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| {{Use dmy dates|date=September 2010}}
| | Hi there. Allow me start psychic solutions by lynne ([http://m-card.co.kr/xe/mcard_2013_promote01/29877 http://m-card.co.kr]) introducing the author, her name is Sophia. She works as a journey agent but soon she'll be on her own. What me and my family adore is doing ballet but I've been taking on new things recently. Some time in the past she chose to live in Alaska and her parents live nearby. |
| {{ SpecialChars
| |
| | special = [[Unicode]] [[mathematical symbols]]
| |
| | fix = Help:Special_characters
| |
| | characters = [[mathematical symbols]]
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| }}
| |
| The following table lists many specialized [[symbols]] commonly used in [[mathematics]], ordered by their introduction date.
| |
| <br clear="all"/>
| |
| {| class="wikitable sortable" border="1"
| |
| |- bgcolor=#a0e0a0
| |
| ! <div style="font-size:130%;">Symbol</div>
| |
| ! Name
| |
| ! Date of earliest use
| |
| ! First author to use
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">+</div>
| |
| | [[plus and minus signs|plus sign]]
| |
| | ca. 1360 (abbreviation for Latin ''et'' resembling the plus sign)
| |
| | [[Nicole Oresme]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">−</div>
| |
| | [[plus and minus signs|minus sign]]
| |
| | 1489 (first appearance of minus sign, and also first appearance of plus sign in print)
| |
| | [[Johannes Widmann]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">√</div>
| |
| | radical symbol (for [[square root]])
| |
| | 1525 (without the [[vinculum (symbol)|vinculum]] above the [[radicand]])
| |
| | [[Christoff Rudolff]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">(…)</div>
| |
| | rowspan=2| [[Bracket#Parentheses ( )|parentheses]] (for precedence grouping)
| |
| | 1544 (in handwritten notes)
| |
| | [[Michael Stifel]]
| |
| |-
| |
| | 1556
| |
| | [[Nicolo Tartaglia]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">=</div>
| |
| | [[equals sign]]
| |
| | 1557
| |
| | [[Robert Recorde]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">×</div>
| |
| | [[multiplication sign]]
| |
| | 1618
| |
| | rowspan=3| [[William Oughtred]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">±</div>
| |
| | [[plus-minus sign]]
| |
| | rowspan=2| 1628
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∷</div>
| |
| | [[proportion sign]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;"><sup>''n''</sup>√<br /> </div>
| |
| | radical symbol (for [[nth root|''n''th root]])
| |
| | 1629
| |
| | [[Albert Girard]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;"><<br />></div>
| |
| | [[strict inequality]] signs (''less-than sign'' and ''greater-than sign'')
| |
| | 1631
| |
| | [[Thomas Harriot]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">''x<sup>y</sup>''<br /> </div>
| |
| | rowspan=2| [[superscript]] notation (for [[exponentiation]])
| |
| | 1636 (using [[Roman numerals]] as superscripts)
| |
| | [[James Hume (mathematician)|James Hume]]
| |
| |-
| |
| | style=border-bottom:none| 1637 (in the modern form)
| |
| | [[René Descartes]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">√ ̅ </div>
| |
| | radical symbol (for [[square root]])
| |
| | style=border-bottom:none| 1637 (with the [[vinculum (symbol)|vinculum]] above the [[radicand]])
| |
| | [[René Descartes]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">%</div>
| |
| | [[percent sign]]
| |
| | ca. 1650
| |
| | unknown
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">÷</div>
| |
| | ''division sign'' (a.k.a. [[obelus]])
| |
| | 1659
| |
| | [[Johann Rahn]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∞</div>
| |
| | [[infinity]] sign
| |
| | 1655
| |
| | rowspan=2| [[John Wallis]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">≤<br /><br />≥</div>
| |
| | rowspan=2| [[inequality (mathematics)|unstrict inequality signs]] (''less-than or equals to sign'' and ''greater-than or equals to sign'')
| |
| | 1670 (with the horizontal bar over the inequality sign, rather than below it)
| |
| |-
| |
| | 1734 (with double horizontal bar below the inequality sign)
| |
| | [[Pierre Bouguer]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">d</div>
| |
| | [[differential (calculus)|differential]] sign
| |
| | rowspan=2| 1675
| |
| | rowspan=4| [[Gottfried Leibniz]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∫</div>
| |
| | [[integral sign]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">:</div>
| |
| | [[colon (punctuation)|colon]] (for [[division (mathematics)|division]])
| |
| | 1684 (deriving from use of colon to denote fractions, dating back to 1633)
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">·</div>
| |
| | [[middle dot]] (for [[multiplication]])
| |
| | 1698 (perhaps deriving from a much earlier use of middle dot to separate juxtaposed numbers)
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">⁄</div>
| |
| | [[slash (punctuation)|division slash]] (a.k.a. ''solidus'')
| |
| | 1718 (deriving from horizontal fraction bar, invented by Arabs in 12th century)
| |
| | [[Thomas Twining]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">≠</div>
| |
| | [[inequality (mathematics)|inequality]] sign (''not equal to'')
| |
| | unknown
| |
| | rowspan=2| [[Leonhard Euler]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∑</div>
| |
| | [[summation]] symbol
| |
| | 1755
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∝</div>
| |
| | [[proportionality (mathematics)|proportionality]] sign
| |
| | 1768
| |
| | [[William Emerson (mathematician)|William Emerson]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∂</div>
| |
| | [[partial differential]] sign (a.k.a. ''curly d'' or ''[[Carl Gustav Jacob Jacobi|Jacobi]]'s delta'')
| |
| | rowspan=2| 1770
| |
| | [[Marquis de Condorcet]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">''x''′</div>
| |
| | [[prime symbol]] (for [[derivative]])
| |
| | [[Joseph Louis Lagrange]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">≡</div>
| |
| | [[Identity (mathematics)|identity]] sign (for [[congruence relation]])
| |
| | 1801 (first appearance in print; used previously in personal writings of Gauss)
| |
| | rowspan=3| [[Carl Friedrich Gauss]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">[''x'']<br /> </div>
| |
| | ''integral part'' (a.k.a. [[floor and ceiling functions|floor]])
| |
| | 1808
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∏</div>
| |
| | [[multiplication|product]] symbol
| |
| | 1812
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">!</div>
| |
| | [[factorial]]
| |
| | 1808
| |
| | [[Christian Kramp]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">⊂<br />⊃</div>
| |
| | rowspan=2| [[set inclusion]] signs (''subset of'', ''superset of'')
| |
| | 1817
| |
| | [[Joseph Gergonne]]
| |
| |-
| |
| | 1890
| |
| | [[Ernst Schröder]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">|…|</div>
| |
| | [[absolute value]] notation
| |
| | rowspan=2| 1841
| |
| | [[Karl Weierstrass]]
| |
| |-
| |
| | [[determinant]] of a matrix
| |
| | rowspan=2| [[Arthur Cayley]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%; white-space:nowrap;">‖…‖</div>
| |
| | [[matrix (mathematics)|matrices]] notation
| |
| | 1843
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∇</div>
| |
| | [[nabla symbol]] (for [[vector differential]])
| |
| | 1846 (previously used by Hamilton as a general-purpose operator sign)
| |
| | [[William Rowan Hamilton]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∩<br /><br />∪</div>
| |
| | [[intersection (set theory)|intersection]] and [[union (set theory)|union]] signs
| |
| | 1888
| |
| | rowspan=3| [[Giuseppe Peano]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∈</div>
| |
| | [[membership sign]] (''is [[element (mathematics)|an element]] of'')
| |
| | 1894
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∃</div>
| |
| | [[existential quantifier]] (''there exists'')
| |
| | 1897
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">ℵ</div>
| |
| | [[aleph number|aleph]] symbol (for [[cardinal number]]s of [[transfinite]] sets)
| |
| | 1893
| |
| | rowspan=2| [[Georg Cantor]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">{…}</div>
| |
| | braces, a.k.a. [[curly bracket]]s (for [[Set (mathematics)|set]] notation)
| |
| | rowspan=2| 1895
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">ℕ</div>
| |
| | [[Blackboard bold]] capital N (for [[natural number]]s set)
| |
| | [[Giuseppe Peano]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">·</div>
| |
| | [[middle dot]] (for [[dot product]])
| |
| | rowspan=2| 1902
| |
| | rowspan=2| [[J. Willard Gibbs]]?
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">×</div>
| |
| | [[multiplication sign]] (for [[cross product]])
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∨</div>
| |
| | [[logical disjunction]] (a.k.a. ''OR'')
| |
| | 1906
| |
| | [[Bertrand Russell]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">(…)</div>
| |
| | rowspan=2| [[matrix (mathematics)|matrices]] notation
| |
| | 1909
| |
| | [[Gerhard Kowalewski]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">[…]<br /> </div>
| |
| | 1913
| |
| | [[Cuthbert Edmund Cullis]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∮</div>
| |
| | [[line integral|contour integral]] sign
| |
| | 1917
| |
| | [[Arnold Sommerfeld]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center rowspan=2| <div style="font-size:200%;" class="Unicode">ℤ</div>
| |
| | rowspan=2| [[Blackboard bold]] capital Z (for [[integer]] numbers set)
| |
| | 1930
| |
| | [[Edmund Landau]]
| |
| |-
| |
| | style=border-bottom:none| 1930s
| |
| | rowspan=2| Commonly accredited to: [[Nicolas Bourbaki]] {{Citation needed|date=November 2013}} <!-- http://en.wikipedia.org/wiki/Blackboard_bold#Origin mentions that Bourbaki did not actually originate the notation, and in fact did not use it in (some?) published books. The mutual consistency, and veracity, of these two WP articles must be worked out -->
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">ℚ</div>
| |
| | [[Blackboard bold]] capital Q (for [[rational number]]s set)
| |
| | style=border-top:none|
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">∀</div>
| |
| | [[universal quantifier]] (''for all'')
| |
| | 1935
| |
| | [[Gerhard Gentzen]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">∅</div>
| |
| | [[empty set]] sign
| |
| | rowspan=2| 1939
| |
| | [[André Weil]] / [[Nicolas Bourbaki]] {{Citation needed|date=November 2013}} <!-- http://en.wikipedia.org/wiki/Blackboard_bold#Origin mentions that Bourbaki did not actually originate the notation, and in fact did not use it in (some?) published books. The mutual consistency, and veracity, of these two WP articles must be worked out -->
| |
| | |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">ℂ</div>
| |
| | [[Blackboard bold]] capital C (for [[complex number]]s set)
| |
| | [[Nathan Jacobson]]
| |
| |-
| |
| | rowspan=2 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">→</div>
| |
| | rowspan=2| [[arrow]] (for [[function (mathematics)|function]] notation)
| |
| | 1936 (to denote images of specific elements)
| |
| | [[Øystein Ore]]
| |
| |-
| |
| | 1940 (in the present form of f: X → Y)
| |
| | [[Witold Hurewicz]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">⌊''x''⌋<br /> </div>
| |
| | ''integral part'' (a.k.a. [[floor and ceiling functions|floor]])
| |
| | 1962
| |
| | [[Kenneth E. Iverson]]
| |
| |-
| |
| | bgcolor=#d0f0d0 align=center| <div style="font-size:200%;" class="Unicode">∎</div>
| |
| | [[end of proof]] sign (a.k.a. [[tombstone (typography)|tombstone]])
| |
| | unknown
| |
| | [[Paul Halmos]]
| |
| | |
| <!--
| |
| |-
| |
| | rowspan=6 bgcolor=#d0f0d0 align=center|<br /><div style="font-size:200%;">⊕</div> <br><br><div style="font-size:200%;">⊻</div> ||[[exclusive or]]
| |
| | rowspan=3| The statement ''A'' ⊕ ''B'' is true when either A or B, but not both, are true. ''A'' ⊻ ''B'' means the same.
| |
| | rowspan=3| (¬''A'') ⊕ ''A'' is always true, ''A'' ⊕ ''A'' is always false.
| |
| |-
| |
| |align=center|xor
| |
| |-
| |
| |align=right|[[propositional logic]], [[Boolean algebra]]
| |
| |-
| |
| ||[[Direct_sum_of_modules|direct sum]]
| |
| |rowspan=3|The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).<br><br>
| |
| |rowspan=3|Most commonly, for vector spaces ''U'', ''V'', and ''W'', the following consequence is used:<br> ''U'' = ''V'' ⊕ ''W'' ⇔ (''U'' = ''V'' + ''W'') ∧ (''V'' ∩ ''W'' = {{0/}})
| |
| |-
| |
| |align=center|direct sum of
| |
| |-
| |
| |align=right|[[Abstract_algebra|Abstract algebra]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∃!</div>
| |
| ||[[uniqueness quantification]]
| |
| | rowspan=3|∃! ''x'': ''P''(''x'') means there is exactly one ''x'' such that ''P''(''x'') is true.
| |
| | rowspan=3|∃! ''n'' ∈ ℕ: ''n'' + 5 = 2''n''.
| |
| |-
| |
| |align=center|there exists exactly one
| |
| |-
| |
| |align=right|[[predicate logic]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">:=<br><br> ≡<br><br>:⇔</div>
| |
| ||[[definition]]
| |
| | rowspan=3|''x'' := ''y'' or ''x'' ≡ ''y'' means ''x'' is defined to be another name for ''y''<br><br>(''Some writers use'' ≡ ''to mean [[congruence]]'').<br><br> ''P'' :⇔ ''Q'' means ''P'' is defined to be logically equivalent to ''Q''.
| |
| | rowspan=3|cosh ''x'' := (1/2)(exp ''x'' + exp (−''x''))<br><br> ''A'' '''xor''' ''B'' :⇔ (''A'' ∨ ''B'') ∧ ¬(''A'' ∧ ''B'')
| |
| |-
| |
| |align=center|is defined as
| |
| |-
| |
| |align=right|everywhere
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">{ , }</div>
| |
| ||[[Set (mathematics)|set]] brackets
| |
| | rowspan=3|{''a'',''b'',''c''} means the set consisting of ''a'', ''b'', and ''c''.
| |
| | rowspan=3|ℕ = { 1, 2, 3, …}
| |
| |-
| |
| |align=center|the set of …
| |
| |-
| |
| |align=right|[[naive set theory|set theory]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> <math>\Delta</math> </div>
| |
| ||[[symmetric difference]]
| |
| | rowspan=3|<math> A\Delta B</math> means the set of elements in exactly one of ''A'' or ''B''.
| |
| | rowspan=3|{1,5,6,8} <math>\Delta</math> {2,5,8} = {1,2,6}
| |
| |-
| |
| |align=center|symmetric difference
| |
| |-
| |
| |align=right|[[naive set theory|set theory]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∖</div>
| |
| ||[[complement (set theory)|set-theoretic complement]]
| |
| | rowspan=3|''A'' ∖ ''B'' means the set that contains all those elements of ''A'' that are not in ''B''.
| |
| | rowspan=3|{1,2,3,4} ∖ {3,4,5,6} = {1,2}
| |
| |-
| |
| |align=center|minus; without
| |
| |-
| |
| |align=right|[[naive set theory|set theory]]
| |
| |-
| |
| | rowspan=6 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">( )</div>
| |
| ||[[function (mathematics)|function]] application
| |
| | rowspan=3|''f''(''x'') means the value of the function ''f'' at the element ''x''.
| |
| | rowspan=3|If ''f''(''x'') := ''x''<sup>2</sup>, then ''f''(3) = 3<sup>2</sup> = 9.
| |
| |-
| |
| |align=center|of
| |
| |-
| |
| |align=right|[[naive set theory|set theory]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"><small>o</small></div>
| |
| ||[[function composition]]
| |
| | rowspan=3|''f''<small>o</small>''g'' is the function, such that (''f''<small>o</small>''g'')(''x'') = ''f''(''g''(''x'')).
| |
| | rowspan=3|if ''f''(''x'') := 2''x'', and ''g''(''x'') := ''x'' + 3, then (''f''<small>o</small>''g'')(''x'') = 2(''x'' + 3).
| |
| |-
| |
| |align=center|composed with
| |
| |-
| |
| |align=right|[[naive set theory|set theory]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> '''''π'''''</div>||pi
| |
| | rowspan=3|'''''π''''' is the ratio of a [[circle]]'s circumference to its diameter. Its value is 3.14159265... .
| |
| | rowspan=3|''A'' = '''''π''''' ''r''² is the area of a circle with radius ''r''<br><br> '''''π''''' [[radian]]s = [[degree (angle)|180°]]<br><br> '''''π''''' ≈ 22 / 7
| |
| |-
| |
| |align=center|pi
| |
| |-
| |
| |align=right|[[Euclidean geometry]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"><nowiki>||</nowiki>…<nowiki>||</nowiki></div>
| |
| ||[[normed vector space|norm]]
| |
| | rowspan=3| <nowiki>||</nowiki> ''x'' <nowiki>||</nowiki> is the [[norm (mathematics)|norm]] of the element ''x'' of a [[normed vector space]].
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| | rowspan=3| <nowiki>||</nowiki> ''x'' + ''y'' <nowiki>||</nowiki> ≤ <nowiki>||</nowiki> ''x'' <nowiki>||</nowiki> + <nowiki>||</nowiki> ''y'' <nowiki>||</nowiki>
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| |-
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| |align=center|norm of<br><br> length of
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| |-
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| |align=right| [[linear algebra]]
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| |-
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| ||[[Cartesian product]]
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| | rowspan=3|
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| <math>\prod_{i=0}^{n}{Y_i}</math> means the set of all [[n-tuple|(n+1)-tuples]]
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| ::(''y''<sub>0</sub>, …, ''y''<sub>''n''</sub>).
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| | rowspan=3|
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| <math>\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3</math>
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| |-
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| |align=center|the Cartesian product of; the direct product of
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| |-
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| |align=right|[[naive set theory|set theory]]
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∐</div>
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| ||[[coproduct]]
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| | rowspan=3|
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| | rowspan=3|
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| |-
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| |align=center|coproduct over … from … to … of
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| |-
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| |align=right|[[category theory]]
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| |-
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| |[[Boundary (topology)|boundary]]
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| | rowspan=3| ∂''M'' means the boundary of ''M''
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| | rowspan=3| ∂{x : <nowiki>||</nowiki>x<nowiki>||</nowiki> ≤ 2} = {x : <nowiki>||</nowiki>x<nowiki>||</nowiki> = 2}
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| |-
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| |align=center|boundary of
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| |-
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| |align=right|[[topology]]
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| |-
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| | rowspan=6 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">⊥</div>
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| ||[[perpendicular]]
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| | rowspan=3|''x'' ⊥ ''y'' means ''x'' is perpendicular to ''y''; or more generally ''x'' is orthogonal to ''y''.
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| | rowspan=3|If ''l'' ⊥ ''m'' and ''m'' ⊥ ''n'' then ''l'' <nowiki>||</nowiki> ''n''.
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| |-
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| |align=center|is perpendicular to
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| |-
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| |align=right|[[geometry]]
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| |-
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| ||[[bottom element]]
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| | rowspan=3|''x'' = ⊥ means ''x'' is the smallest element.
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| | rowspan=3|∀''x'' : ''x'' ∧ ⊥ = ⊥
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| |-
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| |align=center|the bottom element
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| |-
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| |align=right|[[Lattice (order)|lattice theory]]
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"><nowiki>||</nowiki></div>
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| ||[[parallel (geometry)|parallel]]
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| | rowspan=3|''x'' <nowiki>||</nowiki> ''y'' means ''x'' is parallel to ''y''.
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| | rowspan=3|If ''l'' <nowiki>||</nowiki> ''m'' and ''m'' ⊥ ''n'' then ''l'' ⊥ ''n''.
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| |-
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| |align=center|is parallel to
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| |-
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| |align=right|[[geometry]]
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">⊧</div>
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| ||[[entailment]]
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| | rowspan=3| ''A'' ⊧ ''B'' means the sentence ''A'' entails the sentence ''B'', that is every model in which ''A'' is true, ''B'' is also true.
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| | rowspan=3| ''A'' ⊧ ''A'' ∨ ¬''A''
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| |-
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| |align=center|entails
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| |-
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| |align=right| [[model theory]]
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">⊢</div>
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| ||[[inference]]
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| | rowspan=3|''x'' ⊢ ''y'' means ''y'' is derived from ''x''.
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| | rowspan=3| ''A'' → ''B'' ⊢ ¬''B'' → ¬''A''
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| |-
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| |align=center|infers or is derived from
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| |-
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| |align=right|[[propositional logic]], [[predicate logic]]
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;"> ◅ </div>
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| ||[[normal subgroup]]
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| | rowspan=3| ''N'' ◅ ''G'' means that ''N'' is a normal subgroup of group ''G''.
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| | rowspan=3| ''Z''(''G'') ◅ ''G''
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| |-
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| |align=center|is a normal subgroup of
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| |-
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| |align=right|[[group theory]]
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| |-
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| | rowspan=6 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> / </div>
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| ||[[quotient group]]
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| | rowspan=3| ''G''/''H'' means the quotient of group ''G'' [[modulo]] its subgroup ''H''.
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| | rowspan=3| {0, ''a'', 2''a'', ''b'', ''b''+''a'', ''b''+2''a''} / {0, ''b''} = {{0, ''b''}, {''a'', ''b''+''a''}, {2''a'', ''b''+2''a''}}
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| |-
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| |align=center| mod
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| |-
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| |align=right| [[group theory]]
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| |-
| |
| |quotient set
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| | rowspan=3| ''A''/~ means the set of all ~ [[equivalence class]]es in ''A''.
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| |-
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| |align=center|
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| |-
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| |align=right| [[set theory]]
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| |-
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| | rowspan=6 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> ≈ </div>
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| ||[[isomorphism]]
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| | rowspan=3| ''G'' ≈ ''H'' means that group ''G'' is isomorphic to group ''H''
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| | rowspan=3| ''Q'' / {1, −1} ≈ ''V'', <br />where ''Q'' is the [[quaternion group]] and ''V'' is the [[Klein four-group]].
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| |-
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| |align=center | is isomorphic to
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| |-
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| |align=right| [[group theory]]
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| |-
| |
| |approximately equal
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| | rowspan=3|''x'' ≈ ''y'' means ''x'' is approximately equal to ''y''
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| | rowspan=3|π ≈ 3.14159
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| |-
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| |align=center|is approximately equal to
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| |-
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| |align=right|everywhere
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| |-
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| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">~</div>
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| ||same [[order of magnitude]]
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| | rowspan=3| ''m'' ~ ''n'', means the quantities ''m'' and ''n'' have the [[order of magnitude|general size]]. <br><br>(''Note that'' ~ ''is used for an approximation that is poor, otherwise use '' ≈ .)
| |
| | rowspan=3|2 ~ 5<br><br> 8 × 9 ~ 100<br><br> but π<sup>2</sup> ≈ 10
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| |-
| |
| |align=right|roughly similar<br><br> [[approximation|poorly approximates]]
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| |-
| |
| |align=right|[[Approximation theory]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">〈,〉<br/><br/>( | )<br/><br/>·<br/><br/>:</div>
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| ||[[Inner_product_space|inner product]]
| |
| | rowspan=3|〈''x'',''y''〉 means the inner product of ''x'' and ''y'' as defined in an [[Inner_product_space|inner product space]]. <br/>
| |
| For spatial vectors, the [[dot product]] notation, ''x''·''y'' is common. <br/>
| |
| For matricies, the colon notation may be used.
| |
| | rowspan=3|The [[Dot product|standard inner product]] between two vectors ''x'' = (2, 3) and ''y'' = (−1, 5) is:<br />〈x, y〉 = 2×−1 + 3×5 = 13<br/>
| |
| <math>A:B = \sum_{i,j} A_{ij}B_{ij}</math>
| |
| |-
| |
| |align=center|inner product of
| |
| |-
| |
| |align=right|[[Vector_algebra|vector algebra ]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> ⊗ </div>
| |
| ||[[tensor product]]
| |
| | rowspan=3| ''V'' ⊗ ''U'' means the tensor product of ''V'' and ''U''.
| |
| | rowspan=3| {1, 2, 3, 4} ⊗ {1,1,2} = <br /> {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
| |
| |-
| |
| |align=center| tensor product of
| |
| |-
| |
| |align=right| [[linear algebra]]
| |
| |-
| |
| | rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;"> * </div>
| |
| ||[[convolution]]
| |
| | rowspan=3| ''f'' * ''g'' means the convolution of ''f'' and ''g''.
| |
| | rowspan=3| <math>(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau</math>
| |
| |-
| |
| |align=center| convolution
| |
| |-
| |
| |align=right|
| |
| -->
| |
| |}
| |
| | |
| ==See also==
| |
| * [[History of mathematical notation]]
| |
| * [[History of the Hindu-Arabic numeral system]]
| |
| * [[Table of mathematical symbols]]
| |
| | |
| ==Sources==
| |
| * [http://jeff560.tripod.com/mathsym.html Jeff Miller: ''Earliest Uses of Various Mathematical Symbols'']
| |
| | |
| [[Category:Mathematical notation|*]]
| |
| [[Category:Mathematics-related lists|Symbols by introduction date]]
| |
| [[Category:Mathematical symbols| ]]
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| [[Category:Mathematics timelines]]
| |