], but high [[valence (chemistry)|capacity]], and can be released on a signal (see [[inositol trisphosphate]]). Calreticulin is located in storage compartments associated with the [[endoplasmic reticulum]] and is considered an ER resident protein.<ref name="entrez"/>

The term "Mobilferrin"<ref>{{MeSH name|Mobilferrin}}</ref> is considered to be the same as calreticulin by some sources.<ref>{{cite journal | vauthors = Beutler E, West C, Gelbart T | title = HLA-H and associated proteins in patients with hemochromatosis | journal = Molecular Medicine | volume = 3 | issue = 6 | pages = 397–402 | date = June 1997 | doi = 10.1007/BF03401686 | pmid = 9234244 | pmc = 2230203 }}</ref>

== Function ==

Calreticulin binds to misfolded proteins and prevents them from being exported from the [[endoplasmic reticulum]] to the [[Golgi apparatus]].

A similar quality-control [[chaperone (protein)|molecular chaperone]], [[calnexin]], performs the same service for soluble proteins as does calreticulin, however it is a membrane-bound protein. Both proteins, calnexin and calreticulin, have the function of binding to [[oligosaccharide]]s containing terminal glucose residues, thereby targeting them for degradation. Calreticulin and Calnexin's ability to bind carbohydrates associates them with the [[lectin]] protein family. In normal cellular function, trimming of glucose residues off the core oligosaccharide added during N-linked [[glycosylation]] is a part of protein processing. If "overseer" enzymes note that residues are misfolded, proteins within the [[rough endoplasmic reticulum|rER]] will re-add glucose residues so that other calreticulin/calnexin can bind to these proteins and prevent them from proceeding to the Golgi. This leads these aberrantly folded proteins down a path whereby they are targeted for degradation.

Studies on transgenic mice reveal that calreticulin is a cardiac embryonic gene that is essential during development.<ref name="pmid12445456">{{cite journal | vauthors = Michalak M, Lynch J, Groenendyk J, Guo L, Robert Parker JM, Opas M | title = Calreticulin in cardiac development and pathology | journal = Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics | volume = 1600 | issue = 1–2 | pages = 32–7 | date = Nov 2002 | pmid = 12445456 | doi = 10.1016/S1570-9639(02)00441-7 }}</ref>

Calreticulin and calnexin are also integral in the production of [[MHC class I]] proteins. As newly synthesized MHC class I α-chains enter the endoplasmic reticulum, calnexin binds on to them retaining them in a partly folded state.<ref>{{cite book | author = Murphy K |title=Janeway's Immunobiology | date = 2011 | publisher = Taylor & Francis | location = Oxford | isbn = 978-0815342434 | edition = 8th}}</ref> After the β2-microglobulin binds to the [[Peptide loading complex|peptide-loading complex]] (PLC), calreticulin (along with [[ERp57]]) takes over the job of chaperoning the MHC class I protein while the [[tapasin]] links the complex to the [[transporter associated with antigen processing]] (TAP) complex. This association prepares the MHC class I to bind an antigen for presentation on the cell surface.

=== Transcription regulation ===

Calreticulin is also found in the nucleus, suggesting that it may have a role in transcription regulation. Calreticulin binds to the [[synthetic peptide]] KLGFFKR, which is almost identical to an amino acid sequence in the [[DNA-binding domain]] of the superfamily of [[nuclear receptor]]s. The [[N-terminus|amino terminus]] of calreticulin interacts with the DNA-binding domain of the [[glucocorticoid receptor]] and prevents the receptor from binding to its specific [[glucocorticoid response element]]. Calreticulin can inhibit the binding of [[androgen receptor]] to its hormone-responsive DNA element and can inhibit androgen receptor and [[retinoic acid receptor]] transcriptional activities in vivo, as well as retinoic acid-induced neuronal differentiation. Thus, calreticulin can act as an important modulator of the regulation of gene transcription by nuclear hormone receptors.

== Clinical significance ==

Calreticulin binds to [[antibody|antibodies]] in certain area of [[Lupus erythematosus|systemic lupus]] and [[Sjögren syndrome|Sjögren]] patients that contain [[Anti-SSA/Ro autoantibodies|anti-Ro/SSA antibodies]]. Systemic lupus erythematosus is associated with increased autoantibody titers against calreticulin, but calreticulin is not a Ro/SS-A antigen. Earlier papers referred to calreticulin as an Ro/SS-A antigen, but this was later disproven. Increased autoantibody titer against human calreticulin is found in infants with complete congenital heart block of both the [[Immunoglobulin G|IgG]] and [[IgM]] classes.<ref>{{cite web | title = Entrez Gene: CALR calreticulin| url = https://www.ncbi.nlm.nih.gov/gene?Db=gene&Cmd=ShowDetailView&TermToSearch=811}}</ref>

In 2013, two groups detected calreticulin mutations in a majority of [[Janus kinase 2|JAK2]]-negative/[[thrombopoietin receptor|MPL]]-negative patients with [[essential thrombocythemia]] and [[myelofibrosis|primary myelofibrosis]], which makes ''CALR'' mutations the second most common in [[myeloproliferative disease|myeloproliferative neoplasms]]. All mutations (insertions or deletions) affected the last exon, generating a reading [[Frameshift mutation|frame shift]] of the resulting protein, that creates a novel terminal peptide and causes a loss of endoplasmic reticulum [[ER retention|KDEL retention signal]].<ref name="pmid24325359">{{cite journal | vauthors = Nangalia J, Massie CE, Baxter EJ, Nice FL, Gundem G, Wedge DC, Avezov E, Li J, Kollmann K, Kent DG, Aziz A, Godfrey AL, Hinton J, Martincorena I, Van Loo P, Jones AV, Guglielmelli P, Tarpey P, Harding HP, Fitzpatrick JD, Goudie CT, Ortmann CA, Loughran SJ, Raine K, Jones DR, Butler AP, Teague JW, O'Meara S, McLaren S, Bianchi M, Silber Y, Dimitropoulou D, Bloxham D, Mudie L, Maddison M, Robinson B, Keohane C, Maclean C, Hill K, Orchard K, Tauro S, Du MQ, Greaves M, Bowen D, Huntly BJ, Harrison CN, Cross NC, Ron D, Vannucchi AM, Papaemmanuil E, Campbell PJ, Green AR | title = Somatic CALR mutations in myeloproliferative neoplasms with nonmutated JAK2 | journal = The New England Journal of Medicine | volume = 369 | issue = 25 | pages = 2391–405 | date = Dec 2013 | pmid = 24325359 | pmc = 3966280 | doi = 10.1056/NEJMoa1312542 }}</ref><ref name="pmid24325356">{{cite journal | vauthors = Klampfl T, Gisslinger H, Harutyunyan AS, Nivarthi H, Rumi E, Milosevic JD, Them NC, Berg T, Gisslinger B, Pietra D, Chen D, Vladimer GI, Bagienski K, Milanesi C, Casetti IC, Sant'Antonio E, Ferretti V, Elena C, Schischlik F, Cleary C, Six M, Schalling M, Schönegger A, Bock C, Malcovati L, Pascutto C, Superti-Furga G, Cazzola M, Kralovics R | title = Somatic mutations of calreticulin in myeloproliferative neoplasms | journal = The New England Journal of Medicine | volume = 369 | issue = 25 | pages = 2379–90 | date = Dec 2013 | pmid = 24325356 | doi = 10.1056/NEJMoa1311347 | doi-access = free }}</ref>

== Role in cancer ==

Calreticulin (CRT) is expressed in many cancer cells and plays a role to promote [[macrophage]]s to engulf hazardous cancerous cells. The reason why most of the cells are not destroyed is the presence of another molecule with signal [[CD47]], which blocks CRT. Hence antibodies that block CD47 might be useful as a cancer treatment. In mice models of [[myeloid leukemia]] and [[non-Hodgkin lymphoma]], anti-CD47 were effective in clearing cancer cells while normal cells were unaffected.<ref name="pmid21178137">{{cite journal | vauthors = Chao MP, Jaiswal S, Weissman-Tsukamoto R, Alizadeh AA, Gentles AJ, Volkmer J, Weiskopf K, Willingham SB, Raveh T, Park CY, Majeti R, Weissman IL | title = Calreticulin is the dominant pro-phagocytic signal on multiple human cancers and is counterbalanced by CD47 | journal = Science Translational Medicine | volume = 2 | issue = 63 | pages = 63ra94 | date = Dec 2010 | pmid = 21178137 | pmc = 4126904 | doi = 10.1126/scitranslmed.3001375}}
*{{cite web |author=Christopher Vaughan |date=December 22, 2010 |title=Many cancer cells found to have an 'eat me' signal in study |website=Stanford School of Medicine |url=http://med.stanford.edu/ism/2010/december/crt-signal.html |archive-url=https://web.archive.org/web/20131016145928/http://med.stanford.edu/ism/2010/december/crt-signal.html |archive-date=2013-10-16}}</ref>

== Interactions ==

Calreticulin has been shown to [[protein–protein interaction|interact]] with [[Perforin]]<ref name=pmid9671507>{{cite journal | vauthors = Andrin C, Pinkoski MJ, Burns K, Atkinson EA, Krahenbuhl O, Hudig D, Fraser SA, Winkler U, Tschopp J, Opas M, Bleackley RC, Michalak M | title = Interaction between a Ca2+-binding protein calreticulin and perforin, a component of the cytotoxic T-cell granules | journal = Biochemistry | volume = 37 | issue = 29 | pages = 10386–94 | date = Jul 1998 | pmid = 9671507 | doi = 10.1021/bi980595z }}</ref> and [[NK2 homeobox 1]].<ref name=pmid9988700>{{cite journal | vauthors = Perrone L, Tell G, Di Lauro R | title = Calreticulin enhances the transcriptional activity of thyroid transcription factor-1 by binding to its homeodomain | journal = The Journal of Biological Chemistry | volume = 274 | issue = 8 | pages = 4640–5 | date = Feb 1999 | pmid = 9988700 | doi = 10.1074/jbc.274.8.4640 | doi-access = free }}</ref>
{{Clear}}

== References ==
{{Reflist|33em}}

== Further reading ==
{{refbegin|33em}}
* {{cite journal | vauthors = Del Bem LE | title = The evolutionary history of calreticulin and calnexin genes in green plants | journal = Genetica | volume = 139 | issue = 2 | pages = 225–9 | date = Feb 2011 | pmid = 21222018 | doi = 10.1007/s10709-010-9544-y | s2cid = 9228786 }}
* {{cite journal | vauthors = Coppolino MG, Dedhar S | title = Calreticulin | journal = The International Journal of Biochemistry & Cell Biology | volume = 30 | issue = 5 | pages = 553–8 | date = May 1998 | pmid = 9693955 | doi = 10.1016/S1357-2725(97)00153-2 }}
* {{cite journal | vauthors = Brucato A, Grava C, Bortolati M, Ikeda K, Milanesi O, Cimaz R, Ramoni V, Vignati G, Martinelli S, Sadou Y, Borghi A, Tincani A, Chan EK, Ruffatti A | title = Congenital heart block not associated with anti-Ro/La antibodies: comparison with anti-Ro/La-positive cases | journal = The Journal of Rheumatology | volume = 36 | issue = 8 | pages = 1744–8 | date = Aug 2009 | pmid = 19567621 | pmc = 2798588 | doi = 10.3899/jrheum.080737 }}
* {{cite journal | vauthors = Peng RQ, Chen YB, Ding Y, Zhang R, Zhang X, Yu XJ, Zhou ZW, Zeng YX, Zhang XS | title = Expression of calreticulin is associated with infiltration of T-cells in stage IIIB colon cancer | journal = World Journal of Gastroenterology | volume = 16 | issue = 19 | pages = 2428–34 | date = May 2010 | pmid = 20480531 | pmc = 2874150 | doi = 10.3748/wjg.v16.i19.2428 | doi-access = free }}
* {{cite journal | vauthors = Tarr JM, Young PJ, Morse R, Shaw DJ, Haigh R, Petrov PG, Johnson SJ, Winyard PG, Eggleton P | title = A mechanism of release of calreticulin from cells during apoptosis | journal = Journal of Molecular Biology | volume = 401 | issue = 5 | pages = 799–812 | date = Sep 2010 | pmid = 20624402 | doi = 10.1016/j.jmb.2010.06.064 | hdl = 10871/20264 | hdl-access = free }}
* {{cite journal | vauthors = Abd Alla J, Reeck K, Langer A, Streichert T, Quitterer U | title = Calreticulin enhances B2 bradykinin receptor maturation and heterodimerization | journal = Biochemical and Biophysical Research Communications | volume = 387 | issue = 1 | pages = 186–90 | date = Sep 2009 | pmid = 19580784 | doi = 10.1016/j.bbrc.2009.07.011 | url = https://www.zora.uzh.ch/id/eprint/25081/1/Accepted-manuscript-BBRC-387-186-2009.pdf | access-date = 2019-12-09 | archive-date = 2021-05-03 | archive-url = https://web.archive.org/web/20210503073327/https://www.zora.uzh.ch/id/eprint/25081/1/Accepted-manuscript-BBRC-387-186-2009.pdf | url-status = dead }}
* {{cite journal | vauthors = Caramelo JJ, Parodi AJ | title = Getting in and out from calnexin/calreticulin cycles | journal = The Journal of Biological Chemistry | volume = 283 | issue = 16 | pages = 10221–5 | date = Apr 2008 | pmid = 18303019 | pmc = 2447651 | doi = 10.1074/jbc.R700048200 | doi-access = free }}
* {{cite journal | vauthors = Du XL, Yang H, Liu SG, Luo ML, Hao JJ, Zhang Y, Lin DC, Xu X, Cai Y, Zhan QM, Wang MR | title = Calreticulin promotes cell motility and enhances resistance to anoikis through STAT3-CTTN-Akt pathway in esophageal squamous cell carcinoma | journal = Oncogene | volume = 28 | issue = 42 | pages = 3714–22 | date = Oct 2009 | pmid = 19684620 | doi = 10.1038/onc.2009.237 | s2cid = 23543648 | doi-access = }}
* {{cite journal | vauthors = Gelebart P, Opas M, Michalak M | title = Calreticulin, a Ca2+-binding chaperone of the endoplasmic reticulum | journal = The International Journal of Biochemistry & Cell Biology | volume = 37 | issue = 2 | pages = 260–6 | date = Feb 2005 | pmid = 15474971 | doi = 10.1016/j.biocel.2004.02.030 }}
* {{cite journal | vauthors = Qiu Y, Michalak M | title = Transcriptional control of the calreticulin gene in health and disease | journal = The International Journal of Biochemistry & Cell Biology | volume = 41 | issue = 3 | pages = 531–8 | date = Mar 2009 | pmid = 18765291 | doi = 10.1016/j.biocel.2008.06.020 }}
* {{cite journal | vauthors = Zhu Y, Zhang W, Veerapen N, Besra G, Cresswell P | title = Calreticulin controls the rate of assembly of CD1d molecules in the endoplasmic reticulum | journal = The Journal of Biological Chemistry | volume = 285 | issue = 49 | pages = 38283–92 | date = Dec 2010 | pmid = 20861015 | pmc = 2992262 | doi = 10.1074/jbc.M110.170530 | doi-access = free }}
* {{cite journal | vauthors = Talmud PJ, Drenos F, Shah S, Shah T, Palmen J, Verzilli C, Gaunt TR, Pallas J, Lovering R, Li K, Casas JP, Sofat R, Kumari M, Rodriguez S, Johnson T, Newhouse SJ, Dominiczak A, Samani NJ, Caulfield M, Sever P, Stanton A, Shields DC, Padmanabhan S, Melander O, Hastie C, Delles C, Ebrahim S, Marmot MG, Smith GD, Lawlor DA, Munroe PB, Day IN, Kivimaki M, Whittaker J, Humphries SE, Hingorani AD | title = Gene-centric association signals for lipids and apolipoproteins identified via the HumanCVD BeadChip | journal = American Journal of Human Genetics | volume = 85 | issue = 5 | pages = 628–42 | date = Nov 2009 | pmid = 19913121 | pmc = 2775832 | doi = 10.1016/j.ajhg.2009.10.014 }}
* {{cite journal | vauthors = Taner SB, Pando MJ, Roberts A, Schellekens J, Marsh SG, Malmberg KJ, Parham P, Brodsky FM | title = Interactions of NK cell receptor KIR3DL1*004 with chaperones and conformation-specific antibody reveal a functional folded state as well as predominant intracellular retention | journal = Journal of Immunology | volume = 186 | issue = 1 | pages = 62–72 | date = Jan 2011 | pmid = 21115737 | pmc = 3129036 | doi = 10.4049/jimmunol.0903657 }}
* {{cite journal | vauthors = Tarr JM, Winyard PG, Ryan B, Harries LW, Haigh R, Viner N, Eggleton P | title = Extracellular calreticulin is present in the joints of patients with rheumatoid arthritis and inhibits FasL (CD95L)-mediated apoptosis of T cells | journal = Arthritis and Rheumatism | volume = 62 | issue = 10 | pages = 2919–29 | date = Oct 2010 | pmid = 20533543 | doi = 10.1002/art.27602 | hdl = 10871/13850 | url = https://ore.exeter.ac.uk/repository/bitstream/10871/13850/2/AandR%202010.pdf | doi-access = free }}
* {{cite journal | vauthors = Kepp O, Gdoura A, Martins I, Panaretakis T, Schlemmer F, Tesniere A, Fimia GM, Ciccosanti F, Burgevin A, Piacentini M, Eggleton P, Young PJ, Zitvogel L, van Endert P, Kroemer G | title = Lysyl tRNA synthetase is required for the translocation of calreticulin to the cell surface in immunogenic death | journal = Cell Cycle | volume = 9 | issue = 15 | pages = 3072–7 | date = Aug 2010 | pmid = 20699648 | doi = 10.4161/cc.9.15.12459 | s2cid = 1963601 | doi-access = }}
* {{cite journal | vauthors = Sato H, Azuma Y, Higai K, Matsumoto K | title = Altered expression of glycoproteins on the cell surface of Jurkat cells during etoposide-induced apoptosis: shedding and intracellular translocation of glycoproteins | journal = Biochimica et Biophysica Acta (BBA) - General Subjects | volume = 1790 | issue = 10 | pages = 1198–205 | date = Oct 2009 | pmid = 19524015 | doi = 10.1016/j.bbagen.2009.05.019 }}
* {{cite journal | vauthors = Hong C, Qiu X, Li Y, Huang Q, Zhong Z, Zhang Y, Liu X, Sun L, Lv P, Gao XM | title = Functional analysis of recombinant calreticulin fragment 39-272: implications for immunobiological activities of calreticulin in health and disease | journal = Journal of Immunology | volume = 185 | issue = 8 | pages = 4561–9 | date = Oct 2010 | pmid = 20855873 | doi = 10.4049/jimmunol.1000536 | doi-access = free }}
* {{cite journal | vauthors = Alur M, Nguyen MM, Eggener SE, Jiang F, Dadras SS, Stern J, Kimm S, Roehl K, Kozlowski J, Pins M, Michalak M, Dhir R, Wang Z | title = Suppressive roles of calreticulin in prostate cancer growth and metastasis | journal = The American Journal of Pathology | volume = 175 | issue = 2 | pages = 882–90 | date = Aug 2009 | pmid = 19608864 | pmc = 2716982 | doi = 10.2353/ajpath.2009.080417 }}
* {{cite journal | vauthors = Bailey SD, Xie C, Do R, Montpetit A, Diaz R, Mohan V, Keavney B, Yusuf S, Gerstein HC, Engert JC, Anand S | title = Variation at the NFATC2 locus increases the risk of thiazolidinedione-induced edema in the Diabetes REduction Assessment with ramipril and rosiglitazone Medication (DREAM) study | journal = Diabetes Care | volume = 33 | issue = 10 | pages = 2250–3 | date = Oct 2010 | pmid = 20628086 | pmc = 2945168 | doi = 10.2337/dc10-0452 }}
* {{cite journal | vauthors = Nabi MO, Mirabzadeh A, Feizzadeh G, Khorshid HR, Karimlou M, Yeganeh MZ, Asgharian AM, Najmabadi H, Ohadi M | title = Novel mutations in the calreticulin gene core promoter and coding sequence in schizoaffective disorder | journal = American Journal of Medical Genetics Part B | volume = 153B | issue = 2 | pages = 706–9 | date = Mar 2010 | pmid = 19760677 | doi = 10.1002/ajmg.b.31036 | s2cid = 6959934 }}
* {{cite journal | vauthors = Schardt JA, Eyholzer M, Timchenko NA, Mueller BU, Pabst T | title = Unfolded protein response suppresses CEBPA by induction of calreticulin in acute myeloid leukaemia | journal = Journal of Cellular and Molecular Medicine | volume = 14 | issue = 6B | pages = 1509–19 | date = Jun 2010 | pmid = 19659458 | pmc = 3829017 | doi = 10.1111/j.1582-4934.2009.00870.x }}
{{refend}}

== External links ==
* {{MeSH name|Calreticulin}}

{{Antiangiogenics}}
{{PDB Gallery|geneid=811}}
{{Calcium signaling}}
{{Lectins}}

[[Category:C-type lectins]]
[[Category:Immune system]]
[[Category:Transcription coregulators]]
[[Category:Molecular chaperones]]

Explosive

2025-07-09T17:32:39Z

174.138.212.166: fix gap

{{short description|Substance that can explode}}
{{mi|
{{abbreviations|article|date=November 2023}}
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{{Use dmy dates|date=March 2020}}
{{for-multi|the American band|The Explosives|the song by Dr. Dre|Xxplosive|other uses}}
[[File:17. Експлозивни својства на три различни типови експлозиви.webm|thumb|right|upright=1.3|Demonstration of the explosive properties of three different explosives; four explosions are demonstrated. Three are conducted on a solid marble base, and one is conducted on the demonstrator's hand; each is initiated by a match.]]

An '''explosive''' (or '''explosive material''') is a reactive substance that contains a great amount of [[potential energy]] that can produce an [[explosion]] if released suddenly, usually accompanied by the production of [[light]], [[heat]], [[sound]], and [[pressure]]. An '''explosive charge''' is a measured quantity of explosive material, which may either be composed solely of one ingredient or be a mixture containing at least two substances.

The potential energy stored in an explosive material may, for example, be:
* [[chemical energy]], such as [[nitroglycerin]] or [[Dust explosion|grain dust]]
* [[pressure|pressurized]] [[gas compressor|gas]], such as a [[gas cylinder]], [[aerosol can]], or [[boiling liquid expanding vapor explosion]]
* [[nuclear weapon|nuclear]] energy, such as in the [[fissile]] [[isotope]]s [[uranium-235]] and [[plutonium-239]]

Explosive materials may be categorized by the speed at which they expand. Materials that [[detonate]] (the front of the chemical reaction moves faster through the material than the [[speed of sound]]) are said to be "high explosives" and materials that [[deflagrate]] are said to be "low explosives". Explosives may also be categorized by their [[sensitivity (explosives)|sensitivity]]. Sensitive materials that can be initiated by a relatively small amount of heat or pressure are [[primary explosive]]s, and materials that are relatively insensitive are [[secondary explosives|secondary]] or [[tertiary explosive]]s.

A wide variety of chemicals can explode; a smaller number are manufactured specifically for the purpose of being used as explosives. The remainder are too dangerous, sensitive, toxic, expensive, unstable, or prone to decomposition or degradation over short time spans.

In contrast, some materials are merely [[combustible]] or [[flammable]] if they burn without exploding. The distinction, however, is not always clear. Certain materials—dusts, powders, gases, or volatile organic liquids—may be simply combustible or flammable under ordinary conditions, but become explosive in specific situations or forms, such as [[dust explosion|dispersed airborne clouds]], or [[Gas explosion|confinement or sudden release]].

==History==
{{See also|History of gunpowder|Timeline of explosives}}
[[File:Year book - photo flashes showing Toledo's phenomenal progress, thriving industries and wonderful resources - DPLA - ac95c5ef8efd2394c21e2b6edcd01d94 (page 37) (cropped).jpg|thumb|right|The Great Western Powder Company of Toledo, Ohio, a producer of explosives, seen in 1905]]
[[Early thermal weapons]], such as [[Greek fire]], have existed since ancient times. At its roots, the history of chemical explosives lies in the history of [[gunpowder]].<ref>{{Cite book |title=Weapons of Mass Destruction |last= Sastri |first=M.N. |publisher=APH Publishing Corporation |year=2004 |isbn=978-81-7648-742-9 |page=1}}</ref><ref>{{Cite book |title=Chemistry in Daily Life |last=Singh |first=Kirpal |publisher=Prentice-Hall |year=2010 |isbn=978-81-203-4617-8 |page=68}}</ref> During the [[Tang dynasty]] in the 9th century, [[Taoism|Taoist]] Chinese alchemists were eagerly trying to find the elixir of immortality.<ref>{{Cite web |url=https://gbtimes.com/chinas-explosive-history-gunpowder-and-fireworks |title=China's explosive history of gunpowder and fireworks |last=Sigurðsson |first=Albert |date=17 January 2017 |website=GBTimes |url-status=live |archive-url=https://web.archive.org/web/20171201033748/https://gbtimes.com/chinas-explosive-history-gunpowder-and-fireworks |archive-date=1 December 2017 |df=dmy-all}}</ref> In the process, they stumbled upon the explosive invention of black powder made from coal, saltpeter, and sulfur in 1044. Gunpowder was the first form of chemical explosives, and by 1161, the Chinese were using explosives for the first time in warfare.<ref>{{Cite web |url=http://afe.easia.columbia.edu/chinawh/web/s10/0781_sect10_timeline.pdf |title=China and Europe, 1500–2000 and Beyond: What is Modern? |last1=Pomeranz |first1=Ken |last2=Wong |first2=Bin |publisher=Columbia University Press |url-status=live |archive-url=https://web.archive.org/web/20161213050629/http://afe.easia.columbia.edu/chinawh/web/s10/0781_sect10_timeline.pdf |archive-date=13 December 2016 |df=dmy-all}}</ref><ref>{{Cite book |title=A Short History of China |last=Kerr |first=Gordon |publisher=No Exit Press |year=2013 |isbn=978-1-84243-968-5}}</ref><ref>{{Cite book |title=The Ancient World |last1=Takacs |first1=Sarolta Anna |last2=Cline |first2=Eric H. |publisher=Routledge |year=2008 |page=544}}</ref> The Chinese would incorporate explosives fired from bamboo or bronze tubes known as bamboo firecrackers. The Chinese also inserted live rats inside the bamboo firecrackers; when fired toward the enemy, the flaming rats created great psychological ramifications—scaring enemy soldiers away and causing cavalry units to go wild.<ref>{{Cite book |last=Back |first=Fiona |year=2011 |isbn=978-1-86397-826-2 |page=55|title=Australian History Series: The ancient world |publisher=Ready-Ed Publications }}</ref>

The first useful explosive stronger than black powder was [[nitroglycerin]], developed in 1847. Since nitroglycerin is a liquid and highly unstable, it was replaced by [[nitrocellulose]], trinitrotoluene ([[TNT]]) in 1863, [[smokeless powder]], [[dynamite]] in 1867 and [[gelignite]] (the latter two being sophisticated stabilized preparations of nitroglycerine rather than chemical alternatives, both invented by [[Alfred Nobel]]). World War I saw the adoption of TNT in artillery shells. World War II saw extensive use of new explosives {{xref|(see: [[List of explosives used during World War II]])}}.

In turn, these have largely been replaced by more powerful explosives such as [[C-4 (explosive)|C-4]] and [[pentaerythritol tetranitrate]] (PETN) which are also waterproof and malleable though they may catch fire due to reactions with metals.<ref name=ankony1>Ankony, Robert C., ''Lurps: A Ranger's Diary of Tet, Khe Sanh, A Shau, and Quang Tri,'' revised ed., Rowman & Littlefield Publishing Group, Lanham, MD (2009), p.73.</ref>

==Applications==
{{Transcluded section|Explosive device}}
{{:Explosive device}}
==Types==

===Chemical===
[[File:GHS-pictogram-explos.svg|thumb|The [[Globally Harmonized System of Classification and Labelling of Chemicals|international]] [[GHS hazard pictograms|pictogram]] for explosive substances]]

An explosion is a type of spontaneous chemical reaction that, once initiated, is driven by both a large exothermic change (great release of heat) and a large positive [[entropy]] change (great quantities of gases are released) in going from reactants to products, thereby constituting a thermodynamically favorable process in addition to one that propagates very rapidly. Thus, explosives are substances that contain a large amount of energy stored in [[chemical bond]]s. The energetic stability of the gaseous products and hence their generation comes from the formation of strongly bonded species like carbon monoxide, carbon dioxide, and nitrogen gas, which contain strong double and triple bonds having bond strengths of nearly 1 MJ/mole. Consequently, most commercial explosives are organic compounds containing [[nitro compound|–NO2]], [[nitrate|–ONO2]] and [[nitroamine|–NHNO2]] groups that, when detonated, release gases like the aforementioned (e.g., [[nitroglycerin]], [[trinitrotoluene|TNT]], [[HMX]], [[PETN]], [[nitrocellulose]]).<ref>{{cite book |first=W.W. |last=Porterfield |title=Inorganic Chemistry: A Unified Approach |edition=2nd |publisher=Academic Press, Inc. |location=San Diego |pages=479–480 |date=1993}}</ref>

An explosive is classified as a low or high explosive according to its rate of [[combustion]]: low explosives burn rapidly (or [[deflagration|deflagrate]]), while high explosives [[detonation|detonate]]. While these definitions are distinct, the problem of precisely measuring rapid decomposition makes practical classification of explosives difficult. For a reaction to be classified as a detonation as opposed to just a deflagration, the propagation of the reaction shockwave through the material being tested must be faster than the [[speed of sound]] through that material. The speed of sound through a liquid or solid material is usually orders of magnitude faster than the speed of sound through air or other gases.

Traditional explosives mechanics is based on the shock-sensitive rapid oxidation of carbon and hydrogen to carbon dioxide, carbon monoxide, and water in the form of steam. [[Nitrate]]s typically provide the required oxygen to burn the carbon and hydrogen fuel. High explosives tend to have the oxygen, carbon, and hydrogen contained in one organic molecule, and less sensitive explosives like ANFO are combinations of fuel (carbon and hydrogen fuel oil) and [[ammonium nitrate]]. A sensitizer such as powdered aluminum may be added to an explosive to increase the energy of the detonation. Once detonated, the nitrogen portion of the explosive formulation emerges as nitrogen gas and toxic [[NOx|nitric oxides]].

====Decomposition====
The [[chemical decomposition]] of an explosive may take years, days, hours, or a fraction of a second. The slower processes of decomposition take place in storage and are of interest only from a stability standpoint. Of more interest are the other two rapid forms besides decomposition: deflagration and detonation.

====Deflagration====
{{Main|Deflagration}}
In deflagration, decomposition of the explosive material is propagated by a flame front which moves relatively slowly through the explosive material, {{em|i.e.}} at speeds less than the speed of sound within the substance (which is usually still higher than 340 m/s or {{convert|340|m/s|km/h|-1|disp=out}} in most liquid or solid materials)<ref>{{cite web |url=http://www.chem-page.de/publikationen/geschichte-der-sprengstoffe/195-2-wie-unterscheiden-sich-deflagration-detonation-und-explosion.html |access-date=2017-02-05 |url-status=dead |archive-url=https://web.archive.org/web/20170206104750/http://www.chem-page.de/publikationen/geschichte-der-sprengstoffe/195-2-wie-unterscheiden-sich-deflagration-detonation-und-explosion.html |archive-date=6 February 2017 |df=dmy-all |title=2.1 Deflagration |language=de |website=chem-page.de}}</ref> in contrast to detonation, which occurs at speeds greater than the speed of sound. Deflagration is a characteristic of [[low explosive]] material.

====Detonation====
{{Main|Detonation}}
This term is used to describe an explosive phenomenon whereby the decomposition is [[wave propagation|propagated]] by a [[shock wave]] traversing the explosive material at speeds greater than the speed of sound within the substance.<ref>{{cite web |url=http://www.chem-page.de/publikationen/geschichte-der-sprengstoffe/195-2-wie-unterscheiden-sich-deflagration-detonation-und-explosion.html |title=2.2 Detonation |website=chem-page.de |language=de |access-date=2017-02-05 |url-status=dead |archive-url=https://web.archive.org/web/20170206104750/http://www.chem-page.de/publikationen/geschichte-der-sprengstoffe/195-2-wie-unterscheiden-sich-deflagration-detonation-und-explosion.html |archive-date=6 February 2017 |df=dmy-all }}</ref> The shock front is capable of passing through the high explosive material at supersonic {{nowrap|speeds{{tsp}}{{mdash}}{{tsp}}}}typically thousands of metres per second.

===Exotic===
In addition to chemical explosives, there are a number of more exotic explosive materials and exotic methods of causing explosions. Examples include [[nuclear explosive]]s, and abruptly heating a substance to a [[plasma (physics)|plasma]] state with a high-intensity [[laser]] or [[electric arc]].

Laser- and arc-heating are used in laser detonators, [[exploding-bridgewire detonator]]s, and [[slapper detonator|exploding foil initiators]], where a shock wave and then detonation in conventional chemical explosive material is created by laser- or electric-arc heating. Laser and electric energy are not currently used in practice to generate most of the required energy, but only to initiate reactions.

==Properties==
To determine the suitability of an explosive substance for a particular use, its [[physics|physical]] [[Physical property|properties]] must first be known. The usefulness of an explosive can only be appreciated when the properties and the factors affecting them are fully understood. Some of the more important characteristics are listed below:

===Sensitivity===
{{Main|Sensitivity (explosives)}}
Sensitivity refers to the ease with which an explosive can be ignited or detonated, i.e., the amount and intensity of [[electric shock|shock]], [[friction]], or [[heat]] that is required. When the term [[wikt:Sensitivity|sensitivity]] is used, care must be taken to clarify what kind of sensitivity is under discussion. The relative sensitivity of a given explosive to impact may vary greatly from its sensitivity to friction or heat. Some of the test methods used to determine sensitivity relate to:
* '''[[Impact (mechanics)|Impact]]''' – Sensitivity is expressed in terms of the distance through which a standard weight must be dropped onto the material to cause it to explode.
* '''[[Friction]]''' – Sensitivity is expressed in terms of the amount of pressure applied to the material in order to create enough friction to cause a reaction.
* '''[[Heat]]''' – Sensitivity is expressed in terms of the temperature at which decomposition of the material occurs.

Specific explosives (usually but not always highly sensitive on one or more of the three above axes) may be idiosyncratically sensitive to such factors as pressure drop, acceleration, the presence of sharp edges or rough surfaces, incompatible materials, or {{nowrap|even{{hsp}}{{mdash}}{{hsp}}}}in rare {{nowrap|cases{{hsp}}{{mdash}}{{hsp}}}}nuclear or electromagnetic radiation. These factors present special hazards that may rule out any practical utility.

Sensitivity is an important consideration in selecting an explosive for a particular purpose. The explosive in an armor-piercing projectile must be relatively insensitive, or the shock of impact would cause it to detonate before it penetrated to the point desired. The explosive lenses around nuclear charges are also designed to be highly insensitive, to minimize the risk of accidental detonation.

===Sensitivity to initiation===
The index of the capacity of an explosive to be initiated into detonation in a sustained manner. It is defined by the power of the detonator, which is certain to prime the explosive to a sustained and continuous detonation. Reference is made to the [[Sellier-Bellot]] scale that consists of a series of 10 detonators, from {{nowrap|n.{{hsp}}1}} to {{nowrap|n.{{hsp}}10}}, each of which corresponds to an increasing charge weight. In practice, most of the explosives on the market today are sensitive to an {{nowrap|n.{{hsp}}8}} detonator, where the charge corresponds to 2 grams of [[mercury fulminate]].

===Velocity of detonation===
{{Main|Detonation velocity}}
The velocity with which the reaction process propagates in the mass of the explosive. Most commercial mining explosives have detonation velocities ranging from 1,800 m/s to 8,000 m/s. Today, the velocity of detonation can be measured with accuracy. Together with [[density]], it is an important element influencing the yield of the energy transmitted through both atmospheric over-pressure and ground acceleration. By definition, a "low explosive", such as black powder, or smokeless gunpowder has a burn rate of 171–631 m/s.<ref>{{Cite book|url=https://books.google.com/books?id=PmuqCHDC3pwC&pg=PA106|title=History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference|last=Krehl|first=Peter O.K.|date=2008-09-24|publisher=Springer Science & Business Media|isbn=978-3-540-30421-0|page=106|language=en}}</ref> In contrast, a "high explosive", whether a primary, such as [[detonating cord]], or a secondary, such as TNT or C-4, has a significantly higher burn rate about 6900–8092 m/s.<ref>{{Cite book|title=History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference|last=Krehl|first=Peter O.K.|publisher=Springer Science & Business Media|year=2008|isbn=978-3-540-30421-0|page=1970}}</ref>

===Stability===
{{Main|Chemical stability}}
'''Stability ''' is the ability of an explosive to be stored without [[chemical decomposition|deterioration]].

The following factors affect the stability of an explosive:
* '''[[Chemical constitution]].''' In the strictest technical sense, the word "stability" is a thermodynamic term referring to the energy of a substance relative to a reference state or to some other substance. However, in the context of explosives, stability commonly refers to ease of detonation, which is concerned with [[chemical kinetics]] (i.e., rate of decomposition). It is perhaps best, then, to differentiate between the terms thermodynamically stable and kinetically stable by referring to the former as "inert." Contrarily, a kinetically unstable substance is said to be "labile." It is generally recognized that certain groups like nitro (–NO2), [[nitrate]] (–ONO2), and [[azide]] (–N3), are intrinsically labile. Kinetically, there exists a low activation barrier to the decomposition reaction. Consequently, these compounds exhibit high sensitivity to flame or mechanical shock. The chemical bonding in these compounds is characterized as predominantly covalent and thus they are not thermodynamically stabilized by a high ionic-lattice energy. Furthermore, they generally have positive enthalpies of formation and there is little mechanistic hindrance to internal molecular rearrangement to yield the more thermodynamically stable (more strongly bonded) decomposition products. For example, in [[lead azide]], Pb(N3)2, the nitrogen atoms are already bonded to one another, so decomposition into Pb and N2[1] is relatively easy.
* '''[[Temperature]] of storage.''' The rate of decomposition of explosives increases at higher temperatures. All standard military explosives may be considered to have a high degree of stability at temperatures from –10 to +35 °C, but each has a high temperature at which its rate of [[thermal decomposition]] rapidly accelerates and stability is reduced. As a rule of thumb, most explosives become dangerously unstable at temperatures above 70 °C.
* '''Exposure to [[sunlight]].''' When exposed to the [[ultraviolet]] rays of sunlight, many explosive compounds containing [[nitrogen]] groups rapidly decompose, affecting their stability.
* '''[[Electrical discharge]].''' [[Electrostatic discharge|Electrostatic]] or [[electric spark|spark]] sensitivity to initiation is common in a number of explosives. Static or other electrical discharge may be sufficient to cause a reaction, even detonation, under some circumstances. As a result, safe handling of explosives and [[pyrotechnics]] usually requires proper [[ground (electricity)|electrical grounding]] of the operator.

===Power, performance, and strength===
{{Main|Power (physics)|Strength (explosive)}}
The term '''power''' or '''performance''' as applied to an explosive refers to its ability to do work. In practice it is defined as the explosive's ability to accomplish what is intended in the way of energy delivery (i.e., fragment projection, air blast, high-velocity jet, underwater shock and bubble energy, etc.). Explosive power or performance is evaluated by a tailored series of tests to assess the material for its intended use. Of the tests listed below, cylinder expansion and air-blast tests are common to most testing programs, and the others support specific applications.
* '''Cylinder expansion test.''' A standard amount of explosive is loaded into a long hollow [[cylinder (geometry)|cylinder]], usually of copper, and detonated at one end. Data is collected concerning the rate of radial expansion of the cylinder and the maximum cylinder wall velocity. This also establishes the [[Gurney equations|Gurney energy]] or 2''E''.
* '''Cylinder fragmentation.''' A standard steel cylinder is loaded with explosive and detonated in a sawdust pit. The [[fragmentation (weaponry)|fragments]] are collected and the size distribution analyzed.
* '''Detonation pressure ([[Chapman–Jouguet condition]]).''' [[Detonation]] pressure data derived from measurements of shock waves transmitted into water by the detonation of cylindrical explosive charges of a standard size.
* '''Determination of critical diameter.''' This test establishes the minimum physical size a charge of a specific explosive must be to sustain its own detonation wave. The procedure involves the detonation of a series of charges of different diameters until difficulty in detonation wave propagation is observed.
* '''Massive-diameter detonation velocity.''' Detonation velocity is dependent on loading density (c), charge diameter, and grain size. The hydrodynamic theory of detonation used in predicting explosive phenomena does not include the diameter of the charge, and therefore a detonation velocity, for a massive diameter. This procedure requires the firing of a series of charges of the same density and physical structure, but different diameters, and the extrapolation of the resulting detonation velocities to predict the detonation velocity of a charge of a massive diameter.
* '''Pressure versus scaled distance.''' A charge of a specific size is detonated and its pressure effects measured at a standard distance. The values obtained are compared with those for TNT.
* '''Impulse versus scaled distance.''' A charge of a specific size is detonated and its impulse (the area under the pressure-time curve) measured as a function of distance. The results are tabulated and expressed as [[TNT equivalent]]s.
* '''Relative bubble energy (RBE).''' A 5 to 50 kg charge is detonated in water and piezoelectric gauges measure peak pressure, time constant, impulse, and energy.
::The RBE may be defined as ''K''''x'' 3
::RBE = ''K''''s''
::where ''K'' = the bubble expansion period for an experimental (''x'') or a standard (''s'') charge.

===Brisance===
{{Main|Brisance}}
In addition to strength, explosives display a second characteristic, which is their shattering effect or brisance (from the French meaning to "break"). Brisance is important in determining the effectiveness of an explosion in fragmenting shells, bomb casings, and [[grenade]]s. The rapidity with which an explosive reaches its peak pressure ([[power (physics)|power]]) is a measure of its brisance. Brisance values are primarily employed in France and Russia.

The sand crush test is commonly employed to determine the relative brisance in comparison to TNT. No test is capable of directly comparing the explosive properties of two or more compounds; it is important to examine the data from several such tests (sand crush, [[trauzl lead block test|trauzl]], and so forth) in order to gauge relative brisance. True values for comparison require field experiments.

===Density===
Density of loading refers to the mass of an explosive per unit volume. Several methods of loading are available, including pellet loading, cast loading, and press loading, the choice being determined by the characteristics of the explosive. Dependent upon the method employed, an average density of the loaded charge can be obtained that is within 80–99% of the theoretical maximum density of the explosive. High load density can reduce [[sensitivity (explosives)|sensitivity]] by making the [[mass]] more resistant to internal [[friction]]. However, if density is increased to the extent that individual [[crystal]]s are crushed, the explosive may become more sensitive. Increased load density also permits the use of more explosive, thereby increasing the power of the [[warhead]]. It is possible to compress an explosive beyond a point of sensitivity, known also as ''dead-pressing'', in which the material is no longer capable of being reliably initiated, if at all.{{citation needed|date=November 2023}}

===Volatility===
[[Volatility (chemistry)|Volatility]] is the readiness with which a substance [[vaporization|vaporizes]]. Excessive volatility often results in the development of pressure within rounds of ammunition and separation of mixtures into their constituents. Volatility affects the chemical composition of the explosive such that a marked reduction in stability may occur, which results in an increase in the danger of handling.

===Hygroscopicity and water resistance===
The introduction of [[water]] into an explosive is highly undesirable since it reduces the sensitivity, strength, and velocity of detonation of the explosive. [[Hygroscopicity]] is a measure of a material's moisture-absorbing tendencies. Moisture affects explosives adversely by acting as an inert material that absorbs heat when vaporized, and by acting as a solvent medium that can cause undesired chemical reactions. Sensitivity, strength, and velocity of detonation are reduced by inert materials that reduce the continuity of the explosive mass. When the moisture content evaporates during detonation, cooling occurs, which reduces the temperature of reaction. Stability is also affected by the presence of moisture since moisture promotes decomposition of the explosive and, in addition, causes corrosion of the explosive's metal container.

Explosives considerably differ from one another as to their behavior in the presence of water. Gelatin dynamites containing nitroglycerine have a degree of water resistance. Explosives based on [[ammonium nitrate]] have little or no water resistance as ammonium nitrate is highly soluble in water and is hygroscopic.

===Toxicity===
Many explosives are [[toxicity|toxic]] to some extent. Manufacturing inputs can also be organic compounds or hazardous materials that require special handling due to risks (such as [[carcinogen]]s). The decomposition products, residual solids, or gases of some explosives can be toxic, whereas others are harmless, such as carbon dioxide and water.

Examples of harmful by-products are:
* Heavy metals, such as lead, mercury, and barium from primers (observed in high-volume firing ranges)
* Nitric oxides from TNT
* Perchlorates when used in large quantities

"Green explosives" seek to reduce environment and health impacts. An example of such is the lead-free primary explosive copper(I) 5-nitrotetrazolate, an alternative to [[lead azide]].<ref>{{cite magazine |url=https://www.newscientist.com/article/dn8903-green-explosive-is-a-friend-of-the-earth.html |title=Green explosive is a friend of the Earth |magazine=New Scientist |date=27 March 2006 |access-date=12 November 2014 |url-status=live |archive-url=https://web.archive.org/web/20141112220348/http://www.newscientist.com/article/dn8903-green-explosive-is-a-friend-of-the-earth.html |archive-date=12 November 2014 |df=dmy-all}}</ref>

===Explosive train===
{{Main|Explosive train}}
Explosive material may be incorporated in the explosive train of a device or system. An example is a pyrotechnic lead igniting a booster, which causes the main charge to detonate.

===Volume of products of explosion===
The most widely used explosives are condensed liquids or solids converted to gaseous products by explosive chemical reactions and the energy released by those reactions. The gaseous products of complete reaction are typically [[carbon dioxide]], [[steam]], and [[nitrogen]].<ref name="z&k">{{cite book |title=Theory of Detonation |last1=Zel'dovich |first1=Yakov |author-link=Yakov Borisovich Zel'dovich |first2=Alexander Solomonovich |last2=Kompaneets |author-link2=Alexander Solomonovich Kompaneyets |publisher=Academic Press |year=1960|pages=208–210}}</ref> Gaseous volumes computed by the [[ideal gas law]] tend to be too large at high pressures characteristic of explosions.<ref>{{cite book |title=Chemical Process Principles |last1=Hougen |first1=Olaf A. |first2=Kenneth |last2=Watson |first3=Roland |last3=Ragatz |publisher=John Wiley & Sons |year=1954|pages=66–67}}</ref> Ultimate volume expansion may be estimated at three orders of magnitude, or one liter per gram of explosive. Explosives with an oxygen deficit will generate soot or gases like [[carbon monoxide]] and [[hydrogen]], which may react with surrounding materials such as atmospheric [[oxygen]].<ref name="z&k"/> Attempts to obtain more precise volume estimates must consider the possibility of such side reactions, condensation of steam, and aqueous solubility of gases like carbon dioxide.<ref>{{cite book |title=Chemical Calculations |last=Anderson |first=H.V. |publisher=McGraw-Hill |year=1955|page=206}}</ref>

===Oxygen balance (OB% or ''Ω'')===
{{Main|Oxygen balance}}
[[Oxygen balance]] is an expression that is used to indicate the degree to which an explosive can be oxidized. If an explosive molecule contains just enough oxygen to convert all of its carbon to carbon dioxide, all of its hydrogen to water, and all of its metal to metal oxide with no excess, the molecule is said to have a zero oxygen balance. The molecule is said to have a positive oxygen balance if it contains more oxygen than is needed, and a negative oxygen balance if it contains less oxygen than is needed.<ref>{{cite book |last1= Meyer |first1= Rudolf |first2=Josef |last2=Köhler|first3=Axel |last3=Homburg |title= Explosives |edition=6th |publisher= Wiley VCH |year= 2007 |isbn= 978-3-527-31656-4}}</ref> The sensitivity, [[strength (explosive)|strength]], and [[brisance]] of an explosive are all somewhat dependent upon oxygen balance and tend to approach their maxima as oxygen balance approaches zero.

===Chemical composition===
A chemical explosive may consist of either a chemically pure compound, such as [[nitroglycerin]], or a mixture of a [[fuel]] and an [[oxidizer]], such as [[black powder]] or [[grain dust]] and air.

====Pure compounds====
Some chemical compounds are unstable in that, when shocked, they react, possibly to the point of detonation. Each molecule of the compound dissociates into two or more new molecules (generally gases) with the release of energy.
* '''[[Nitroglycerin]]''': A highly sensitive colorless liquid
* '''[[Acetone peroxide]]''': A very unstable white [[organic peroxide]]
* '''[[TNT]]''': Yellow insensitive crystals that can be melted and cast without detonation
* '''[[Cellulose nitrate]]''': A nitrated polymer which can be a high or low explosive depending on nitration level and conditions
* '''[[RDX]]''', '''[[PETN]]''', '''[[HMX]]''': Very powerful explosives which can be used pure or in plastic explosives
** '''[[C4 explosive|C-4]]''' (or Composition C-4): An [[RDX]] [[plastic explosive]] plasticized to be adhesive and malleable

The above compositions may describe most of the explosive material, but a practical explosive will often include small percentages of other substances. For example, [[dynamite]] is a mixture of highly sensitive nitroglycerin with [[sawdust]], powdered [[silica]], or most commonly [[diatomaceous earth]], which act as stabilizers. Plastics and polymers may be added to bind powders of explosive compounds; waxes may be incorporated to make them safer to handle; [[aluminium]] powder may be introduced to increase total energy and blast effects. Explosive compounds are also often "alloyed": HMX or RDX powders may be mixed (typically by melt-casting) with TNT to form [[Octol]] or [[Cyclotol]].

====Oxidized fuel====
An [[oxidizer]] is a pure substance ([[molecule]]) that in a chemical reaction can contribute some atoms of one or more oxidizing elements, in which the [[fuel]] component of the explosive burns. On the simplest level, the oxidizer may itself be an oxidizing [[Chemical element|element]], such as [[gas state|gaseous]] or [[liquid]] [[oxygen]].
* '''[[Black powder]]''': [[Potassium nitrate]], [[charcoal]] and [[sulfur]]
* '''[[Flash powder]]''': Fine metal powder (usually [[aluminium]] or [[magnesium]]) and a strong oxidizer (e.g. [[potassium chlorate]] or [[potassium perchlorate|perchlorate]])
* '''[[Ammonal]]''': [[Ammonium nitrate]] and aluminium powder
* '''[[Armstrong's mixture]]''': [[Potassium chlorate]] and [[red phosphorus]]. This is a very sensitive mixture. It is a primary high explosive in which sulfur is substituted for some or all of the phosphorus to slightly decrease sensitivity.
* '''[[Sprengel explosive]]s''': A very general class incorporating any strong oxidizer and highly reactive fuel, although in practice the name was most commonly applied to mixtures of [[chlorate]]s and [[nitroaromatic]]s.
** '''[[ANFO]]''': Ammonium nitrate and [[fuel oil]]
** '''[[Cheddite]]s''': [[Chlorate]]s or [[perchlorate]]s and oil
** '''[[Oxyliquit]]s''': Mixtures of organic materials and [[liquid oxygen]]
** '''[[Panclastite]]s''': Mixtures of organic materials and [[dinitrogen tetroxide]]

===Availability and cost===
The availability and cost of explosives are determined by the availability of the raw materials and the cost, complexity, and safety of the manufacturing operations.

==Classification==

===By sensitivity===

====Primary====
A '''primary explosive''' is an explosive that is extremely sensitive to stimuli such as [[impact (mechanics)|impact]], [[friction]], [[heat]], [[electrostatic|static electricity]], or [[electromagnetic radiation]]. Some primary explosives are also known as [[contact explosive]]s. A relatively small amount of energy is required for [[initiation (chemistry)|initiation]]. As a very general rule, primary explosives are considered to be those compounds that are more sensitive than [[Pentaerythritol tetranitrate|PETN]]. As a practical measure, primary explosives are sufficiently sensitive that they can be reliably initiated with a blow from a hammer; however, PETN can also usually be initiated in this manner, so this is only a very broad guideline. Additionally, several compounds, such as [[nitrogen triiodide]], are so sensitive that they cannot even be handled without detonating. Nitrogen triiodide is so sensitive that it can be reliably detonated by exposure to [[Alpha particle|alpha radiation]].<ref>{{Citation |title=Nitrogen triiodide |date=2024-06-08 |work=Wikipedia |url=https://en.m.wikipedia.org/wiki/Nitrogen_triiodide |access-date=2024-09-02 |language=en}}</ref><ref>{{Cite web |last=UCL |date=2020-06-23 |title=Nitrogen Triiodide (NI3) |url=https://www.ucl.ac.uk/safety-services/policies/2021/jun/nitrogen-triiodide-ni3 |access-date=2024-09-02 |website=Safety Services |language=en}}</ref>

Primary explosives are often used in [[detonator]]s or to [[trigger (firearms)|trigger]] larger charges of less sensitive [[secondary explosives]]. Primary explosives are commonly used in [[blasting cap]]s and [[percussion cap]]s to translate a physical shock signal. In other situations, different signals such as electrical or physical shock, or, in the case of laser detonation systems, light, are used to initiate an action, i.e., an explosion. A small quantity, usually milligrams, is sufficient to initiate a larger charge of explosive that is usually safer to handle.

Examples of primary high explosives are:
{{Div col|colwidth=17em}}
* [[Acetone peroxide]]
* Alkali metal [[ozonide]]s
* [[Ammonium permanganate]]
* [[Ammonium chlorate]]
* [[Azidotetrazolates]]
* Azoclathrates
* [[Benzoyl peroxide]]
* [[Benzvalene]]
* 3,5-Bis(trinitromethyl)tetrazole<ref>{{Cite web|url=https://www.science.org/content/blog-post/can-t-stop-nitro-groups|title=Can't Stop the Nitro Groups |work=Science.org |last=Lowe |first=Derek |date=15 August 2019 |access-date=22 August 2022}}</ref>
* [[Chlorine oxide]]s
* [[Copper(I) acetylide]]
* [[Copper(II) azide]]
* [[Cumene hydroperoxide]]
* Cycloprop(-2-)enyl nitrate (CXP or CPN)
* [[Cyanogen azide]]
* [[Cyanuric triazide]]
* [[Diacetyl peroxide]]
* [[1-Diazidocarbamoyl-5-azidotetrazole]]
* [[Diazodinitrophenol]]
* [[Diazomethane]]
* [[Diethyl ether peroxide]]
* [[4-Dimethylaminophenylpentazole]]
* [[Disulfur dinitride]]
* [[Ethyl azide]]
* [[Explosive antimony]]
* [[Fluorine perchlorate]]
* [[Fulminic acid]]
* Halogen azides:
** [[Fluorine azide]]
** [[Chlorine azide]]
** [[Bromine azide]]
** [[Iodine azide]]
* [[Hexamethylene triperoxide diamine]]
* [[Hydrazoic acid]]
* [[Hypofluorous acid]]
* [[Lead azide]]
* [[Lead styphnate]]
* [[Lead picrate]]<ref>{{cite web|url=http://www.powerlabs.org/chemlabs/lead_picrate.htm|title=PowerLabs Lead Picrate Synthesis|first=Sam |last=Barros |website=powerlabs.org|url-status=live|archive-url=http://arquivo.pt/wayback/20160522174456/http://www.powerlabs.org/chemlabs/lead_picrate.htm|archive-date=22 May 2016|df=dmy-all}}</ref>
* [[Manganese heptoxide]]
* [[Mercury(II) fulminate]]
* [[Mercury nitride]]
* [[Methyl ethyl ketone peroxide]]
* [[Nickel hydrazine nitrate]]<ref>{{cite book |first1=Robert |last1=Matyáš |first2=Jiří |last2=Pachman |title=Primary Explosives |publisher=Springer-Verlag Berlin Heidelberg |date=2013 |page=331}}</ref>
* Nickel hydrazine perchlorate
* Nitrogen trihalides:
** [[Nitrogen trichloride]]
** [[Nitrogen tribromide]]
** [[Nitrogen triiodide]]
* [[Nitroglycerin]]
* [[Nitronium perchlorate]]
* [[Nitrosyl perchlorate]]
* Nitrotetrazolate-''N''-oxides
* [[Pentazenium]] [[hexafluoroarsenate]]
* [[Peroxy acid]]s
* [[Peroxymonosulfuric acid]]
* [[Selenium tetraazide]]
* [[Silicon tetraazide]]
* [[Silver azide]]
* [[Silver acetylide]]
* [[Silver fulminate]]
* [[Silver nitride]]
* [[Tellurium tetraazide]]
* [[tert-Butyl hydroperoxide|''tert''-Butyl hydroperoxide]]
* Tetraamine copper complexes
* [[Tetraazidomethane]]
* [[Tetrazene explosive]]
* [[Tetrazole]]s
* [[Titanium tetraazide]]
* Triazidomethane
* Oxides of xenon:
** [[Xenon dioxide]]
** [[Xenon oxytetrafluoride]]
** [[Xenon tetroxide]]
** [[Xenon trioxide]]
{{div col end}}

====Secondary====
A '''secondary explosive''' is less sensitive than a primary explosive and requires substantially more energy to be initiated. Because they are less sensitive, they are usable in a wider variety of applications and are safer to handle and store. Secondary explosives are used in larger quantities in an explosive train and are usually initiated by a smaller quantity of a primary explosive.

Examples of secondary explosives include [[TNT]] and [[RDX]].

====Tertiary====
'''Tertiary explosives''', also called '''blasting agents''', are so insensitive to shock that they cannot be reliably detonated by practical quantities of [[primary explosive]], and instead require an intermediate [[explosive booster]] of [[secondary explosive]]. These are often used for safety and the typically lower costs of material and handling. The largest consumers are large-scale [[mining]] and [[construction]] operations.

Most tertiaries include a fuel and an oxidizer. [[ANFO]] can be a tertiary explosive if its reaction rate is slow.

===By velocity===

====Low====
Low explosives (or low-order explosives) are compounds wherein the rate of decomposition proceeds through the material at less than the [[speed of sound]]. The decomposition is propagated by a flame front ([[deflagration]]) that travels much more slowly through the explosive material than the [[shock wave]] of a high explosive. [[Standard temperature and pressure|Under normal conditions]], low explosives undergo deflagration at rates that vary from a few centimetres per second to approximately {{Convert|0.4|km/s|ft/s}}. It is possible for them to deflagrate very quickly, producing an effect similar to a [[detonation]]. This can happen under higher [[pressure]] (such as when [[gunpowder]] deflagrates inside the confined space of a bullet casing, accelerating the bullet to well beyond the speed of sound) or [[temperature]].

A low explosive is usually a mixture of a [[combustible]] substance and an [[oxidant]] that decomposes rapidly (deflagration); however, they burn more slowly than a high explosive, which has an extremely fast burn rate.<ref>{{Cite book |url=https://nap.nationalacademies.org/read/5966/chapter/22 |title=Read "Containing the Threat from Illegal Bombings: An Integrated National Strategy for Marking, Tagging, Rendering Inert, and Licensing Explosives and Their Precursors" at NAP.edu |date=1998 |doi=10.17226/5966 |isbn=978-0-309-06126-1 |language=en}}</ref>

Low explosives are normally employed as [[propellant]]s. Included in this group are petroleum products such as [[propane]] and [[gasoline]], [[gunpowder]] (including [[smokeless powder]]), and light [[pyrotechnics]] such as [[flare]]s and [[fireworks]], but they can replace high explosives in certain applications, including gas pressure blasting.<ref>{{Cite journal |date=1958-07-29 |title=The initiation of explosion by neutrons, α -particles and fission products |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1958.0123 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |language=en |volume=246 |issue=1245 |pages=216–219 |doi=10.1098/rspa.1958.0123 |bibcode=1958RSPSA.246..216B |s2cid=137728239 |issn=0080-4630|last1=Bowden |first1=F. P. |url-access=subscription }}</ref>

====High====
High explosives (HE, or high-order explosives) are explosive materials that [[Detonation|detonate]], meaning that the [[explosion|explosive]] [[shock front]] passes through the material at a [[supersonic]] speed. High explosives detonate with [[explosive velocity]] of about {{Convert|3|-|9|km/s|ft/s}}. For instance, TNT has a detonation (burn) rate of approximately 6.9 km/s (22,600 feet per second), detonating cord of 6.7 km/s (22,000 feet per second), and C-4 about 8.0 km/s (26,000 feet per second). They are normally employed in mining, demolition, and military applications. The term ''high explosive'' is in contrast with the term ''low explosive'', which explodes ([[Deflagration|deflagrates]]) at a lower rate.

High explosives can be divided into two explosives classes differentiated by [[sensitivity (explosives)|sensitivity]]: [[#Primary|primary explosive]] and [[#Secondary|secondary explosive]]. Although tertiary explosives (such as ANFO at 3,200 m/s) can technically meet the explosive velocity definition, they are not considered high explosives in regulatory contexts.

Countless high-explosive compounds are chemically possible, but commercially and militarily important ones have included [[nitroglycerin|NG]], [[TNT]], [[Trinitrophenol|TNP]], TNX, [[RDX]], [[HMX]], [[pentaerythritol tetranitrate|PETN]], [[TATP]], [[TATB]], and [[hexanitrostilbene|HNS]].

===By physical form===
{{Main|Use forms of explosives}}
Explosives are often characterized by the physical form that the explosives are produced or used in. These use forms are commonly categorized as:<ref name="Cooper">{{cite book
|last=Cooper
|first=Paul W.
|title=Explosives Engineering
|year=1996
|publisher=Wiley-VCH
|pages=51–66
|chapter=Chapter 4: Use forms of explosives
|isbn=978-0-471-18636-6}}</ref>
* Pressings
* Castings
* [[Polymer-bonded explosive|Plastic or polymer bonded]]
* [[Plastic explosive]]s, a.k.a. putties
* Rubberized
* Extrudable
* [[Binary explosive|Binary]]
* Blasting agents
* Slurries and gels
* Dynamites

===Shipping label classifications===
Shipping labels and tags may include both [[United Nations]] and national markings.

United Nations markings include numbered Hazard Class and Division (HC/D) codes and alphabetic Compatibility Group codes. Though the two are related, they are separate and distinct. Any Compatibility Group designator can be assigned to any Hazard Class and Division. An example of this hybrid marking would be a consumer [[firework]], which is labeled as 1.4G or 1.4S.

Examples of national markings would include [[United States Department of Transportation]] (U.S. DOT) codes.

====United Nations (UN) GHS Hazard Class and Division====
[[File:UN transport pictogram - 1.svg|thumb|GHS Explosives transport pictogram]]
{{See also|HAZMAT Class 1 Explosives}}

The UN [[Globally Harmonized System of Classification and Labelling of Chemicals|GHS]] Hazard Class and Division (HC/D) is a numeric designator within a hazard class indicating the character, predominance of associated hazards, and potential for causing personnel casualties and property damage. It is an internationally accepted system that communicates using the minimum amount of markings the primary hazard associated with a substance.<ref>[http://www.tpub.com/content/aviation/14313/css/14313_288.htm Table 12-4. – United Nations Organization Hazard Classes] {{webarchive|url=https://web.archive.org/web/20100605034021/http://www.tpub.com/content/aviation/14313/css/14313_288.htm |date=5 June 2010}}. Tpub.com. Retrieved on 2010-02-11.</ref>

Listed below are the Divisions for Class 1 (Explosives):
* '''1.1''' Mass Detonation Hazard. With HC/D 1.1, it is expected that if one item in a container or pallet inadvertently detonates, the explosion will [[sympathetic detonation|sympathetically detonate]] the surrounding items. The explosion could propagate to all or the majority of the items stored together, causing a mass detonation. There will also be fragments from the item's casing and/or structures in the blast area.
* '''1.2''' Non-mass explosion, fragment-producing. HC/D 1.2 is further divided into three subdivisions, HC/D 1.2.1, 1.2.2 and 1.2.3, to account for the magnitude of the effects of an explosion.
* '''1.3''' Mass fire, minor blast or fragment hazard. Propellants and many pyrotechnic items fall into this category. If one item in a package or stack initiates, it will usually propagate to the other items, creating a mass fire.
* '''1.4''' Moderate fire, no blast or fragment. HC/D 1.4 items are listed in the table as explosives with no significant hazard. Most small arms ammunition (including loaded weapons) and some pyrotechnic items fall into this category. If the energetic material in these items inadvertently initiates, most of the energy and fragments will be contained within the storage structure or the item containers themselves.
* '''1.5''' mass detonation hazard, very insensitive.
* '''1.6''' [[detonation]] hazard without mass detonation hazard, extremely insensitive.

To see an entire UNO Table, browse Paragraphs 3–8 and 3–9 of NAVSEA OP 5, Vol. 1, Chapter 3.

====Class 1 Compatibility Group====
Compatibility Group codes are used to indicate storage compatibility for HC/D Class 1 (explosive) materials. Letters are used to designate 13 compatibility groups as follows.
* '''A''': Primary explosive substance (1.1A).
* '''B''': An article containing a primary explosive substance and not containing two or more effective protective features. Some articles, such as detonator assemblies for blasting and primers, cap-type, are included. (1.1B, 1.2B, 1.4B).
* '''C''': Propellant explosive substance or other deflagrating explosive substance or article containing such explosive substance (1.1C, 1.2C, 1.3C, 1.4C). These are bulk [[propellant]]s, propelling charges, and devices containing propellants with or without means of ignition. Examples include single-based propellant, double-based propellant, triple-based propellant, and [[composite propellant]]s, [[solid propellant]] [[rocket motor]]s and ammunition with inert projectiles.
* '''D''': Secondary detonating explosive substance or black powder or article containing a secondary detonating explosive substance, in each case without means of initiation and without a propelling charge, or article containing a primary explosive substance and containing two or more effective protective features. (1.1D, 1.2D, 1.4D, 1.5D).
* '''E''': Article containing a secondary detonating explosive substance without means of initiation, with a propelling charge (other than one containing flammable liquid, gel or [[hypergolic]] liquid) (1.1E, 1.2E, 1.4E).
* '''F''' containing a [[secondary explosive|secondary]] [[detonating explosive]] substance with its means of initiation, with a propelling charge (other than one containing flammable liquid, gel or hypergolic liquid) or without a propelling charge (1.1F, 1.2F, 1.3F, 1.4F).
* '''G''': Pyrotechnic substance or article containing a pyrotechnic substance, or article containing both an explosive substance and an illuminating, incendiary, tear-producing or smoke-producing substance (other than a water-activated article or one containing white phosphorus, phosphide or flammable liquid or gel or hypergolic liquid) (1.1G, 1.2G, 1.3G, 1.4G). Examples include Flares, signals, incendiary or illuminating ammunition and other smoke and tear producing devices.
* '''H''': Article containing both an explosive substance and white phosphorus (1.2H, 1.3H). These articles will spontaneously combust when exposed to the atmosphere.
* '''J''': Article containing both an explosive substance and flammable liquid or gel (1.1J, 1.2J, 1.3J). This excludes liquids or gels which are spontaneously flammable when exposed to water or the atmosphere, which belong in group H. Examples include liquid or gel filled incendiary ammunition, fuel-air explosive (FAE) devices, and flammable liquid fueled missiles.
* '''K''': Article containing both an explosive substance and a toxic chemical agent (1.2K, 1.3K)
* '''L''' Explosive substance or article containing an explosive substance and presenting a special risk (e.g., due to water-activation or presence of hypergolic liquids, phosphides, or [[pyrophoricity|pyrophoric]] substances) needing isolation of each type (1.1L, 1.2L, 1.3L). Damaged or suspect ammunition of any group belongs in this group.
* '''N''': Articles containing only extremely insensitive detonating substances (1.6N).
* '''S''': Substance or article so packed or designed that any hazardous effects arising from accidental functioning are limited to the extent that they do not significantly hinder or prohibit fire fighting or other emergency response efforts in the immediate vicinity of the package (1.4S).

==Regulation==
The legality of possessing or using explosives varies by jurisdiction. Various countries around the world have enacted explosives law and require licenses to manufacture, distribute, store, use, possess explosives or ingredients.

===Netherlands===
In the [[Netherlands]], the civil and commercial use of explosives is covered under the ''Wet explosieven voor civiel gebruik'' (explosives for civil use Act), in accordance with EU directive nr. 93/15/EEG<ref>{{cite web|url=http://wetten.overheid.nl/BWBR0006803/geldigheidsdatum_23-12-2013|title=wetten.nl – Wet- en regelgeving – Wet explosieven voor civiel gebruik – BWBR0006803|url-status=live|archive-url=https://web.archive.org/web/20131225005146/http://wetten.overheid.nl/BWBR0006803/geldigheidsdatum_23-12-2013|archive-date=25 December 2013|df=dmy-all}}</ref> (Dutch). The illegal use of explosives is covered under the ''Wet Wapens en Munitie'' (Weapons and Munition Act)<ref>{{cite web|url=http://wetten.overheid.nl/BWBR0008804/geldigheidsdatum_23-12-2013|title=wetten.nl – Wet- en regelgeving – Wet wapens en munitie – BWBR0008804|url-status=live|archive-url=https://web.archive.org/web/20131225005149/http://wetten.overheid.nl/BWBR0008804/geldigheidsdatum_23-12-2013|archive-date=25 December 2013|df=dmy-all}}</ref> (Dutch).

===United Kingdom===
{{further|Explosive Substances Act 1883}}
The new Explosives Regulations 2014 (ER 2014)<ref name=":1">{{OGL-attribution|version=3.0|{{Cite web|url=http://www.legislation.gov.uk/uksi/2014/1638/made/data.htm|title=The Explosives Regulations 2014|website=www.legislation.gov.uk|access-date=2019-02-16 |archive-url=https://web.archive.org/web/20190212014704/https://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/ |archive-date=12 February 2019}}}}</ref> came into force on 1 October 2014 and defines "explosive" as:

{{blockquote|"a) any explosive article or explosive substance which would —

(i) if packaged for transport, be classified in accordance with the [[United Nations]] Recommendations as falling within Class 1; or

(ii) be classified in accordance with the United Nations Recommendations as —

(aa) being unduly sensitive or so reactive as to be subject to spontaneous reaction and accordingly too dangerous to transport, and

(bb) falling within Class 1; or

(b) a desensitised explosive,

but it does not include an explosive substance produced as part of a manufacturing process which thereafter reprocesses it in order to produce a substance or preparation which is not an explosive substance"<ref name=":1" />}}

{{blockquote|"Anyone who wishes to acquire and or keep relevant explosives needs to contact their local police explosives liaison officer. All explosives are relevant explosives apart from those listed under Schedule 2 of Explosives Regulations 2014."<ref>{{Cite web|url=http://www.hse.gov.uk/explosives/licensing/index.htm|title=HSE Explosives – Licensing|website=www.hse.gov.uk|access-date=2019-02-16|archive-url=https://web.archive.org/web/20190421213250/http://www.hse.gov.uk/explosives/licensing/index.htm|archive-date=21 April 2019|url-status=live}}</ref>}}

===United States===
During [[World War I]], numerous laws were created to regulate war related industries and increase security within the United States. In 1917, the [[65th United States Congress]] created [[List of United States federal legislation, 1901–2001#65th United States Congress|many laws]], including the ''[[Espionage Act of 1917]]'' and ''Explosives Act of 1917''.

The ''Explosives Act of 1917'' (session 1, chapter 83, {{USStat|40|385}}) was signed on 6 October 1917 and went into effect on 16 November 1917. The legal summary is "An Act to prohibit the manufacture, distribution, storage, use, and possession in '''time of war''' of explosives, providing regulations for the safe manufacture, distribution, storage, use, and possession of the same, and for other purposes". This was the first federal regulation of licensing explosives purchases. The act was deactivated after World War I ended.<ref>{{cite web|url=https://www.ime.org/content/1913_1919|title=1913–1919|url-status=live|archive-url=https://web.archive.org/web/20160201054704/https://www.ime.org/content/1913_1919|archive-date=1 February 2016|df=dmy-all}}</ref>

After the United States entered [[World War II]], the Explosives Act of 1917 was reactivated. In 1947, the act was deactivated by [[President Truman]].<ref>{{cite web|url=https://www.ime.org/content/1940_1949|title=1940–1949|url-status=live|archive-url=https://web.archive.org/web/20160304075716/https://www.ime.org/content/1940_1949|archive-date=4 March 2016|df=dmy-all}}</ref>

The ''[[Organized Crime Control Act of 1970]]'' ({{USPL|91|452}}) transferred many explosives regulations to the [[Bureau of Alcohol, Tobacco, Firearms and Explosives|Bureau of Alcohol, Tobacco and Firearms]] (ATF) of the [[United States Department of the Treasury|Department of Treasury]]. The bill became effective in 1971.<ref>{{cite web|url=https://www.ime.org/content/1970_1979|title=1970–1979|url-status=live|archive-url=https://web.archive.org/web/20151117031212/https://www.ime.org/content/1970_1979|archive-date=17 November 2015|df=dmy-all}}</ref>

Currently, regulations are governed by [[Title 18 of the United States Code]] and [[Title 27 of the Code of Federal Regulations]]:
* "Importation, Manufacture, Distribution and Storage of Explosive Materials" (18 U.S.C. Chapter 40).<ref>{{cite web |url=http://www.gpo.gov/fdsys/pkg/USCODE-2011-title18/pdf/USCODE-2011-title18-partI-chap40.pdf |title=Federal Explosives Laws |publisher=U.S. Department of Justice, Bureau of Alcohol, Tobacco, Firearms and Explosives |access-date=1 February 2016 |url-status=live |archive-url=https://web.archive.org/web/20160306153102/https://www.gpo.gov/fdsys/pkg/USCODE-2011-title18/pdf/USCODE-2011-title18-partI-chap40.pdf |archive-date=6 March 2016 |df=dmy-all}}</ref>
* "Commerce in Explosives" (27 C.F.R. Chapter II, Part 555).<ref>{{cite web |url=http://www.atf.gov/content/library/codified-regulations |title=Regulations for Alcohol, Tobacco, Firearms and Explosives | Bureau of Alcohol, Tobacco, Firearms and Explosives |access-date=2014-12-13 |url-status=live |archive-url=https://web.archive.org/web/20141215074434/http://www.atf.gov/content/library/codified-regulations |archive-date=15 December 2014 |df=dmy-all}} ATF Regulations</ref>

==List of explosives==

===[[Chemical compound|Compounds]]===

====[[Acetylide]]s====
* [[Copper(I) acetylide]], [[Dichloroacetylene]], [[Silver acetylide]]

====[[Fulminate]]s====
* [[Fulminic Acid]], [[Fulminating Gold]], [[Mercury(II) fulminate]], [[Platinum fulminate]], [[Potassium fulminate]], [[Silver fulminate]]

====Nitro====
* '''MonoNitro:''' [[Nitroguanidine]], [[Nitroethane]], [[Nitromethane]], [[Nitropropane]], [[Nitrourea]]
* '''DiNitro''': [[Diazo dinitro phenol]], [[Dinitrobenzene]], [[Dinitroethylene urea]], [[1,5-Dinitro naphthalene|DNN]], [[Dinitrophenol]], [[Dinitrophenolate]], [[2,4-Dinitrophenylhydrazine|DNPH]], [[Dinitroresorcinol]], [[Dinitropentano nitrile]], [[Polydinitropropyl acrylate]], [[Dinitro cerine]], [[Dipicryl sulfone]], [[Dipicrylamine]], [[Ethyl 4,4-Dinitropentanate|EDNP]], [[Potassium dinitrobenzofuroxan|KDNBF]], [[1,2-Bis 2,2-difluro 2-nitroacetoxythane|BEAF]], [[DADNE]]
* '''TriNitro:''' [[Cyclotrimethylene trinitramine|RDX]], [[Diaminotrinitrobenzene]], [[Triaminotrinitrobenzene]], [[Lead styphnate]], [[Lead picrate]], [[Trinitroaniline]], [[Trinitroanisole]], [[1,3,3-Trinitroazetidine|TNAS]], [[1,3,5-Trinitrobenzene|TNB]], [[2,4,6-Trinitrobenzoic acid|TNBA]], [[Styphnic acid]], [[Metacresol|MC]], [[Trinitroethyl formal]], [[Trinitroethylorthocarbonate|TNOC]], [[Trinitroethylorthoformate|TNOF]], [[Trinitrophenol|TNP]], [[Trinitrotoluene|TNT]], [[Trinitronaphthalene|TNN]], [[Trinitrophloro glucinol|TNPG]], [[Trinitroresorcinol|TNR]], [[Bistrinitroethyl nitramine|BTNEN]], [[Bistrinitroethyl carbonate|BTNEC]], [[Ammonium picrate]], [[Trinitrosorcinate|TNS]]
* '''TetraNitro''': [[Tetryl]], [[HMX]]
* '''HexaNitro''': [[Hexanitrostilbene|HNS]], [[HNIW]], [[Hexanitrohexaazatricyclododecanedione|HHTDD]]
* '''HeptaNitro''': [[Heptanitrocubane]]
* '''OctaNitro:''' [[Octanitrocubane]]

====[[Nitroso]]s====
* '''Trinitrosos:''' [[R-salt]]

====[[Nitrate]]s====
* '''Mononitrates:''' [[Ammonium nitrate]], [[Methyl ammonium nitrate]], [[Urea nitrate|Urea Nitrate]]
* '''Dinitrates:''' [[Diethyleneglycol dinitrate]], [[Ethylenediamine dinitrate]], [[Ethylene dinitramine]], [[Ethylene glycol dinitrate]], [[Hexamethylenetetramine dinitrate]], [[Triethylene glycol dinitrate]]
* '''Trinitrates:''' [[1,2,4-Butanetriol trinitrate]], [[Trimethylolethane trinitrate]], [[Nitroglycerin]]
* '''Tetranitrates:''' [[Erythritol tetranitrate]], [[Pentaerythritol tetranitrate]], [[Tetranitratoxycarbon]]
* '''Pentanitrates:''' [[Xylitol pentanitrate]]
* '''Polynitrates:''' [[Nitrocellulose]], [[Nitrostarch]], [[Mannitol hexanitrate]]

====[[Amine]]s====
* '''Tertiary Amines:''' [[Nitrogen tribromide]], [[Nitrogen trichloride]], [[Nitrogen triiodide]], [[Nitrogen trisulfide]], [[Selenium nitride]], [[Silver nitride]]
* '''Diamines:''' [[Disulfur dinitride]]
* '''Tetramines:''' [[Tetrazene]], [[Tetrazole]], [[Azidoazide azide]]
* '''Pentamines:''' [[Pentazenium]]
* '''Octamines:''' [[Octaazacubane]], [[1,1'-Azobis-1,2,3-triazole]]

====[[Azide]]s====
* '''Inorganic:''' [[Chlorine azide]], [[Copper(II) azide]], [[Fluorine azide]], [[Hydrazoic acid]], [[Lead(II) azide]], [[Silver azide]], [[Sodium azide]], [[Rubidium azide]], [[Selenium tetraazide]], [[Silicon tetraazide]], [[Tellurium tetraazide]], [[Titanium tetraazide]]
* '''Organic:''' [[Cyanuric triazide]], [[Cyanogen azide]], [[Ethyl azide]], [[Tetraazidomethane]]

====[[Peroxide]]s====
* [[Acetone peroxide|Acetone peroxide (TATP)]], [[Cumene hydroperoxide]], [[Diacetyl peroxide]], [[Dibenzoyl peroxide]], [[Diethyl ether peroxide]], [[Hexamethylene triperoxide diamine]], [[Methyl ethyl ketone peroxide]], [[Tert-butyl hydroperoxide]], [[Tetramethylene diperoxide dicarbamide]]

====[[Oxide]]s====
* [[Xenon oxytetrafluoride]], [[Xenon dioxide]], [[Xenon trioxide]], [[Xenon tetroxide]]

====Unsorted====
* [[Alkali metal Ozonides]]
* [[Ammonium chlorate]]
* [[Ammonium perchlorate]]
* [[Ammonium permanganate]]
* [[Azidotetrazolates]]
* Azoclathrates
* [[Benzvalene]]
* [[Chlorine oxide]]s
* [[4-Dimethylaminophenylpentazole|DMAPP]]
* [[Fluorine perchlorate]]
* [[Fulminating gold]]
* [[Fulminating silver]] (several substances)
* [[Hexafluoroantimonate]]
* [[Hexafluoroarsenate]]
* [[Hypofluorous acid]]
* [[Manganese heptoxide]]
* [[Mercury nitride]]
* [[Nitronium perchlorate]]
* [[Nitrotetrazolate-N-Oxides]]
* [[Peroxy acid]]s
* [[Peroxymonosulfuric acid]]
* [[Tetramine copper complexes]]
* [[Tetrasulfur tetranitride]]

===Mixtures===
* [[Aluminum Orphorite]], [[Amatex]], [[Amatol]], [[Ammonal]], [[Armstrong's mixture]], [[ANFO]], [[ANNMAL]], [[Astrolite]]
* [[Baranol]], [[Baratol]], [[Ballistite]], [[Butyl tetryl]]
* [[Carbonite (explosive)|Carbonite]], [[Composition A]], [[Composition B]], [[Composition C]], [[Composition 1]], [[Composition 2]], [[Composition 3]], [[Composition 4]], [[Composition 5]], [[Composition H6]], [[Cordtex]], [[Cyclotol]]
* [[Danubit]], [[Detasheet]], [[Detonating cord]], [[Dualin]], [[Dunnite]], [[Dynamite]]
* [[Ecrasite]], [[Ednatol]]
* [[Flash powder]]
* [[Gelignite]], [[Gunpowder]]
* [[Hexanite]], [[Hydromite 600]]
* [[Kinetite]]
* [[Minol (explosive)|Minol]]
* [[Octol]], [[Oxyliquit]]
* [[Panclastite]], [[Pentolite]], [[Picratol]], [[PNNM]], [[Pyrotol]]
* [[Schneiderite]], [[Semtex]], [[Shellite (explosive)|Shellite]]
* [[Tannerit simply]], [[Tannerite]], [[Titadine]], [[Tovex]], [[Torpex]], [[Tritonal]]

===Elements and isotopes===
* [[Alkali metals]]
* [[Explosive antimony]]
* [[Plutonium-239]]
* [[Uranium-235]]

==See also==
* [[Blast injury]]
* [[Detection dog]]
* [[Flame speed]]
* [[Improvised explosive device]]
* [[Insensitive munition]]
* [[Largest artificial non-nuclear explosions]]
* [[Nuclear weapon]]
* [[Orica]]; largest supplier of commercial explosives
* [[TM 31-210 Improvised Munitions Handbook]]
* [[Total body disruption]]

==References==
{{Reflist}}

==Further reading==
===U.S. Government===
* [https://archive.org/details/FM_5_250_Explosives_And_Demolitions_ ''Explosives and Demolitions''] FM 5–250; U.S. Department of the Army; 274 pp.; 1992.
* [https://archive.org/details/TM_9_1300_214_Military_Explosives_ ''Military Explosives''] TM 9–1300–214; U.S. Department of the Army; 355 pp.; 1984.
* [https://archive.org/details/explosivesblasti00dick ''Explosives and Blasting Procedures Manual'']; U.S. Department of Interior; 128 pp.; 1982.
* ''Safety and Performance Tests for Qualification of Explosives''; Commander, Naval Ordnance Systems Command; NAVORD OD 44811. Washington, DC: GPO, 1972.
* ''Weapons Systems Fundamentals''; Commander, Naval Ordnance Systems Command. NAVORD OP 3000, vol. 2, 1st rev. Washington, DC: GPO, 1971.
* ''Elements of Armament Engineering – Part One''; Army Research Office. Washington, D.C.: [[U.S. Army Materiel Command]], 1964.
* Hazardous Materials Transportation Plaecards; USDOT.

===Institute of Makers of Explosives===
* [https://archive.org/details/gov.law.ime.explosives.1940 ''Safety in the Handling and Use of Explosives''] SLP 17; Institute of Makers of Explosives; 66 pp.; 1932 / 1935 / 1940.
* ''History of the Explosives Industry in America''; Institute of Makers of Explosives; 37 pp.; 1927.
* [https://archive.org/details/clearinglandofst00matt ''Clearing Land of Stumps'']; Institute of Makers of Explosives; 92 pp.; 1917.
* [https://archive.org/details/useofexplosivesf00inst ''The Use of Explosives for Agricultural and Other Purposes'']; Institute of Makers of Explosives; 190 pp.; 1917.
* [https://archive.org/details/cu31924080100872 ''The Use of Explosives in making Ditches'']; Institute of Makers of Explosives; 80 pp.; 1917.

===Other historical===
* [https://archive.org/details/farmershandbook00deptgoog ''Farmers' Hand Book of Explosives'']; duPont; 113 pp.; 1920.
* [https://archive.org/details/ashortaccountex00marsgoog ''A Short Account of Explosives'']; Arthur Marshall; 119 pp.; 1917.
* [https://archive.org/details/historicalpapers00macdrich ''Historical Papers on Modern Explosives'']; George MacDonald; 216 pp.; 1912.
* [https://archive.org/details/riseprogressofbr00interich ''The Rise and Progress of the British Explosives Industry'']; International Congress of Pure and Applied Chemistry; 450 pp.; 1909.
* [https://archive.org/details/explosivestheirp00bertrich ''Explosives and their Power'']; M. Berthelot; 592 pp.; 1892.

==External links==
Listed in alphabetical order:
* [https://web.archive.org/web/20190803013902/http://blasterexchange.com/ Blaster Exchange – Explosives Industry Portal]
* [http://environmentalchemistry.com/yogi/hazmat/placards/class1.html Class 1 Hazmat Placards]
* [http://www.explosivesacademy.org/ Explosives Academy]
* [https://web.archive.org/web/20131012060546/http://www.explosives.org.au/ Explosives info]
* [http://www.tandf.co.uk/journals/titles/07370652.asp Journal of Energetic Materials]
* [https://fas.org/man/dod-101/navy/docs/fun/part12.htm Military Explosives]
* [http://www.roguesci.org/theforum The Explosives and Weapons Forum]
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Cauchy sequence

2025-07-01T01:34:13Z

174.138.212.166: /* Non-example: open interval */ MOS:EMDASH

{{Short description|Sequence of points that get progressively closer to each other}}
{{Use shortened footnotes|date=November 2022}}
{{multiple image
|align = right
|direction = vertical
|width = 250
|image1 = Cauchy sequence illustration.svg
|caption1 = (a) The plot of a Cauchy [[sequence (mathematics)|sequence]] <math>(x_n),</math> shown in blue, as <math>x_n</math> versus <math>n.</math> If the [[Space (mathematics)|space]] containing the sequence is [[Complete metric space|complete]], then the sequence has a [[Limit (mathematics)|limit]].
|image2 = Cauchy sequence illustration2.svg
|caption2 = (b) A sequence that is not Cauchy. The [[Element (mathematics)|elements]] of the sequence do not get arbitrarily close to each other as the sequence progresses.
}}
In [[mathematics]], a '''Cauchy sequence''' is a [[sequence]] whose [[Element (mathematics)|elements]] become arbitrarily close to each other as the sequence progresses.{{sfn|Lang|1992}} More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after [[Augustin-Louis Cauchy]]; they may occasionally be known as '''fundamental sequences'''.<ref>{{cite book|first=Heinz-Dieter |last=Ebbinghaus|title=Numbers|place=New York|publisher=Springer|date=1991|page=40}}</ref>

It is not sufficient for each term to become arbitrarily close to the {{em|preceding}} term. For instance, in the sequence of square roots of natural numbers:
<math display="block">a_n=\sqrt n,</math>
the consecutive terms become arbitrarily close to each other – their differences
<math display="block">a_{n+1}-a_n = \sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2\sqrt n}</math>
tend to zero as the index {{mvar|n}} grows. However, with growing values of {{mvar|n}}, the terms <math>a_n</math> become arbitrarily large. So, for any index {{mvar|n}} and distance {{mvar|d}}, there exists an index {{mvar|m}} big enough such that <math>a_m - a_n > d.</math> As a result, no matter how far one goes, the remaining terms of the sequence never get close to {{em|each other}}; hence the sequence is not Cauchy.

The utility of Cauchy sequences lies in the fact that in a [[complete metric space]] (one where all such sequences are known to [[Limit of a sequence|converge to a limit]]), the criterion for [[Convergence (mathematics)|convergence]] depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in [[algorithm]]s, both theoretical and applied, where an [[Iterative method|iterative process]] can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.

Generalizations of Cauchy sequences in more abstract [[uniform spaces]] exist in the form of [[Cauchy filter]]s and [[Cauchy net]]s.

==In real numbers==
A sequence
<math display="block">x_1, x_2, x_3, \ldots</math>
of real numbers is called a Cauchy sequence if for every [[Positive and negative numbers|positive]] real number <math>\varepsilon,</math> there is a positive [[integer]] ''N'' such that for all [[natural numbers]] <math>m, n > N,</math>
<math display="block">|x_m - x_n| < \varepsilon,</math>
where the vertical bars denote the [[absolute value]]. In a similar way one can define Cauchy sequences of rational or [[complex number]]s. Cauchy formulated such a condition by requiring <math>x_m - x_n</math> to be [[infinitesimal]] for every pair of infinite ''m'', ''n''.

For any real number ''r'', the sequence of truncated decimal expansions of ''r'' forms a Cauchy sequence. For example, when <math>r = \pi,</math> this sequence is (3, 3.1, 3.14, 3.141, ...). The ''m''th and ''n''th terms differ by at most <math>10^{1-m}</math> when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number <math>\varepsilon.</math>

===Modulus of Cauchy convergence===

If <math>(x_1, x_2, x_3, ...)</math> is a sequence in the set <math>X,</math> then a ''modulus of Cauchy convergence'' for the sequence is a [[Function (mathematics)|function]] <math>\alpha</math> from the set of [[natural number]]s to itself, such that for all natural numbers <math>k</math> and natural numbers <math>m, n > \alpha(k),</math> <math>|x_m - x_n| < 1/k.</math>

Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the [[well-ordering property]] of the natural numbers (let <math>\alpha(k)</math> be the smallest possible <math>N</math> in the definition of Cauchy sequence, taking <math>\varepsilon</math> to be <math>1/k</math>). The existence of a modulus also follows from the principle of [[countable choice]]. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually <math>\alpha(k) = k</math> or <math>\alpha(k) = 2^k</math>). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice.

Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by {{harvtxt|Bishop|2012}} and by {{harvtxt|Bridges|1997}} in constructive mathematics textbooks.

==In a metric space==

Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space ''X''.
To do so, the [[absolute difference]] <math>\left|x_m - x_n\right|</math> is replaced by the distance <math>d\left(x_m, x_n\right)</math> (where ''d'' denotes a [[Metric (mathematics)|metric]]) between <math>x_m</math> and <math>x_n.</math>

Formally, given a [[metric space]] <math>(X, d),</math> a sequence of elements of <math>X</math>
<math display="block">x_1, x_2, x_3, \ldots</math>
is Cauchy, if for every positive [[real number]] <math>\varepsilon > 0</math> there is a positive [[integer]] <math>N</math> such that for all positive integers <math>m, n > N,</math> the distance
<math display="block">d\left(x_m, x_n\right) < \varepsilon.</math>

Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a [[Limit of a sequence|limit]] in ''X''.
Nonetheless, such a limit does not always exist within ''X'': the property of a space that every Cauchy sequence converges in the space is called ''completeness'', and is detailed below.

==Completeness==
A metric space (''X'', ''d'') in which every Cauchy sequence converges to an element of ''X'' is called [[Complete metric space|complete]].

===Examples===
The [[real number]]s are complete under the metric induced by the usual absolute value, and one of the standard [[Construction of the real numbers|constructions of the real numbers]] involves Cauchy sequences of [[rational number]]s. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number.

A rather different type of example is afforded by a metric space ''X'' which has the [[discrete space|discrete metric]] (where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of ''X'' must be constant beyond some fixed point, and converges to the eventually repeating term.

===Non-example: rational numbers===
The [[rational number]]s <math>\Q</math> are not complete (for the usual distance): 
There are sequences of rationals that converge (in <math>\R</math>) to [[irrational number]]s; these are Cauchy sequences having no limit in <math>\Q.</math> In fact, if a real number ''x'' is irrational, then the sequence (''x''''n''), whose ''n''-th term is the truncation to ''n'' decimal places of the decimal expansion of ''x'', gives a Cauchy sequence of rational numbers with irrational limit ''x''. Irrational numbers certainly exist in <math>\R,</math> for example:

* The sequence defined by <math>x_0=1, x_{n+1}=\frac{x_n+2/x_n}{2}</math> consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational [[square root of 2]], see [[Methods of computing square roots#Heron's method|Babylonian method of computing square root]].
* The sequence <math>x_n = F_n / F_{n-1}</math> of ratios of consecutive [[Fibonacci number]]s which, if it converges at all, converges to a limit <math>\phi</math> satisfying <math>\phi^2 = \phi+1,</math> and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number <math>\varphi = (1+\sqrt5)/2,</math> the [[Golden ratio]], which is irrational.
* The values of the exponential, sine and cosine functions, exp(''x''), sin(''x''), cos(''x''), are known to be irrational for any rational value of <math>x \neq 0,</math> but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the [[Maclaurin series]].

===Non-example: open interval===
The open interval <math>X = (0, 2)</math> in the set of real numbers with an ordinary distance in <math>\R</math> is not a complete space: there is a sequence <math>x_n = 1/n</math> in it, which is Cauchy (for arbitrarily small distance bound <math>d > 0</math> all terms <math>x_n</math> of <math>n > 1/d</math> fit in the <math>(0, d)</math> interval), however does not converge in <math>X</math>—its 'limit', number 0, does not belong to the space <math>X .</math>

===Other properties===
* Every convergent sequence (with limit ''s'', say) is a Cauchy sequence, since, given any real number <math>\varepsilon > 0,</math> beyond some fixed point, every term of the sequence is within distance <math>\varepsilon/2</math> of ''s'', so any two terms of the sequence are within distance <math>\varepsilon</math> of each other.
* In any metric space, a Cauchy sequence <math>x_n</math> is [[Bounded function|bounded]] (since for some ''N'', all terms of the sequence from the ''N''-th onwards are within distance 1 of each other, and if ''M'' is the largest distance between <math>x_N</math> and any terms up to the ''N''-th, then no term of the sequence has distance greater than <math>M + 1</math> from <math>x_N</math>).
* In any metric space, a Cauchy sequence which has a convergent subsequence with limit ''s'' is itself convergent (with the same limit), since, given any real number ''r'' > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance ''r''/2 of ''s'', and any two terms of the original sequence are within distance ''r''/2 of each other, so every term of the original sequence is within distance ''r'' of ''s''.

These last two properties, together with the [[Bolzano–Weierstrass theorem]], yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the [[Heine–Borel theorem]]. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the [[least upper bound axiom]]. The alternative approach, mentioned above, of {{em|constructing}} the real numbers as the [[Completion (metric space)|completion]] of the rational numbers, makes the completeness of the real numbers tautological.

One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an [[infinite series]] of real numbers
(or, more generally, of elements of any complete [[normed linear space]], or [[Banach space]]). Such a series
<math display="inline">\sum_{n=1}^{\infty} x_n</math> is considered to be convergent if and only if the sequence of [[partial sum]]s <math>(s_{m})</math> is convergent, where <math display="inline">s_m = \sum_{n=1}^{m} x_n.</math> It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers <math>p > q,</math>
<math display="block">s_p - s_q = \sum_{n=q+1}^p x_n.</math>

If <math>f : M \to N</math> is a [[uniformly continuous]] map between the metric spaces ''M'' and ''N'' and (''x''''n'') is a Cauchy sequence in ''M'', then <math>(f(x_n))</math> is a Cauchy sequence in ''N''. If <math>(x_n)</math> and <math>(y_n)</math> are two Cauchy sequences in the rational, real or complex numbers, then the sum <math>(x_n + y_n)</math> and the product <math>(x_n y_n)</math> are also Cauchy sequences.

==Generalizations==

===In topological vector spaces===
There is also a concept of Cauchy sequence for a [[topological vector space]] <math>X</math>: Pick a [[local base]] <math>B</math> for <math>X</math> about 0; then (<math>x_k</math>) is a Cauchy sequence if for each member <math>V\in B,</math> there is some number <math>N</math> such that whenever
<math>n,m > N, x_n - x_m</math> is an element of <math>V.</math> If the topology of <math>X</math> is compatible with a [[translation-invariant metric]] <math>d,</math> the two definitions agree.

===In topological groups===

Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a [[topological group]]: A sequence <math>(x_k)</math> in a topological group <math>G</math> is a Cauchy sequence if for every open neighbourhood <math>U</math> of the [[Identity element|identity]] in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n x_m^{-1} \in U.</math> As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in <math>G.</math>

As in the [[Complete metric space#Completion|construction of the completion of a metric space]], one can furthermore define the binary relation on Cauchy sequences in <math>G</math> that <math>(x_k)</math> and <math>(y_k)</math> are equivalent if for every open [[Neighbourhood (mathematics)|neighbourhood]] <math>U</math> of the identity in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n y_m^{-1} \in U.</math> This relation is an [[equivalence relation]]: It is reflexive since the sequences are Cauchy sequences. It is symmetric since <math>y_n x_m^{-1} = (x_m y_n^{-1})^{-1} \in U^{-1}</math> which by continuity of the inverse is another open neighbourhood of the identity. It is [[Transitive relation|transitive]] since <math>x_n z_l^{-1} = x_n y_m^{-1} y_m z_l^{-1} \in U' U''</math> where <math>U'</math> and <math>U''</math> are open neighbourhoods of the identity such that <math>U'U'' \subseteq U</math>; such pairs exist by the continuity of the group operation.

===In groups===

There is also a concept of Cauchy sequence in a [[group (mathematics)|group]] <math>G</math>:
Let <math>H=(H_r)</math> be a decreasing sequence of [[normal subgroup]]s of <math>G</math> of finite [[Index of a subgroup|index]].
Then a sequence <math>(x_n)</math> in <math>G</math> is said to be Cauchy (with respect to <math>H</math>) if and only if for any <math>r</math> there is <math>N</math> such that for all <math>m, n > N, x_n x_m^{-1} \in H_r.</math>

Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on <math>G,</math> namely that for which <math>H</math> is a local base.

The set <math>C</math> of such Cauchy sequences forms a group (for the componentwise product), and the set <math>C_0</math> of null sequences (sequences such that <math>\forall r, \exists N, \forall n > N, x_n \in H_r</math>) is a normal subgroup of <math>C.</math> The [[factor group]] <math>C/C_0</math> is called the completion of <math>G</math> with respect to <math>H.</math>

One can then show that this completion is isomorphic to the [[inverse limit]] of the sequence <math>(G/H_r).</math>

An example of this construction familiar in [[number theory]] and [[algebraic geometry]] is the construction of the [[p-adic number|<math>p</math>-adic completion]] of the integers with respect to a [[prime number|prime]] <math>p.</math> In this case, <math>G</math> is the integers under addition, and <math>H_r</math> is the additive subgroup consisting of integer multiples of <math>p_r.</math>

If <math>H</math> is a [[Cofinal (mathematics)|cofinal]] sequence (that is, any normal subgroup of finite index contains some <math>H_r</math>), then this completion is [[Canonical form|canonical]] in the sense that it is isomorphic to the inverse limit of <math>(G/H)_H,</math> where <math>H</math> varies over {{em|all}} normal subgroups of finite [[Index of a subgroup|index]]. For further details, see Ch. I.10 in [[Serge Lang|Lang]]'s "Algebra".

===In a hyperreal continuum===
A real sequence <math>\langle u_n : n \in \N \rangle</math> has a natural [[Hyperreal number|hyperreal]] extension, defined for [[hypernatural]] values ''H'' of the index ''n'' in addition to the usual natural ''n''. The sequence is Cauchy if and only if for every infinite ''H'' and ''K'', the values <math>u_H</math> and <math>u_K</math> are infinitely close, or [[Adequality|adequal]], that is,
:<math>\mathrm{st}(u_H-u_K)= 0</math>
where "st" is the [[standard part function]].

===Cauchy completion of categories===
{{harvtxt|Krause|2020}} introduced a notion of Cauchy completion of a [[Category (mathematics)|category]]. Applied to <math>\Q</math> (the category whose [[object (category theory)|objects]] are rational numbers, and there is a [[morphism]] from ''x'' to ''y'' if and only if <math>x \leq y</math>), this Cauchy completion yields <math>\R\cup\left\{\infty\right\}</math> (again interpreted as a category using its natural ordering).

==See also==
* {{annotated link|Modes of convergence (annotated index)}}
* {{annotated link|Dedekind cut}}

==References==
{{reflist}}

==Further reading==
*{{cite book
|last= Bishop
|first= Errett Albert
|author-link= Errett Bishop
|title=Foundations of Constructive Analysis
|publisher=Ishi Press
|year=2012
|isbn=9784871877145
}}

*{{cite book
|last= Bourbaki
|first= Nicolas
|author-link= Nicolas Bourbaki
|title=Commutative Algebra
|url=https://archive.org/details/commutativealgeb0000bour
|url-access=registration
|edition=English translation
|publisher=Addison-Wesley / Hermann
|year=1972
|isbn=0-201-00644-8
}}

*{{cite book
|last= Bridges
|first= Douglas Sutherland
|title=Foundations of Constructive Analysis
|publisher=Springer
|year=1997
|isbn=978-0-387-98239-7
}}

*{{cite journal
|last=Krause
|first=Henning
|title=Completing perfect complexes: With appendices by Tobias Barthel and Bernhard Keller
|journal=Mathematische Zeitschrift
|volume=296
|issue=3–4
|year=2020
|pages=1387–1427
|doi=10.1007/s00209-020-02490-z
|doi-access=free
|arxiv=1805.10751
}}

*{{cite book
|last= Lang
|first= Serge
|author-link= Serge Lang
|title= Algebra
|edition= 3d
|publisher=Addison Wesley Publishing Company
|location=Reading, Mass.
|year=1992
|isbn=978-0-201-55540-0
|zbl=0848.13001
}}

*{{cite book
|last=Spivak
|first=Michael
|author-link=Michael Spivak
|title=Calculus
|year=1994
|edition=3rd
|location=Berkeley, CA
|publisher=Publish or Perish
|isbn=0-914098-89-6
|url=http://www.mathpop.com/bookhtms/cal.htm
|access-date=2007-05-26
|archive-url=https://web.archive.org/web/20070517171054/http://www.mathpop.com/bookhtms/cal.htm
|archive-date=2007-05-17
|url-status=dead
}}

*{{cite book
|last1=Troelstra
|first1=A. S.
|author-link1=Anne Sjerp Troelstra
|last2=van Dalen
|first2=D.
|author-link2=Dirk van Dalen
|title=Constructivism in Mathematics: An Introduction
|date=1988
|url=https://archive.org/details/constructivismin0002troe
|url-access=registration
}} (for uses in constructive mathematics)

==External links==

* {{springer|title=Cauchy sequence|id=p/f042240}}

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