43:
100:
295:
470:(that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers.
465:
such that every non-negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in
1031:, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by
1022:
427:
is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the
1212:
The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is a
625:, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle.
1378:
1338:
1129:, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.
1127:
1073:
1180:
767:
1599:
Marshall, Murray
Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp.
690:, and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly
793:
154:
2; for instance, the square of 3 may be written as 3, which is the number 9. In some cases when superscripts are not available, as for instance in
1220:. Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of
1604:
359:. On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on
541:
916:
64:
1474:
1515:
1673:
1631:
1612:
86:
1493:
1442:
1404:
351:
The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a
253:
One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers
1090:
1483:
1663:
1380:. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the
509: times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a
800:
432:
function, which associates with a non-negative real number the non-negative number whose square is the original number.
717:
More generally, in rings, the square function may have different properties that are sometimes used to classify rings.
57:
51:
1623:
1393:
222:
1678:
1469:
31:
68:
892:
218:
840:. A commutative ring in which every element is equal to its square (every element is idempotent) is called a
443:. The lack of real square roots for the negative numbers can be used to expand the real number system to the
1538:
1343:
1520:
1503:
1417:
1266:
1668:
1280:
1221:
907:
833:
810:. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in
1257:
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in
221:, or values in systems of mathematical values other than the numbers. For instance, the square of the
1316:
1217:
860:
337:
232:
155:
1408:
1262:
1240:
1105:
1051:
707:
634:
525:
1547:(has square charge in the denominator, and may be expressed with square distance in the numerator)
1110:
1056:
457:
The property "every non-negative real number is a square" has been generalized to the notion of a
1432:
1295:
1291:
815:
796:
638:
622:
533:
514:
352:
1152:
739:
548:, with a non-smooth point at the tip of the cone. However, the square of the distance (denoted
1627:
1608:
1600:
1544:
1412:
1206:
853:
676:
611:
565:
529:
467:
458:
517:
describing how the strength of physical forces such as gravity varies according to distance.
1658:
1637:
1458:
1209:, and equals the sum of the squares of the real and imaginary parts of the complex number.
845:
772:
721:
576:
345:
262:
1641:
1556:
1437:
1299:
1214:
1183:
811:
537:
142:
by itself. The verb "to square" is used to denote this operation. Squaring is the same as
513:
is proportional to the square of its radius, a fact that is manifested physically by the
1550:
1422:
1398:
1236:
1190:
1146:
1140:
1076:
1028:
603:
591:
448:
444:
143:
135:
1401:, the representation of a non-negative polynomial as the sum of squares of polynomials
1089:, and then doubling again to obtain quaternions. The doubling procedure is called the
1652:
1427:
1276:
1032:
849:
729:
711:
599:
462:
410:
317:
283:
201:
1578:
416:. This implies that the square of an integer is never less than the original number
1532:
1245:
1225:
1044:
841:
824:
725:
669:
655:
595:
440:
1048:
647:
572:
436:
429:
424:
151:
127:
610:
is an example of a quadratic form. It demonstrates a quadratic relation of the
1487:
1287:
1186:
in the sense that each non-zero complex number has exactly two square roots.
1040:
807:
710:
under multiplication. The properties of quadratic residues are widely used in
561:
481:
The name of the square function shows its importance in the definition of the
214:
188:
159:
147:
17:
607:
341:
856:
as the multiplication operation and bitwise XOR as the addition operation.
99:
1497:
1381:
1229:
1036:
521:
309:
1093:, and has been generalized to form algebras of dimension 2 over a field
1258:
368:
210:
196:
435:
No square root can be taken of a negative number within the system of
294:
282:. This can also be expressed by saying that the square function is an
724:
such that the square of a non zero element is never zero is called a
615:
510:
486:
139:
111:
1017:{\displaystyle \forall x,y\in A\quad (xy)^{2}=xyxy=xxyy=x^{2}y^{2}.}
1622:. London Mathematical Society Lecture Note Series. Vol. 171.
806:
An element of a ring that is equal to its own square is called an
499:. The area depends quadratically on the size: the area of a shape
170:
98:
478:
There are several major uses of the square function in geometry.
1310:
545:
482:
364:
36:
654:
The notion of squaring is particularly important in the
30:"²" redirects here. For typography of superscripts, see
641:. An element in the image of this function is called a
271:. That is, the square function satisfies the identity
706:
quadratic non-residues. The quadratic residues form a
1346:
1319:
1155:
1113:
1059:
919:
775:
742:
454:, which is one of the square roots of −1.
1205:. It is the product of the complex number with its
720:
Zero may be the square of some non-zero elements. A
213:, the operation of squaring is often generalized to
1075:and the square function, doubling it to obtain the
1043:by doubling. The doubling method was formalized by
1372:
1332:
1174:
1121:
1067:
1016:
787:
761:
579:with itself is equal to the square of its length:
1445:(disambiguation page with various relevant links)
1384:, and its square root is the standard deviation.
186:. The adjective which corresponds to squaring is
1270:
1141:Exponentiation § Powers of complex numbers
645:, and the inverse images of a square are called
1239:, the dot product can be defined involving the
675:. A non-zero element of this field is called a
110:(5 squared), can be shown graphically using a
485:: it comes from the fact that the area of a
8:
728:. More generally, in a commutative ring, a
1559:, a (square velocity)-dimensioned quantity
439:, because squares of all real numbers are
1360:
1351:
1345:
1320:
1318:
1166:
1154:
1115:
1114:
1112:
1061:
1060:
1058:
1005:
995:
952:
918:
895:where 2 is invertible, the square of any
774:
747:
741:
87:Learn how and when to remove this message
1340:of the set is defined as the difference
348:is the set of nonnegative real numbers.
293:
50:This article includes a list of general
1569:
544:of distance from a fixed point forms a
633:The square function is defined in any
1373:{\displaystyle x_{i}-{\overline {x}}}
814:. However, the ring of the integers
629:In abstract algebra and number theory
7:
668:formed by the numbers modulo an odd
1302:. The deviation of each value
371:of the square function. The square
118:, and the entire square represents
1553:(quadratic dependence on velocity)
1516:Pythagorean trigonometric identity
1193:of a complex number is called its
920:
520:The square function is related to
114:. Each block represents one unit,
56:it lacks sufficient corresponding
25:
1279:is the standard method used with
590:. This is further generalised to
316:The squaring operation defines a
298:The graph of the square function
261:is the same as the square of its
795:. Both notions are important in
41:
1333:{\displaystyle {\overline {x}}}
938:
848:is the ring whose elements are
698:quadratic residues and exactly
1541:, an area-dimensioned quantity
1475:Brahmagupta–Fibonacci identity
1265:are defined using squares and
1159:
949:
939:
564:as its graph, is a smooth and
1:
1494:Degen's eight-square identity
1478:
1477:, related to complex numbers
1405:Hilbert's seventeenth problem
27:Product of a number by itself
1484:Euler's four-square identity
1479:in the sense discussed above
1407:, for the representation of
1365:
1325:
1122:{\displaystyle \mathbb {C} }
1068:{\displaystyle \mathbb {R} }
528:and its generalization, the
122:, or the area of the square.
1527:Related physical quantities
1091:Cayley–Dickson construction
489:with sides of length
1695:
1624:Cambridge University Press
1394:Exponentiation by squaring
1224:are involved (for example
1175:{\displaystyle z\to z^{2}}
1138:
1079:field with quadratic form
832:is the number of distinct
762:{\displaystyle x^{2}\in I}
621:There are infinitely many
29:
1470:Difference of two squares
1298:of a set of values, or a
801:Hilbert's Nullstellensatz
32:subscript and superscript
1674:Squares in number theory
1535:, length per square time
893:supercommutative algebra
180:may be used in place of
1618:Rajwade, A. R. (1993).
1539:cross section (physics)
1411:as a sum of squares of
542:three-dimensional graph
71:more precise citations.
1418:Square-free polynomial
1374:
1334:
1281:overdetermined systems
1176:
1149:, the square function
1123:
1069:
1018:
789:
788:{\displaystyle x\in I}
763:
503: times larger is
313:
150:, and is denoted by a
123:
1664:Elementary arithmetic
1583:mathworld.wolfram.com
1375:
1335:
1222:mathematical analysis
1177:
1124:
1104:is the "norm" of the
1070:
1047:who started with the
1019:
908:commutative semigroup
899:element equals zero.
790:
764:
679:if it is a square in
447:, by postulating the
297:
199:may also be called a
162:files, the notations
156:programming languages
102:
1409:positive polynomials
1344:
1317:
1286:Squaring is used in
1218:real-valued function
1153:
1111:
1100:The square function
1057:
917:
861:totally ordered ring
773:
740:
233:quadratic polynomial
1577:Weisstein, Eric W.
1521:Parseval's identity
1504:Lagrange's identity
1294:in determining the
1241:conjugate transpose
1106:composition algebra
1027:In the language of
828:idempotents, where
623:Pythagorean triples
526:Pythagorean theorem
1450:Related identities
1433:Quadratic equation
1413:rational functions
1370:
1330:
1296:standard deviation
1292:probability theory
1189:The square of the
1172:
1133:In complex numbers
1119:
1065:
1014:
844:; an example from
797:algebraic geometry
785:
759:
536:distance is not a
515:inverse-square law
353:monotonic function
314:
124:
1605:978-0-8218-4402-1
1579:"Absolute Square"
1545:coupling constant
1368:
1328:
1243:, leading to the
1207:complex conjugate
1203:squared magnitude
1097:with involution.
732:is an ideal
677:quadratic residue
612:moment of inertia
566:analytic function
530:parallelogram law
468:first-order logic
459:real closed field
397:) if and only if
332:squaring function
257:), the square of
223:linear polynomial
195:The square of an
134:is the result of
97:
96:
89:
16:(Redirected from
1686:
1679:Unary operations
1645:
1587:
1586:
1574:
1462:
1459:commutative ring
1379:
1377:
1376:
1371:
1369:
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1356:
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1128:
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1125:
1120:
1118:
1088:
1074:
1072:
1071:
1066:
1064:
1023:
1021:
1020:
1015:
1010:
1009:
1000:
999:
957:
956:
887:
880:
873:
869:
846:computer science
839:
831:
827:
821:
812:integral domains
794:
792:
791:
786:
768:
766:
765:
760:
752:
751:
735:
722:commutative ring
705:
697:
674:
589:
577:Euclidean vector
559:
553:
508:
502:
498:
492:
453:
419:
415:
408:
404:
396:
386:
382:
376:
367:is the (global)
362:
358:
355:on the interval
334:
333:
326:
325:
307:
281:
270:
263:additive inverse
260:
256:
249:
230:
121:
117:
109:
105:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
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1596:
1594:Further reading
1591:
1590:
1576:
1575:
1571:
1566:
1557:specific energy
1529:
1500:in the same way
1490:in the same way
1456:
1452:
1443:Sums of squares
1438:Polynomial ring
1390:
1347:
1342:
1341:
1315:
1314:
1307:
1303:
1300:random variable
1255:
1237:complex vectors
1199:squared modulus
1195:absolute square
1162:
1151:
1150:
1147:complex numbers
1143:
1135:
1109:
1108:
1080:
1055:
1054:
1035:to produce the
1029:quadratic forms
1001:
991:
948:
915:
914:
910:, then one has
882:
881:if and only if
875:
871:
864:
837:
829:
823:
819:
771:
770:
743:
738:
737:
733:
699:
691:
672:
631:
592:quadratic forms
580:
560:), which has a
555:
549:
538:smooth function
504:
500:
494:
490:
476:
451:
445:complex numbers
423:Every positive
417:
413:
409:belongs to the
406:
398:
388:
384:
378:
372:
360:
356:
331:
330:
324:square function
323:
322:
299:
292:
290:In real numbers
272:
265:
258:
254:
235:
225:
185:
179:
168:
146:the power
119:
115:
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103:
93:
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63:Please help to
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1595:
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1551:kinetic energy
1548:
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1423:Cube (algebra)
1420:
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1402:
1399:Polynomial SOS
1396:
1389:
1386:
1367:
1364:
1359:
1354:
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1327:
1324:
1305:
1254:
1251:
1191:absolute value
1169:
1165:
1161:
1158:
1134:
1131:
1117:
1077:complex number
1063:
1025:
1024:
1013:
1008:
1004:
998:
994:
990:
987:
984:
981:
978:
975:
972:
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960:
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925:
922:
850:binary numbers
784:
781:
778:
758:
755:
750:
746:
630:
627:
604:inertia tensor
475:
472:
461:, which is an
449:imaginary unit
405:, that is, if
291:
288:
207:perfect square
181:
174:
163:
95:
94:
49:
47:
40:
26:
24:
18:Square modulus
14:
13:
10:
9:
6:
4:
3:
2:
1691:
1680:
1677:
1675:
1672:
1670:
1667:
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1662:
1660:
1657:
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1654:
1643:
1639:
1635:
1633:0-521-42668-5
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1621:
1616:
1614:
1613:0-8218-4402-4
1610:
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1602:
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1593:
1584:
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1573:
1570:
1563:
1558:
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1540:
1537:
1534:
1531:
1530:
1526:
1522:
1519:
1517:
1514:
1513:
1509:
1508:
1505:
1502:
1499:
1496:, related to
1495:
1492:
1489:
1486:, related to
1485:
1482:
1480:
1476:
1473:
1471:
1468:
1467:
1464:
1460:
1454:
1453:
1449:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1428:Metric tensor
1426:
1424:
1421:
1419:
1416:
1414:
1410:
1406:
1403:
1400:
1397:
1395:
1392:
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1362:
1357:
1352:
1348:
1322:
1312:
1301:
1297:
1293:
1289:
1284:
1282:
1278:
1277:Least squares
1274:
1272:
1269:squares: see
1268:
1264:
1260:
1252:
1250:
1248:
1247:
1242:
1238:
1233:
1231:
1227:
1223:
1219:
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1210:
1208:
1204:
1200:
1196:
1192:
1187:
1185:
1182:is a twofold
1167:
1163:
1156:
1148:
1142:
1137:
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1130:
1107:
1103:
1098:
1096:
1092:
1087:
1083:
1078:
1053:
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1046:
1042:
1038:
1034:
1033:L. E. Dickson
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900:
898:
894:
889:
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878:
867:
862:
857:
855:
851:
847:
843:
835:
834:prime factors
826:
817:
813:
809:
804:
802:
799:, because of
798:
782:
779:
776:
756:
753:
748:
744:
731:
730:radical ideal
727:
723:
718:
715:
713:
712:number theory
709:
703:
695:
689:
686:
682:
678:
671:
667:
664:
660:
657:
656:finite fields
652:
650:
649:
644:
640:
636:
628:
626:
624:
619:
617:
614:to the size (
613:
609:
605:
601:
600:inner product
597:
596:linear spaces
593:
587:
583:
578:
574:
569:
567:
563:
558:
552:
547:
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539:
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531:
527:
523:
518:
516:
512:
507:
497:
488:
484:
479:
473:
471:
469:
464:
463:ordered field
460:
455:
450:
446:
442:
438:
433:
431:
426:
421:
412:
411:open interval
402:
395:
391:
383:is less than
381:
375:
370:
366:
354:
349:
347:
343:
340:is the whole
339:
335:
327:
319:
318:real function
311:
306:
302:
296:
289:
287:
285:
284:even function
279:
275:
269:
264:
251:
247:
243:
239:
234:
228:
224:
220:
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212:
208:
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203:
202:square number
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177:
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149:
145:
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137:
133:
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101:
91:
88:
80:
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
1669:Exponentials
1619:
1582:
1572:
1533:acceleration
1285:
1275:
1256:
1246:squared norm
1244:
1234:
1226:optimization
1211:
1202:
1198:
1194:
1188:
1144:
1136:
1101:
1099:
1094:
1085:
1081:
1045:A. A. Albert
1026:
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890:
883:
876:
874:. Moreover,
865:
858:
842:Boolean ring
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726:reduced ring
719:
716:
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693:
687:
684:
680:
670:prime number
665:
662:
658:
653:
648:square roots
646:
642:
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620:
585:
581:
570:
556:
550:
524:through the
519:
505:
495:
493:is equal to
480:
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441:non-negative
437:real numbers
434:
422:
400:
393:
389:
379:
377:of a number
373:
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273:
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1488:quaternions
1261:where many
1230:integration
1049:real number
1041:quaternions
854:bitwise AND
573:dot product
474:In geometry
430:square root
425:real number
357:[0, +∞)
320:called the
219:expressions
215:polynomials
152:superscript
136:multiplying
128:mathematics
77:August 2015
69:introducing
1653:Categories
1642:0785.11022
1288:statistics
1253:Other uses
1139:See also:
808:idempotent
736:such that
562:paraboloid
361:(−∞,0]
344:, and its
160:plain text
144:raising to
52:references
1564:Footnotes
1498:octonions
1455:Algebraic
1366:¯
1358:−
1326:¯
1309:from the
1160:→
1037:octonions
933:∈
921:∀
780:∈
754:∈
608:mechanics
534:Euclidean
387:(that is
363:. Hence,
342:real line
189:quadratic
120:5⋅5
116:1⋅1
104:5⋅5
1457:(need a
1388:See also
1382:variance
870:for any
836:of
769:implies
598:via the
522:distance
310:parabola
217:, other
1659:Algebra
1620:Squares
1267:inverse
1259:physics
1039:out of
852:, with
584:⋅
399:0 <
369:minimum
328:or the
266:−
240:+ 1) =
231:is the
211:algebra
197:integer
65:improve
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1215:smooth
818:
816:modulo
704:− 1)/2
696:− 1)/2
643:square
616:length
602:. The
540:: the
511:sphere
487:square
403:< 1
338:domain
336:. Its
140:number
132:square
112:square
54:, but
1510:Other
1271:below
1263:units
1201:, or
1184:cover
1052:field
906:is a
891:In a
859:In a
708:group
635:field
575:of a
414:(0,1)
392:<
346:image
308:is a
209:. In
205:or a
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171:caret
106:, or
1628:ISBN
1609:ISBN
1601:ISBN
1311:mean
1290:and
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639:ring
571:The
546:cone
483:area
365:zero
276:= (−
130:, a
1638:Zbl
1232:).
1228:or
1145:On
902:If
897:odd
886:= 0
879:= 0
868:≥ 0
637:or
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606:in
594:in
588:= v
554:or
248:+ 1
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229:+ 1
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