1500:
895:
1125:
1263:
1201:
308:
713:
363:
498:
1310:
1344:
589:
1679:. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of
1482:
Orders of magnitude denote differences in numeric quantities, usually measurements, by a factor of 10—that is, a difference of one digit in the location of the decimal point.
53:
is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an
705:
626:
934:
1637:
of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in
661:
1023:
237:
1595:
1552:
1572:
1525:
197:
1035:
967:. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector
1217:
1933:
1904:
1832:
260:
1776:
1133:
890:{\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}-abi+abi-b^{2}i^{2}}}={\sqrt {a^{2}+b^{2}}}}
1948:
1983:
314:
445:
1464:
1696:
88:
is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.
31:
1282:
1988:
1658:
405:
1316:
535:
1272:: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector
1455:
1415:
1610:
1492:
66:
1614:
1362:
1354:
980:
666:
1601:
is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.
1379:
By definition, all
Euclidean vectors have a magnitude (see above). However, a vector in an abstract
1395:
1374:
58:
50:
1894:
1638:
1477:
1391:
1269:
1026:
85:
1700:
1499:
602:
1787:
The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece.
1929:
1900:
1828:
1772:
1650:
1504:
1431:
903:
596:
1766:
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1654:
1459:
956:
631:
1966:. Osmania University, Digital Library Of India. Cambridge University Press. pp. 91–98.
1002:
1718:
1662:
964:
436:
396:
151:
119:
212:
1799:
1577:
1534:
1961:
1684:
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1668:
1646:
1557:
1510:
1419:
1350:
950:
516:
432:
385:
201:
182:
171:
129:
96:
77:
70:
62:
1868:
1977:
1850:
1748:
1688:
400:
17:
80:
is a measure of magnitude used to define a distance between two points in space. In
1910:
1782:
1713:
1692:
1387:
1380:
109:
73:
of a number is commonly applied as the measure of units between a number and zero.
54:
1120:{\displaystyle \|\mathbf {x} \|={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}
1743:
1634:
1606:
1358:
1349:
A disadvantage of the second notation is that it can also be used to denote the
1208:
1204:
373:
248:
38:
1672:
1447:
147:
1598:
512:
1258:{\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}.}
1723:
1642:
1458:
of earthquake intensity. Logarithmic magnitudes can be negative. In the
1435:
369:
104:
1130:
For instance, in a 3-dimensional space, the magnitude of is 13 because
1443:
81:
1626:
1618:
1528:
417:
from the origin of that space. The formula for the absolute value of
177:
133:
113:
595:
may be defined as the square root of the product of itself and its
1498:
1439:
1402:
in a normed vector space can be considered to be the magnitude of
975:-dimensional Euclidean space can be defined as an ordered list of
139:
1630:
1622:
1451:
159:
123:
61:
of objects to which it belongs. Magnitude as a concept dates to
1196:{\displaystyle {\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.}
146:
They proved that the first two could not be the same, or even
1926:
The Linear
Algebra a Beginning Graduate Student Ought to Know
162:
is either the smallest size or less than all possible sizes.
99:
distinguished between several types of magnitude, including:
376:. For example, the absolute value of both 70 and −70 is 70.
303:{\displaystyle \left|r\right|=r,{\text{ if }}r{\text{ ≥ }}0}
1800:"Magnitude Definition (Illustrated Mathematics Dictionary)"
1268:
The
Euclidean norm of a vector is just a special case of
69:
of distance from one object to another. For numbers, the
1462:, a logarithmic magnitude is typically referred to as a
84:, magnitude can be defined as quantity or distance. An
368:
Absolute value may also be thought of as the number's
358:{\displaystyle \left|r\right|=-r,{\text{ if }}r<0.}
1580:
1560:
1537:
1513:
1319:
1285:
1220:
1136:
1038:
1005:
906:
716:
669:
634:
605:
538:
448:
317:
263:
215:
185:
591:. Alternatively, the magnitude of a complex number
493:{\displaystyle \left|z\right|={\sqrt {a^{2}+b^{2}}}}
1849:
1754:(2nd ed. ed.). New York: Dover Publications.
1747:
1589:
1566:
1546:
1519:
1338:
1304:
1257:
1195:
1119:
1017:
928:
889:
699:
655:
620:
583:
492:
357:
302:
231:
191:
1960:Heath, T. L. (1897). "Measurement of a Circle".
1667:The intuition behind this concept dates back to
1503:Informally, a measure has the property of being
1418:, the magnitude of a vector is the value of the
1896:Elementary Linear Algebra: Applications Version
1629:) and other common notions, such as magnitude,
523:, respectively. For instance, the modulus of
158:is still primarily used in contexts in which
8:
1893:Howard Anton; Chris Rorres (12 April 2010).
1365:, which introduces an element of ambiguity.
1305:{\displaystyle \left\|\mathbf {x} \right\|,}
1229:
1221:
1047:
1039:
1012:
1006:
150:systems of magnitude. They did not consider
27:Property determining comparison and ordering
1394:, such as the Euclidean space, is called a
1339:{\displaystyle \left|\mathbf {x} \right|.}
584:{\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5}
1613:is a generalization and formalization of
1579:
1559:
1536:
1512:
1324:
1318:
1290:
1284:
1245:
1237:
1235:
1224:
1219:
1180:
1169:
1156:
1143:
1137:
1135:
1106:
1101:
1082:
1077:
1064:
1059:
1053:
1042:
1037:
1004:
911:
905:
879:
866:
860:
849:
839:
802:
796:
753:
737:
736:
731:
715:
671:
670:
668:
633:
607:
606:
604:
567:
554:
539:
537:
482:
469:
463:
447:
391:may be viewed as the position of a point
341:
316:
292:
284:
262:
224:
216:
214:
184:
1653:. Far-reaching generalizations (such as
1574:is less than or equal to the measure of
1827:. McGraw-Hill Professional. p. 2.
1750:The Thirteen Books of Euclid's Elements
1735:
1825:Schaum's Outline of Beginning Calculus
413:may be thought of as the distance of
7:
1924:Golan, Johnathan S. (January 2007),
1434:is often used. Examples include the
1645:, and can be generalized to assume
959:represents the position of a point
1869:"Magnitude of a vector definition"
1768:The Real Numbers and Real Analysis
1025:, is most commonly defined as its
25:
154:magnitudes to be meaningful, and
1661:) of measure are widely used in
1597:Furthermore, the measure of the
1491:This section is an excerpt from
1325:
1291:
1246:
1238:
1225:
1043:
1856:. Tokyo: McGraw Hill Kogakusha.
700:{\displaystyle {\bar {z}}=a-bi}
628:, where for any complex number
435:of a vector in a 2-dimensional
1383:does not possess a magnitude.
1295:
1287:
788:
773:
770:
755:
742:
676:
612:
551:
541:
225:
217:
142:(ordered by angular magnitude)
1:
1430:When comparing magnitudes, a
1823:Mendelson, Elliott (2008).
1486:Other mathematical measures
1211:of the vector with itself:
663:, its complex conjugate is
431:is similar to that for the
2005:
1928:(2nd ed.), Springer,
1659:projection-valued measures
1490:
1475:
1372:
1203:This is equivalent to the
948:
621:{\displaystyle {\bar {z}}}
169:
65:and has been applied as a
32:Magnitude (disambiguation)
29:
1899:. John Wiley & Sons.
1848:Ahlfors, Lars V. (1953).
403:. The absolute value (or
1771:, Springer, p. 52,
1765:Bloch, Ethan D. (2011),
1665:and physics in general.
929:{\displaystyle i^{2}=-1}
247:The absolute value of a
1963:The Works Of Archimedes
1697:Constantin Carathéodory
1675:tried to calculate the
1398:. The norm of a vector
1029:(or Euclidean length):
503:where the real numbers
1984:Elementary mathematics
1602:
1591:
1568:
1548:
1521:
1426:Logarithmic magnitudes
1416:pseudo-Euclidean space
1410:Pseudo-Euclidean space
1340:
1306:
1259:
1197:
1121:
1019:
945:Euclidean vector space
930:
891:
701:
657:
656:{\displaystyle z=a+bi}
622:
585:
494:
372:from zero on the real
359:
304:
233:
199:is usually called its
193:
76:In vector spaces, the
1592:
1569:
1549:
1522:
1507:in the sense that if
1502:
1493:Measure (mathematics)
1341:
1307:
1260:
1198:
1122:
1020:
1018:{\displaystyle \|x\|}
981:Cartesian coordinates
931:
892:
702:
658:
623:
586:
495:
360:
305:
234:
194:
176:The magnitude of any
18:Logarithmic magnitude
1949:Measuring the Circle
1615:geometrical measures
1578:
1558:
1535:
1511:
1369:Normed vector spaces
1317:
1283:
1218:
1134:
1036:
1003:
904:
714:
667:
632:
603:
536:
446:
315:
294: ≥
261:
213:
183:
57:(or ranking) of the
30:For other uses, see
1609:, the concept of a
1396:normed vector space
1375:Normed vector space
1111:
1087:
1069:
397:2-dimensional space
232:{\displaystyle |x|}
51:mathematical object
1744:Heath, Thomas Smd.
1643:integration theory
1639:probability theory
1603:
1590:{\displaystyle B.}
1587:
1564:
1547:{\displaystyle B,}
1544:
1517:
1478:Order of magnitude
1472:Order of magnitude
1336:
1302:
1270:Euclidean distance
1255:
1193:
1117:
1097:
1073:
1055:
1015:
979:real numbers (the
926:
887:
697:
653:
618:
581:
490:
355:
300:
229:
189:
86:order of magnitude
1935:978-1-4020-5494-5
1906:978-0-470-43205-1
1834:978-0-07-148754-2
1655:spectral measures
1651:electrical charge
1567:{\displaystyle A}
1520:{\displaystyle A}
1432:logarithmic scale
1422:for that vector.
1250:
1185:
1175:
1112:
885:
855:
791:
748:
745:
679:
615:
597:complex conjugate
573:
488:
344:
295:
287:
192:{\displaystyle x}
16:(Redirected from
1996:
1989:Unary operations
1968:
1967:
1957:
1951:
1945:
1939:
1938:
1921:
1915:
1914:
1890:
1884:
1883:
1881:
1879:
1864:
1858:
1857:
1855:
1852:Complex Analysis
1845:
1839:
1838:
1820:
1814:
1813:
1811:
1810:
1796:
1790:
1789:
1762:
1756:
1755:
1753:
1740:
1677:area of a circle
1596:
1594:
1593:
1588:
1573:
1571:
1570:
1565:
1553:
1551:
1550:
1545:
1526:
1524:
1523:
1518:
1460:natural sciences
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1110:
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1063:
1054:
1046:
1024:
1022:
1021:
1016:
957:Euclidean vector
935:
933:
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927:
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915:
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754:
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430:
364:
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288:
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238:
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21:
2004:
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1958:
1954:
1946:
1942:
1936:
1923:
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1918:
1907:
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1887:
1877:
1875:
1867:Nykamp, Duane.
1866:
1865:
1861:
1847:
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1822:
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1764:
1763:
1759:
1742:
1741:
1737:
1732:
1719:Vector notation
1710:
1705:
1704:
1703:, among others.
1701:Maurice Fréchet
1663:quantum physics
1647:negative values
1576:
1575:
1556:
1555:
1554:the measure of
1533:
1532:
1509:
1508:
1496:
1488:
1480:
1474:
1428:
1412:
1390:endowed with a
1377:
1371:
1320:
1315:
1314:
1286:
1281:
1280:
1216:
1215:
1165:
1152:
1139:
1132:
1131:
1034:
1033:
1001:
1000:
965:Euclidean space
953:
947:
942:
907:
902:
901:
875:
862:
845:
835:
798:
717:
712:
711:
665:
664:
630:
629:
601:
600:
563:
550:
534:
533:
530:
524:
478:
465:
449:
444:
443:
437:Euclidean space
418:
382:
380:Complex numbers
318:
313:
312:
264:
259:
258:
254:is defined by:
245:
211:
210:
181:
180:
174:
168:
94:
35:
28:
23:
22:
15:
12:
11:
5:
2002:
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1991:
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1969:
1952:
1940:
1934:
1916:
1905:
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1840:
1833:
1815:
1804:mathsisfun.com
1791:
1777:
1757:
1734:
1733:
1731:
1728:
1727:
1726:
1721:
1716:
1709:
1706:
1685:Henri Lebesgue
1669:ancient Greece
1586:
1583:
1563:
1543:
1540:
1516:
1497:
1489:
1487:
1484:
1476:Main article:
1473:
1470:
1427:
1424:
1420:quadratic form
1411:
1408:
1373:Main article:
1370:
1367:
1351:absolute value
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1346:
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1327:
1323:
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1301:
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1146:
1142:
1128:
1127:
1116:
1109:
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1085:
1080:
1076:
1072:
1067:
1062:
1058:
1052:
1049:
1045:
1041:
1027:Euclidean norm
1014:
1011:
1008:
951:Euclidean norm
949:Main article:
946:
943:
941:
938:
925:
922:
919:
914:
910:
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897:
882:
878:
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730:
726:
723:
720:
696:
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690:
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652:
649:
646:
643:
640:
637:
614:
611:
580:
577:
570:
566:
562:
557:
553:
549:
546:
543:
526:
517:imaginary part
501:
500:
485:
481:
477:
472:
468:
462:
458:
455:
452:
433:Euclidean norm
386:complex number
381:
378:
366:
365:
354:
351:
348:
343: if
340:
337:
334:
331:
327:
324:
321:
310:
299:
291:
286: if
283:
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273:
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244:
241:
227:
223:
219:
202:absolute value
188:
172:Absolute value
170:Main article:
167:
164:
144:
143:
137:
127:
117:
107:
97:Ancient Greeks
93:
90:
78:Euclidean norm
71:absolute value
63:Ancient Greece
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2001:
1990:
1987:
1985:
1982:
1981:
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1965:
1964:
1956:
1953:
1950:
1944:
1941:
1937:
1931:
1927:
1920:
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1902:
1898:
1897:
1889:
1886:
1874:
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1841:
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1816:
1805:
1801:
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1780:
1778:9780387721774
1774:
1770:
1769:
1761:
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1745:
1739:
1736:
1729:
1725:
1722:
1720:
1717:
1715:
1712:
1711:
1707:
1702:
1698:
1694:
1690:
1689:Nikolai Luzin
1686:
1682:
1678:
1674:
1670:
1666:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
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1632:
1628:
1624:
1620:
1616:
1612:
1608:
1600:
1584:
1581:
1561:
1541:
1538:
1530:
1514:
1506:
1501:
1494:
1485:
1483:
1479:
1471:
1469:
1467:
1466:
1461:
1457:
1456:Richter scale
1453:
1449:
1445:
1442:(measured in
1441:
1437:
1433:
1425:
1423:
1421:
1417:
1409:
1407:
1405:
1401:
1397:
1393:
1389:
1384:
1382:
1376:
1368:
1366:
1364:
1360:
1356:
1352:
1333:
1329:
1321:
1313:
1299:
1279:
1278:
1277:
1275:
1271:
1252:
1242:
1232:
1214:
1213:
1212:
1210:
1206:
1190:
1187:
1182:
1177:
1170:
1166:
1162:
1157:
1153:
1149:
1144:
1140:
1114:
1107:
1102:
1098:
1094:
1091:
1088:
1083:
1078:
1074:
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1050:
1032:
1031:
1030:
1028:
1009:
999:, denoted by
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
958:
952:
944:
940:Vector spaces
939:
937:
923:
920:
917:
912:
908:
880:
876:
872:
867:
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761:
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739:
733:
728:
724:
721:
718:
710:
709:
708:
694:
691:
688:
685:
682:
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638:
635:
609:
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594:
578:
575:
568:
564:
560:
555:
547:
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522:
518:
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510:
506:
483:
479:
475:
470:
466:
460:
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453:
450:
442:
441:
440:
438:
434:
429:
425:
421:
416:
412:
408:
407:
402:
401:complex plane
399:, called the
398:
394:
390:
387:
379:
377:
375:
371:
352:
349:
346:
338:
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332:
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297:
289:
281:
278:
275:
271:
268:
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257:
256:
255:
253:
250:
242:
240:
221:
209:, denoted by
208:
204:
203:
186:
179:
173:
165:
163:
161:
157:
153:
149:
141:
138:
135:
131:
128:
125:
121:
120:Plane figures
118:
115:
111:
110:Line segments
108:
106:
102:
101:
100:
98:
91:
89:
87:
83:
79:
74:
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1962:
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1943:
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1911:Google Books
1909:– via
1895:
1888:
1876:. Retrieved
1873:Math Insight
1872:
1862:
1851:
1843:
1824:
1818:
1807:. Retrieved
1803:
1794:
1786:
1783:Google Books
1781:– via
1767:
1760:
1749:
1738:
1714:Number sense
1693:Johann Radon
1604:
1481:
1463:
1429:
1413:
1403:
1399:
1388:vector space
1385:
1381:vector space
1378:
1359:determinants
1348:
1273:
1267:
1129:
996:
992:
988:
984:
976:
972:
968:
960:
954:
899:
592:
527:
520:
508:
504:
502:
427:
423:
419:
414:
410:
404:
392:
388:
383:
367:
251:
246:
243:Real numbers
206:
200:
175:
155:
145:
132:(ordered by
122:(ordered by
112:(ordered by
95:
75:
46:
42:
36:
1947:Archimedes
1681:Émile Borel
1635:probability
1607:mathematics
1209:dot product
1205:square root
374:number line
249:real number
39:mathematics
1978:Categories
1878:August 23,
1809:2020-08-23
1730:References
1673:Archimedes
1649:, as with
1454:, and the
1448:brightness
148:isomorphic
1599:empty set
1243:⋅
1230:‖
1222:‖
1092:⋯
1048:‖
1040:‖
1013:‖
1007:‖
993:magnitude
921:−
833:−
809:−
780:−
743:¯
689:−
677:¯
613:¯
545:−
513:real part
333:−
156:magnitude
105:fractions
103:Positive
43:magnitude
1746:(1956).
1724:Set size
1708:See also
1505:monotone
1444:decibels
1436:loudness
1363:matrices
1357:and the
1296:‖
1288:‖
991:= . Its
515:and the
511:are the
370:distance
152:negative
55:ordering
1671:, when
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1446:), the
1355:scalars
1207:of the
900:(where
406:modulus
207:modulus
166:Numbers
92:History
82:physics
67:measure
1932:
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1831:
1775:
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1633:, and
1627:volume
1619:length
1529:subset
997:length
971:in an
525:−3 + 4
178:number
140:Angles
134:volume
130:Solids
114:length
41:, the
1527:is a
1465:level
1450:of a
1440:sound
1438:of a
1414:In a
963:in a
409:) of
395:in a
59:class
49:of a
1930:ISBN
1901:ISBN
1880:2020
1829:ISBN
1773:ISBN
1657:and
1631:mass
1623:area
1452:star
1392:norm
507:and
350:<
160:zero
124:area
47:size
1605:In
1531:of
1361:of
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987:):
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936:).
707:.
532:is
519:of
205:or
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384:A
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239:.
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1108:2
1103:n
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989:x
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913:2
909:i
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858:=
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777:a
774:(
771:)
768:i
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751:=
740:z
734:z
729:=
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645:+
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639:=
636:z
610:z
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576:=
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336:r
330:=
326:|
323:r
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272:|
269:r
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226:|
222:x
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187:x
136:)
126:)
116:)
34:.
20:)
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