1430:
1113:
1517:, then at least one container must contain more than one item. For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into.
1927:
is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. Uniformly distributed digits would each occur about 11.1% of the
1333:
1016:
1326:
is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:
1285:
436:
1934:: (Humerous) Any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few.
1667:
1736:
1324:
893:
694:
253:
1148:
195:
1859:
with", often read as "if and only if". For any combination of true/false values for P and Q, the left and right sides of the arrow will hold the same truth value after evaluation.
1425:{\displaystyle \left\vert \langle {\mathbf {u}},{\mathbf {v}}\rangle \right\vert \leq \langle {\mathbf {u}},{\mathbf {u}}\rangle \cdot \langle {\mathbf {v}},{\mathbf {v}}\rangle }
1108:{\displaystyle \left\vert \langle {\mathbf {u}},{\mathbf {v}}\rangle \right\vert \leq \langle {\mathbf {u}},{\mathbf {u}}\rangle \cdot \langle {\mathbf {v}},{\mathbf {v}}\rangle }
591:
490:
753:
1477:
1455:
1225:
1005:
983:
105:
805:
351:
931:
831:
776:
718:
1798:
1848:
1822:
1774:
1187:
2270:
Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
1233:
2325:
374:
624:
These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the
2008:
Pratt, Vaughan, "Algebra", The
Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL:
1602:
1674:
1293:
836:
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity
839:
2394:
2368:
2293:
2104:
2544:
638:
621:, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
2459:
200:
2529:
2184:
2156:
2565:
2513:
2343:
2263:
2238:
2211:
2163:...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
2146:
1121:
949:
604:
1954:
123:
941:
2575:
2021:
1931:
540:
1910:
1159:
629:
2417:
933:), which can be used to break down expressions of larger angles into those with smaller constituents.
466:
2064:
726:
1906:
1880:
957:
67:
55:
39:
1460:
1438:
1208:
988:
966:
364:
and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (
2069:
77:
781:
321:
1887:
1531:
898:
810:
618:
610:
117:
63:
59:
47:
1197:
2092:
2077:
761:
703:
2300:
This inequality is an equality if and only if one of u, v is a scalar multiple of the other.
1783:
1949:
1886:
The law of excluded middle: 'Everything must either be or not be.' In accordance with the
1831:
1807:
1759:
1492:
1165:
20:
8:
2058:
1891:
1890:
or excluded third, for every proposition, either its positive or negative form is true: A
1804:
1780:
1562:
1558:
1527:
1480:
1008:
945:
500:
289:
43:
35:
2148:
The Cauchy–Schwarz Master Class: an
Introduction to the Art of Mathematical Inequalities
1554:
1189:
953:
365:
361:
71:
1280:{\displaystyle \|\mathbf {u} \|:={\sqrt {\langle \mathbf {u} ,\mathbf {u} \rangle }},}
2570:
2525:
2509:
2400:
2390:
2364:
2339:
2289:
2259:
2234:
2207:
2180:
2152:
2100:
1964:
1550:
635:
One of the most prominent examples of trigonometric identities involves the equation
441:
where the length of the third side has been replaced by the length of the vector sum
268:
431:{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,}
2501:
2331:
2009:
1916:
1566:
1547:
628:
of non-trigonometric functions: a common technique which involves first using the
2283:
2228:
2201:
1873:
1592:
1535:
2473:
2121:
2460:
Benford's Law
Strikes Back: No Simple Explanation in Sight for Mathematical Gem
1867:
1828:
493:
256:
2550:
515:) is equal to the sum of the areas of the squares on the other two sides. The
2559:
2404:
1924:
1151:
632:, and then simplifying the resulting integral with a trigonometric identity.
2033:
1959:
1883:(alternately the 'law of contradiction'): 'Nothing can both be and not be.'
1856:
1584:
1576:
51:
2384:
2335:
1557:, a 19th-century British mathematician. The rules allow the expression of
1985:
1749:
1155:
697:
512:
260:
108:
27:
508:
264:
2227:
Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06).
2022:
http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
1852:
1756:
625:
520:
311:
2179:(4th ed.). Stamford, CT: Cengage Learning. pp. 154–155.
2097:
Roots to research: a vertical development of mathematical problems
356:
with equality only in the degenerate case of a triangle with zero
1944:
516:
272:
952:
in an inner product space in terms of the product of the vector
2436:
2434:
2432:
504:
492:, and the triangle inequality expresses a relationship between
2256:
Mathematical
Physics: A Modern Introduction to Its Foundations
2443:
956:. It is considered one of the most important and widely used
614:
2429:
1920:
357:
116:, and is therefore a law. Laws over an equality are called
1919:: An observation that in many real-life sets of numerical
1662:{\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),}
1575:
not (A and B) = (not A) or (not B) where "A or B" is an "
1154:. Examples of inner products include the real and complex
963:
The Cauchy–Schwarz inequality states that for all vectors
613:
are identities involving certain functions of one or more
2547:
Online encyclopedia of mathematical identities (archived)
2422:
2389:. Richard Parker (10th ed.). New York: McGraw-Hill.
2226:
2090:
1731:{\displaystyle \neg (P\land Q)\iff (\neg P)\lor (\neg Q)}
1319:{\displaystyle \langle \mathbf {u} ,\mathbf {u} \rangle }
255:
are identities. Mathematical laws are distinguished from
888:{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta }
2324:
Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016).
2099:. American Mathematical Society Bookstore. p. 63.
2056:
2151:. The Mathematical Association of America. p. 1.
2233:. Springer Science & Business Media. p. 14.
2020:
1834:
1810:
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813:
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729:
706:
641:
543:
469:
377:
324:
203:
126:
80:
689:{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
1546:, are a pair of transformation rules that are both
778:, not all. For example, this equation is true when
463:are real numbers, they can be viewed as vectors in
248:{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
70:, or between formulas themselves, for instance, in
2288:. Springer International Publishing. p. 172.
2199:
2057:
1842:
1816:
1792:
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1471:
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688:
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430:
345:
247:
189:
99:
2508:(1912), Oxford University Press, New York, 1997,
1435:Moreover, the two sides are equal if and only if
2557:
2323:
2010:https://plato.stanford.edu/entries/algebra/#Laws
1162:. Every inner product gives rise to a Euclidean
16:A mathematical statement which always holds true
630:substitution rule with a trigonometric function
2471:
2452:
2200:Hunter, John K.; Nachtergaele, Bruno (2001).
1143:{\displaystyle \langle \cdot ,\cdot \rangle }
535:, sometimes called the Pythagorean equation:
271:. The more significant laws are often called
1569:. The rules can be expressed in English as:
1419:
1399:
1393:
1373:
1362:
1342:
1313:
1297:
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1245:
1237:
1137:
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2305:
1839:
1835:
1700:
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1628:
1624:
2524:, Waltham: Blaisdell Publishing Company,
472:
190:{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}
2519:
2311:
1900:
1801:is the conjunction logic operator (AND),
314:then the triangle inequality states that
2253:
2175:Strang, Gilbert (19 July 2005). "3.2".
1825:is the disjunction logic operator (OR),
2558:
2358:
2174:
2168:
2144:
1876:: 'Whatever is, is.' For all a: a = a.
2382:
2281:
2138:
2084:
1983:
1855:symbol meaning "can be replaced in a
1777:is the negation logic operator (NOT),
1486:
2551:A Collection of Algebraic Identities
2091:Judith D. Sally; Paul Sally (2007).
2078:participating institution membership
2363:(12th ed.), Cengage Learning,
2177:Linear Algebra and its Applications
758:is only true for certain values of
278:
13:
2458:Arno Berger and Theodore P. Hill,
2193:
2119:
1763:
1719:
1704:
1678:
1647:
1632:
1606:
1572:not (A or B) = (not A) and (not B)
1565:purely in terms of each other via
720:. On the other hand, the equation
586:{\displaystyle a^{2}+b^{2}=c^{2}.}
523:relating the lengths of the sides
310:are the lengths of the sides of a
14:
2587:
2538:
2478:MathWorld, A Wolfram web resource
2285:Linear Algebra Done Right, 3rd Ed
936:
503:: It states that the area of the
283:
2093:"Chapter 3: Pythagorean triples"
1465:
1443:
1414:
1404:
1388:
1378:
1357:
1347:
1309:
1301:
1265:
1257:
1241:
1213:
1097:
1087:
1071:
1061:
1040:
1030:
993:
971:
605:List of trigonometric identities
485:{\displaystyle \mathbb {R} ^{1}}
418:
404:
390:
382:
2465:
2411:
2376:
2361:A Concise Introduction to Logic
2352:
2317:
2275:
2247:
2220:
1955:List of mathematical identities
1521:
2449:Russell 1912:72, 1997 edition.
2113:
2050:
2026:
2014:
2002:
1977:
1836:
1725:
1716:
1710:
1701:
1697:
1693:
1681:
1653:
1644:
1638:
1629:
1625:
1621:
1609:
1583:one of A or B rather than an "
920:
908:
858:
849:
748:{\displaystyle \cos \theta =1}
140:
127:
1:
2495:
2440:Russell 1912:72,1997 edition.
1205:, where the norm of a vector
2545:The Encyclopedia of Equation
2230:Fourier and Wavelet Analysis
1970:
1472:{\displaystyle \mathbf {v} }
1450:{\displaystyle \mathbf {u} }
1220:{\displaystyle \mathbf {u} }
1000:{\displaystyle \mathbf {v} }
978:{\displaystyle \mathbf {u} }
7:
2383:Moore, Brooke Noel (2012).
2359:Hurley, Patrick J. (2015),
2145:Steele, J. Michael (2004).
1938:
1932:Strong law of small numbers
895:, the addition formula for
100:{\displaystyle a^{2}\geq 0}
74:. For example, the formula
10:
2592:
2506:The Problems of Philosophy
2122:"Trigonometric Identities"
1911:Empirical statistical laws
1904:
1595:, the rules are written as
1227:is denoted and defined by
800:{\displaystyle \theta =0,}
602:
346:{\displaystyle c\leq a+b,}
18:
2065:Oxford English Dictionary
1160:examples in inner product
942:Cauchy–Schwarz inequality
926:{\displaystyle \tan(x+y)}
826:{\displaystyle \theta =2}
617:. They are distinct from
263:, and try to describe or
2566:Mathematical terminology
2520:Herstein, I. N. (1964),
2258:. Springer. p. 29.
1907:Mathematical coincidence
1881:law of non-contradiction
944:: An upper bound on the
611:trigonometric identities
599:Trigonometric identities
2282:Axler, Sheldon (2015).
2254:Hassani, Sadri (1999).
2070:Oxford University Press
1553:. They are named after
771:{\displaystyle \theta }
713:{\displaystyle \theta }
511:(the side opposite the
1888:law of excluded middle
1844:
1818:
1794:
1793:{\displaystyle \land }
1770:
1732:
1663:
1473:
1451:
1426:
1320:
1281:
1221:
1183:
1144:
1109:
1001:
979:
927:
889:
827:
801:
772:
749:
714:
696:which is true for all
690:
587:
486:
432:
347:
249:
191:
101:
50:, between two or more
2336:10.4324/9781315510897
2327:Introduction to Logic
2034:"Mathwords: Identity"
1990:mathworld.wolfram.com
1901:Phinominological laws
1845:
1843:{\displaystyle \iff }
1819:
1817:{\displaystyle \lor }
1795:
1771:
1769:{\displaystyle \neg }
1733:
1664:
1474:
1452:
1427:
1321:
1282:
1222:
1184:
1182:{\displaystyle l_{2}}
1145:
1110:
1002:
980:
928:
890:
828:
802:
773:
750:
715:
691:
588:
519:can be written as an
487:
433:
348:
250:
192:
102:
2206:. World Scientific.
1950:List of inequalities
1832:
1808:
1784:
1760:
1675:
1603:
1493:Pigeonhole principle
1461:
1439:
1334:
1294:
1234:
1209:
1166:
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1017:
989:
967:
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811:
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704:
639:
541:
467:
375:
322:
201:
124:
78:
21:Law (disambiguation)
19:For other uses, see
2472:Weisstein, Eric W.
2120:Stapel, Elizabeth.
2068:(Online ed.).
1984:Weisstein, Eric W.
1544:De Morgan's theorem
1532:propositional logic
1501:items are put into
1009:inner product space
619:triangle identities
531:and the hypotenuse
501:Pythagorean theorem
290:Triangle inequality
259:which are based on
58:(which may contain
2576:Mathematical logic
2418:DeMorgan's Theorem
1840:
1814:
1790:
1766:
1728:
1659:
1591:one of A or B. In
1555:Augustus De Morgan
1551:rules of inference
1487:Combinatorial laws
1481:linearly dependent
1469:
1447:
1422:
1316:
1277:
1217:
1179:
1140:
1105:
997:
975:
923:
885:
823:
797:
768:
745:
710:
686:
583:
507:whose side is the
482:
428:
362:Euclidean geometry
343:
245:
187:
97:
72:mathematical logic
46:. Laws describe a
2522:Topics In Algebra
2396:978-0-07-803828-0
2386:Critical thinking
2370:978-1-285-19654-1
2295:978-3-319-11079-0
2106:978-0-8218-4403-8
2076:(Subscription or
2038:www.mathwords.com
1965:Statement (logic)
1507:containers, with
1272:
269:natural phenomena
62:), usually using
2583:
2534:
2502:Bertrand Russell
2489:
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2203:Applied Analysis
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1660:
1542:, also known as
1540:De Morgan's laws
1528:De Morgan's laws
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960:in mathematics.
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279:Notable Examples
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107:is true for all
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2480:
2474:"Benford's Law"
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2457:
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2159:
2143:
2139:
2130:
2128:
2118:
2114:
2107:
2089:
2085:
2075:
2059:"law of nature"
2055:
2051:
2042:
2040:
2032:
2031:
2027:
2019:
2015:
2007:
2003:
1994:
1992:
1982:
1978:
1973:
1941:
1913:
1903:
1874:law of identity
1868:Laws of thought
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1761:
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1593:formal language
1536:Boolean algebra
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841:
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837:
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808:
807:but false when
783:
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759:
728:
725:
724:
705:
702:
701:
665:
661:
646:
642:
640:
637:
636:
609:Geometrically,
607:
601:
574:
570:
561:
557:
548:
544:
542:
539:
538:
494:absolute values
476:
471:
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417:
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125:
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85:
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76:
75:
24:
17:
12:
11:
5:
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2539:External links
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995:
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937:Algebraic laws
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284:Geometric laws
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2312:Herstein 1964
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1925:leading digit
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1917:Benford's law
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1857:logical proof
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1587:" that means
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1201:
1198:
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1192:, called the
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1157:
1153:
1152:inner product
1134:
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1128:
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1021:
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947:
946:inner product
943:
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911:
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902:
882:
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723:
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662:
658:
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647:
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606:
580:
575:
571:
567:
562:
558:
554:
549:
545:
537:
536:
534:
530:
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522:
518:
514:
510:
506:
502:
499:
498:
497:
495:
477:
461:
455:
449:
445:
425:
411:
397:
386:
371:
370:
369:
367:
363:
359:
340:
337:
334:
331:
328:
325:
318:
317:
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309:
303:
297:
291:
288:
287:
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270:
266:
262:
258:
242:
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236:
233:
228:
224:
220:
217:
214:
209:
205:
182:
178:
174:
171:
168:
165:
162:
157:
153:
149:
144:
136:
133:
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119:
115:
110:
94:
91:
86:
82:
73:
69:
65:
61:
57:
53:
49:
45:
44:given context
41:
37:
33:
29:
22:
2521:
2505:
2481:. Retrieved
2477:
2467:
2454:
2445:
2421:
2413:
2385:
2378:
2360:
2354:
2326:
2319:
2314:, p. 90
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2255:
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2229:
2222:
2202:
2195:
2176:
2170:
2162:
2147:
2140:
2129:. Retrieved
2125:
2115:
2096:
2086:
2063:
2052:
2041:. Retrieved
2037:
2028:
2016:
2004:
1993:. Retrieved
1989:
1979:
1960:List of laws
1750:propositions
1747:
1743:
1588:
1585:exclusive or
1580:
1577:inclusive or
1563:disjunctions
1559:conjunctions
1543:
1539:
1522:Logical laws
1513:
1509:
1502:
1496:
1434:
1289:
1117:
962:
958:inequalities
948:between two
940:
835:
757:
634:
623:
608:
532:
528:
524:
459:
453:
447:
443:
440:
355:
305:
299:
293:
261:observations
111:
109:real numbers
48:relationship
31:
25:
1853:metalogical
1156:dot product
626:integration
513:right angle
267:a range of
118:indentities
56:expressions
40:always true
28:mathematics
2560:Categories
2496:References
2420: [
2131:2019-12-01
2126:Purplemath
2080:required.)
2043:2019-12-01
1995:2024-08-19
1905:See also:
1866:The three
1579:" meaning
1158:; see the
700:values of
509:hypotenuse
68:inequality
2405:689858599
1971:Citations
1837:⟺
1812:∨
1788:∧
1764:¬
1720:¬
1714:∨
1705:¬
1698:⟺
1688:∧
1679:¬
1648:¬
1642:∧
1633:¬
1626:⟺
1616:∨
1607:¬
1420:⟩
1400:⟨
1397:⋅
1394:⟩
1374:⟨
1371:≤
1363:⟩
1343:⟨
1314:⟩
1298:⟨
1270:⟩
1254:⟨
1246:‖
1238:‖
1194:canonical
1138:⟩
1135:⋅
1129:⋅
1126:⟨
1103:⟩
1083:⟨
1080:⋅
1077:⟩
1057:⟨
1054:≤
1046:⟩
1026:⟨
906:
883:θ
880:
874:θ
871:
856:θ
847:
815:θ
786:θ
766:θ
737:θ
734:
708:θ
675:θ
672:
656:θ
653:
423:‖
415:‖
409:‖
401:‖
398:≤
395:‖
379:‖
329:≤
237:θ
234:
218:θ
215:
92:≥
60:variables
42:within a
2571:Theorems
1939:See also
1581:at least
1567:negation
521:equation
312:triangle
273:theorems
64:equality
38:that is
2462:, 2011.
1945:Formula
1589:exactly
1199:induced
1150:is the
950:vectors
517:theorem
451:. When
265:predict
36:formula
2528:
2512:
2483:7 June
2403:
2393:
2367:
2342:
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1923:, the
1748:Q are
1530:: In
1290:where
1118:where
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615:angles
505:square
304:, and
2426:]
2074:
1986:"Law"
1928:time.
1851:is a
1738:where
1548:valid
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1495:: If
954:norms
366:norms
360:. In
292:: If
52:terms
34:is a
2526:ISBN
2510:ISBN
2485:2015
2401:OCLC
2391:ISBN
2365:ISBN
2340:ISBN
2290:ISBN
2260:ISBN
2235:ISBN
2208:ISBN
2181:ISBN
2153:ISBN
2101:ISBN
1921:data
1909:and
1879:The
1872:The
1746:and
1561:and
1534:and
1479:are
1457:and
1202:norm
1190:norm
985:and
698:real
457:and
358:area
197:and
30:, a
2423:sic
2332:doi
1894:¬A.
1669:and
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877:cos
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663:cos
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54:or
32:law
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2476:.
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