1268:
435:
447:
483:
459:
495:
38:
1101:
471:
745:
549:
865:
684:
511:
881:
1775:
427:
can be performed as in the following example: To divide 6 by 2—that is, to find out how many times 2 goes into 6—note that the length from 0 to 2 lies at the beginning of the length from 0 to 6; pick up the former length and put it down again to the right of its original position, with
420:
as in this example: To multiply 5 × 3, note that this is the same as 5 + 5 + 5, so pick up the length from 0 to 5 and place it to the right of 5, and then pick up that length again and place it to the right of the previous result. This gives a result that is 3 combined lengths of 5 each; since the
394:
If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number
373:
always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions. Another convention uses only one arrowhead which indicates the direction in which numbers grow. The line continues
428:
the end formerly at 0 now placed at 2, and then move the length to the right of its latest position again. This puts the right end of the length 2 at the right end of the length from 0 to 6. Since three lengths of 2 filled the length 6, 2 goes into 6 three times (that is, 6 ÷ 2 = 3).
731:. Positive numbers are closer to the viewer's eyes than the screen is, while negative numbers are "behind the screen"; larger numbers are farther from the screen. Then any point in the three-dimensional space that we live in represents the values of a trio of real numbers.
609:
numbers on a line, such that the distance of two points is the unit length, if the ratio of the represented numbers has a fixed value, typically 10. In such a logarithmic scale, the origin represents 1; one inch to the right, one has 10, one inch to the right of 10 one has
727:, and any point in the plane represents the value of a pair of real numbers. Further, the Cartesian coordinate system can itself be extended by visualizing a third number line "coming out of the screen (or page)", measuring a third variable called
522:. If the section includes both numbers it is said to be a closed interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, it is called a half-open interval.
348:
does not feature a number line, defined as we use it today, though it does use a coordinate system. In particular, Descartes's work does not contain specific numbers mapped onto lines, only abstract quantities.
434:
972:
1476:
253:
598:
On the number line, the distance between two points is the unit length if and only if the difference of the represented numbers equals 1. Other choices are possible.
1513:
322:(1685). In his treatise, Wallis describes addition and subtraction on a number line in terms of moving forward and backward, under the metaphor of a person walking.
1857:
838:
asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to
446:
413:
by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number.
395:
minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of
1938:
458:
374:
indefinitely in the positive and negative directions according to the rules of geometry which define a line without endpoints as an
637:. For example, one requires a logarithmic scale for representing simultaneously the size of the different bodies that exist in the
1902:
494:
1807:
482:
1633:
715:
Alternatively, one real number line can be drawn horizontally to denote possible values of one real number, commonly called
134:
Using a number line, numerical concepts can be interpreted geometrically and geometric concepts interpreted numerically. An
1933:
1703:
719:, and another real number line can be drawn vertically to denote possible values of another real number, commonly called
1241:
1762:
1740:
845:
1852:
1847:
17:
1928:
1566:
801:. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is
225:
of real numbers, with which it is often conflated; both the real numbers and the real line are commonly denoted
1105:
1779:
1617:
1556:
701:
1948:
724:
362:
1232:), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two
1800:
1256:
1219:
773:
470:
1685:
1267:
1862:
1842:
1581:
1551:
1520:
809:
696:
A line drawn through the origin at right angles to the real number line can be used to represent the
633:
This approach is useful, when one wants to represent, on the same figure, values with very different
151:
1953:
1653:
917:
135:
1446:
1247:
In some contexts, it is helpful to place other topologies on the set of real numbers, such as the
525:
All the points extending forever in one direction from a particular point are together known as a
236:
1169:
1669:
1300:
1161:
1120:, which can be introduced in two different, equivalent ways. First, since the real numbers are
1036:
821:
519:
424:
179:
175:
147:
139:
46:
1218:
As a locally compact space, the real line can be compactified in several different ways. The
1943:
1892:
1793:
1597:
1586:
1489:
1441:
1368:, the products of real numbers with 1 is a real line within the algebra. For example, in the
1349:
1320:
1180:
1081:
1066:
529:. If the ray includes the particular point, it is a closed ray; otherwise it is an open ray.
261:
117:
82:
680:
for multiplying or dividing numbers by adding or subtracting lengths on logarithmic scales.
1887:
1365:
1248:
1229:
1150:
997:
identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.
705:
8:
1832:
1280:
1203:
as well, though it can be disconnected by removing any one point. The real line is also
1088:
873:
203:
138:
between numbers corresponds to a left-or-right order relation between points. Numerical
1535:
1304:
1237:
1022:
891:
634:
159:
89:
relations between points, and provides a conceptual scaffold for learning mathematics.
1684:
Minnesota
Symposia on Child Psychology: Culture and Developmental Systems, Volume 38.
1160:. Up to homeomorphism, it is one of only two different connected 1-manifolds without
1872:
1758:
1736:
1629:
1184:
1117:
1109:
908:
543:
526:
343:
287:
222:
187:
66:
337:
37:
1328:
1324:
1252:
1212:
1188:
1176:, there is only one differentiable structure that the topological space supports.)
982:
802:
780:
761:
753:
697:
358:
295:
270:
when comparing it to higher-dimensional spaces. The real line is a one-dimensional
109:
104:. As students progress, more kinds of numbers can be placed on the line, including
78:
31:
77:, imagined to extend infinitely. The metaphorical association between numbers and
30:
This article is about the mathematical concept. For the typesetting practice, see
1882:
1707:
1576:
1561:
1539:
1531:
1244:
of the real line, which involves adding an infinite number of additional points.
1200:
1129:
1121:
1070:
798:
370:
366:
271:
191:
101:
440:
The ordering on the number line: Greater elements are in direction of the arrow.
1867:
1700:
1308:
1233:
1208:
1196:
1173:
1125:
1077:
835:
709:
578:
Note the logarithmic scale markings on each of the axes, and that the log
406:
between 0 and some other number represents the magnitude of the latter number.
365:
the vertical axis (y-axis) is also a number line. According to one convention,
303:
171:
1922:
1877:
1732:
1724:
1369:
1332:
1296:
1100:
978:
813:
784:
314:
The first mention of the number line used for operation purposes is found in
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54:
1750:
1345:
1276:
1204:
1192:
1139:
904:
869:
831:
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666:
403:
283:
279:
275:
143:
113:
1132:
from the metric defined above. The order topology and metric topology on
1816:
1625:
1591:
769:
396:
326:
315:
218:
128:
97:
325:
An earlier depiction without mention to operations, though, is found in
1907:
1837:
1571:
1392:
1344:
Lebesgue measure on the real line is one of the simplest examples of a
1292:
1157:
989:-dimensional Euclidean metric can be represented in matrix form as the
849:
677:
417:
183:
1168:. It also has a standard differentiable structure on it, making it a
864:
683:
548:
1897:
1289:
1154:
791:
788:
333:(1616), which shows values 1 through 12 lined up from left to right.
258:
167:
510:
1827:
1686:
http://www.cogsci.ucsd.edu/~nunez/COGS152_Readings/Nunez_ch3_MN.pdf
1341:, where the measure of any interval is the length of the interval.
1061:
This real line has several important properties as a metric space:
880:
817:
654:
646:
638:
199:
105:
93:
86:
74:
194:
are number lines which associate points in a geometric space with
198:
of numbers, so geometric shapes can be described using numerical
1654:
http://lhldigital.lindahall.org/cdm/ref/collection/math/id/11231
518:
The section of the number line between two numbers is called an
1785:
1774:
1165:
794:
670:
642:
209:
In advanced mathematics, the number line is usually called the
155:
58:
662:
658:
410:
195:
163:
62:
1670:
https://www.math.ru.nl/werkgroepen/gmfw/bronnen/napier1.html
302:, used as a two-dimensional geometric representation of the
1538:
that has a real line in the form of real products with the
650:
70:
779:
In addition to the above properties, the real line has no
1271:
The bijection between points on the real line and vectors
853:
127:: Every point of the number line corresponds to a unique
73:
and evenly spaced marks in either direction representing
41:
The order of the natural numbers shown on the number line
1138:
are the same. As a topological space, the real line is
1255:. For the real numbers, the latter is the same as the
121:
1492:
1449:
920:
622:, etc. Similarly, one inch to the left of 1, one has
239:
1667:
A description of the admirable table of logarithmes
1616:
331:
A description of the admirable table of logarithmes
274:using the difference between numbers to define the
1682:How Much Mathematics Is "Hardwired", If Any at All
1507:
1470:
966:
452:The difference 3-2=3+(-2) on the real number line.
247:
1920:
1236:, and the resulting end compactification is the
748:Each set on the real number line has a supremum.
1391:= 0} is a real line. Similarly, the algebra of
723:. Together these lines form what is known as a
505:
357:A number line is usually represented as being
1801:
421:process ends at 15, we find that 5 × 3 = 15.
1519:. In this way the real line consists of the
1307:defined by this inner product is simply the
586:axes (where the logarithms are 0) are where
532:
341:
278:on the line. It can also be thought of as a
131:, and every real number to a unique point.
92:The number line is initially used to teach
1808:
1794:
1288:of real numbers (that is, over itself) of
760:ordering. Specifically, the real line is
687:The two logarithmic scales of a slide rule
352:
1620:; Redlin, Lothar; Watson, Saleem (2008).
1095:
1080:and is one of the simplest examples of a
691:
241:
57:that serves as spatial representation of
1696:
1694:
1295:. It has the usual multiplication as an
1266:
1099:
879:
863:
743:
739:
682:
547:
509:
500:The division 3÷2 on the real number line
464:The addition 1+2 on the real number line
36:
1723:
844:. This statement has been shown to be
298:. The real line can be embedded in the
14:
1921:
1327:. This measure can be defined as the
1128:. Second, the real numbers inherit a
601:One of the most common choices is the
369:always lie on the right side of zero,
154:of the line. Wrapping the line into a
1789:
1749:
1691:
1314:
382:, and a line with two endpoints as a
1858:Decidability of first-order theories
1262:
859:
848:of the standard axiomatic system of
734:
537:
389:
1355:
605:, which is a representation of the
402:Thus, for example, the length of a
24:
1939:One-dimensional coordinate systems
1717:
1319:The real line carries a canonical
100:of integers, especially involving
25:
1965:
1710:"Purplemath" Retrieved 2015-11-13
1421:has a real line in the subspace {
1116:The real line carries a standard
1091:of the real line is equal to one.
808:The real line also satisfies the
1815:
1773:
493:
481:
469:
457:
445:
433:
378:, a line with one endpoint as a
162:to the geometric composition of
1567:Projectively extended real line
812:: every collection of mutually
704:, extends the number line to a
676:Logarithmic scales are used in
202:and numerical functions can be
182:, the principle underlying the
27:Line formed by the real numbers
1674:
1659:
1642:
1610:
1496:
957:
943:
936:
924:
911:given by absolute difference:
488:The multiplication 2 times 1.5
170:spaced graduations associates
142:are associated to geometrical
69:point representing the number
13:
1:
1701:Introduction to the x,y-plane
1603:
1486:is introduced by the mapping
1149:The real line is trivially a
981:is clearly the 1-dimensional
967:{\displaystyle d(x,y)=|x-y|.}
1557:Imaginary line (mathematics)
1471:{\displaystyle A=R\oplus V,}
673:, and the visible Universe.
336:Contrary to popular belief,
248:{\displaystyle \mathbb {R} }
146:of the line. Operations and
7:
1545:
1440:When the real algebra is a
1242:Stone–Čech compactification
725:Cartesian coordinate system
708:, with points representing
506:Portions of the number line
61:, usually graduated like a
10:
1970:
1934:Mathematical manipulatives
1257:finite complement topology
1220:one-point compactification
774:least-upper-bound property
541:
363:Cartesian coordinate plane
309:
264:, so is sometimes denoted
217:, and is a geometric line
29:
1863:Extended real number line
1823:
1755:Real and Complex Analysis
1594:(neurological phenomenon)
1582:Extended real number line
1228:is a circle (namely, the
1207:, and as such all of its
810:countable chain condition
533:Extensions of the concept
257:. The real line is a one-
152:geometric transformations
150:on numbers correspond to
85:operations on numbers to
1069:, in the sense that any
476:The absolute difference.
166:. Marking the line with
1508:{\displaystyle v\to -v}
1170:differentiable manifold
768:, and this ordering is
353:Drawing the number line
276:distance between points
1929:Elementary mathematics
1680:Núñez, Rafael (2017).
1509:
1472:
1301:Euclidean vector space
1272:
1164:, the other being the
1113:
1096:As a topological space
968:
903:The real line forms a
900:
877:
749:
692:Combining number lines
688:
595:
515:
342:
249:
118:transcendental numbers
47:elementary mathematics
42:
1903:Tarski axiomatization
1893:Real coordinate space
1843:Cantor–Dedekind axiom
1665:Napier, John (1616).
1648:Wallis, John (1685).
1598:One-dimensional space
1587:Hyperreal number line
1552:Cantor–Dedekind axiom
1510:
1473:
1350:locally compact group
1270:
1240:. There is also the
1181:locally compact space
1142:to the open interval
1104:The real line can be
1103:
1082:geodesic metric space
1067:complete metric space
969:
883:
867:
747:
740:As a linear continuum
686:
620:10×1000 = 10,000 = 10
551:
514:The closed interval .
513:
262:real coordinate space
250:
40:
1888:Rational zeta series
1782:at Wikimedia Commons
1523:of the conjugation.
1490:
1447:
1249:lower limit topology
1230:real projective line
1151:topological manifold
1073:of points converges.
918:
872:on the real line is
797:, namely the set of
706:complex number plane
700:. This line, called
237:
1833:Absolute difference
1650:Treatise of algebra
1275:The real line is a
1199:, and is therefore
1179:The real line is a
1089:Hausdorff dimension
1065:The real line is a
1035:is simply the open
874:absolute difference
756:under the standard
752:The real line is a
568: (green), and
416:Two numbers can be
409:Two numbers can be
320:Treatise of algebra
1949:Topological spaces
1706:2015-11-09 at the
1628:. pp. 13–19.
1505:
1468:
1315:As a measure space
1273:
1238:extended real line
1114:
964:
901:
878:
830:is countable. In
805:to the real line.
750:
689:
635:order of magnitude
616:10×100 = 1000 = 10
596:
552:A log-log plot of
516:
245:
160:modular arithmetic
81:on the line links
65:with a particular
53:is a picture of a
43:
1916:
1915:
1873:Irrational number
1778:Media related to
1635:978-0-495-56521-5
1618:Stewart, James B.
1263:As a vector space
1215:groups are zero.
1185:paracompact space
1110:point at infinity
1076:The real line is
909:distance function
860:As a metric space
787:. It also has a
735:Advanced concepts
698:imaginary numbers
603:logarithmic scale
594:themselves are 1.
544:Logarithmic scale
538:Logarithmic scale
390:Comparing numbers
288:topological space
188:analytic geometry
110:decimal fractions
16:(Redirected from
1961:
1810:
1803:
1796:
1787:
1786:
1777:
1768:
1746:
1731:(2nd ed.).
1711:
1698:
1689:
1678:
1672:
1663:
1657:
1646:
1640:
1639:
1624:(5th ed.).
1614:
1526:For a dimension
1514:
1512:
1511:
1506:
1477:
1475:
1474:
1469:
1383:, the subspace {
1356:In real algebras
1340:
1325:Lebesgue measure
1287:
1253:Zariski topology
1227:
1213:reduced homology
1189:second-countable
1145:
1137:
1124:, they carry an
1057:
1034:
1030:
1020:
1016:
1009:
996:
992:
988:
983:Euclidean metric
973:
971:
970:
965:
960:
946:
899:
894:around a number
889:
843:
829:
803:order-isomorphic
799:rational numbers
767:
762:linearly ordered
759:
754:linear continuum
629:
625:
621:
617:
613:
497:
485:
473:
461:
449:
437:
371:negative numbers
367:positive numbers
347:
296:linear continuum
269:
256:
254:
252:
251:
246:
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230:
215:real number line
125:
122:circle constant
102:negative numbers
21:
18:Real number line
1969:
1968:
1964:
1963:
1962:
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1959:
1958:
1954:Line (geometry)
1919:
1918:
1917:
1912:
1883:Rational number
1819:
1814:
1765:
1757:. McGraw-Hill.
1743:
1720:
1718:Further reading
1715:
1714:
1708:Wayback Machine
1699:
1692:
1679:
1675:
1664:
1660:
1647:
1643:
1636:
1622:College Algebra
1615:
1611:
1606:
1577:Cuisenaire rods
1562:Line (geometry)
1548:
1540:identity matrix
1532:square matrices
1491:
1488:
1487:
1448:
1445:
1444:
1358:
1336:
1317:
1283:
1265:
1223:
1209:homotopy groups
1143:
1133:
1130:metric topology
1122:totally ordered
1098:
1071:Cauchy sequence
1039:
1032:
1026:
1018:
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1001:
994:
990:
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785:minimum element
765:
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742:
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710:complex numbers
694:
641:, typically, a
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611:
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498:
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477:
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392:
355:
312:
304:complex numbers
272:Euclidean space
265:
240:
238:
235:
234:
232:
226:
192:coordinate axes
178:with geometric
168:logarithmically
123:
35:
28:
23:
22:
15:
12:
11:
5:
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1875:
1870:
1868:Gregory number
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1725:Munkres, James
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1461:
1458:
1455:
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1419:
1418:
1357:
1354:
1316:
1313:
1309:absolute value
1299:, making it a
1264:
1261:
1197:path-connected
1195:. It is also
1174:diffeomorphism
1126:order topology
1097:
1094:
1093:
1092:
1085:
1078:path-connected
1074:
975:
974:
963:
959:
955:
952:
949:
945:
941:
938:
935:
932:
929:
926:
923:
861:
858:
836:Suslin problem
741:
738:
736:
733:
702:imaginary line
693:
690:
560: (blue),
542:Main article:
539:
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534:
531:
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442:
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391:
388:
354:
351:
338:René Descartes
311:
308:
243:
172:multiplication
26:
9:
6:
4:
3:
2:
1966:
1955:
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1901:
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1886:
1884:
1881:
1879:
1878:Normal number
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1874:
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1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
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1836:
1834:
1831:
1829:
1826:
1825:
1822:
1818:
1811:
1806:
1804:
1799:
1797:
1792:
1791:
1788:
1781:
1776:
1772:
1771:
1766:
1764:0-07-100276-6
1760:
1756:
1752:
1751:Rudin, Walter
1748:
1744:
1742:0-13-181629-2
1738:
1734:
1733:Prentice Hall
1730:
1726:
1722:
1721:
1709:
1705:
1702:
1697:
1695:
1687:
1683:
1677:
1671:
1668:
1662:
1655:
1651:
1645:
1637:
1631:
1627:
1623:
1619:
1613:
1609:
1599:
1596:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1549:
1543:
1542:in the ring.
1541:
1537:
1533:
1529:
1524:
1522:
1518:
1502:
1499:
1493:
1485:
1481:
1465:
1462:
1459:
1456:
1453:
1450:
1443:
1438:
1436:
1432:
1428:
1424:
1416:
1412:
1408:
1404:
1400:
1397:
1396:
1395:
1394:
1390:
1386:
1382:
1378:
1374:
1371:
1370:complex plane
1367:
1363:
1353:
1351:
1347:
1342:
1339:
1334:
1333:Borel measure
1330:
1326:
1323:, namely the
1322:
1312:
1310:
1306:
1302:
1298:
1297:inner product
1294:
1291:
1286:
1282:
1278:
1269:
1260:
1258:
1254:
1250:
1245:
1243:
1239:
1235:
1231:
1226:
1221:
1216:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1187:, as well as
1186:
1182:
1177:
1175:
1171:
1167:
1163:
1159:
1156:
1152:
1147:
1141:
1136:
1131:
1127:
1123:
1119:
1111:
1108:by adding a
1107:
1102:
1090:
1086:
1083:
1079:
1075:
1072:
1068:
1064:
1063:
1062:
1059:
1055:
1051:
1047:
1043:
1038:
1029:
1024:
1014:
1008:
1004:
998:
984:
980:
979:metric tensor
961:
953:
950:
947:
939:
933:
930:
927:
921:
914:
913:
912:
910:
906:
898:
893:
888:
882:
875:
871:
866:
857:
855:
851:
847:
842:
837:
834:, the famous
833:
828:
823:
819:
815:
811:
806:
804:
800:
796:
793:
790:
786:
782:
777:
775:
771:
763:
755:
746:
732:
730:
726:
722:
718:
713:
711:
707:
703:
699:
685:
681:
679:
674:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
631:
608:
604:
599:
593:
589:
585:
582:and log
581:
575:
572: =
571:
567:
564: =
563:
559:
556: =
555:
550:
545:
530:
528:
523:
521:
512:
496:
491:
484:
479:
472:
467:
460:
455:
448:
443:
436:
431:
430:
429:
426:
422:
419:
414:
412:
407:
405:
400:
398:
387:
385:
381:
377:
376:infinite line
372:
368:
364:
360:
350:
346:
345:
339:
334:
332:
328:
323:
321:
317:
307:
305:
301:
300:complex plane
297:
293:
292:measure space
289:
285:
281:
277:
273:
268:
263:
260:
229:
224:
220:
216:
212:
207:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
145:
141:
137:
132:
130:
126:
119:
115:
111:
107:
103:
99:
95:
90:
88:
84:
80:
76:
72:
68:
64:
60:
56:
55:straight line
52:
48:
39:
33:
32:Printer's key
19:
1944:Real numbers
1853:Construction
1848:Completeness
1817:Real numbers
1780:Number lines
1754:
1728:
1681:
1676:
1666:
1661:
1649:
1644:
1621:
1612:
1527:
1525:
1521:fixed points
1516:
1515:of subspace
1483:
1479:
1439:
1434:
1430:
1426:
1422:
1420:
1414:
1410:
1406:
1402:
1398:
1388:
1384:
1380:
1376:
1372:
1366:real algebra
1364:is a unital
1361:
1359:
1346:Haar measure
1343:
1337:
1318:
1284:
1277:vector space
1274:
1246:
1224:
1217:
1205:contractible
1178:
1148:
1140:homeomorphic
1134:
1115:
1106:compactified
1060:
1053:
1049:
1045:
1041:
1031:centered at
1027:
1012:
1006:
1002:
999:
985:. Since the
976:
905:metric space
902:
896:
886:
840:
832:order theory
826:
807:
778:
772:and has the
751:
728:
720:
716:
714:
695:
675:
667:Solar System
632:
606:
602:
600:
597:
591:
587:
583:
579:
576: (red).
573:
569:
565:
561:
557:
553:
524:
517:
423:
415:
408:
404:line segment
401:
393:
384:line segment
383:
379:
375:
356:
344:La Géométrie
340:'s original
335:
330:
324:
319:
313:
284:metric space
280:vector space
266:
227:
214:
210:
208:
180:translations
133:
120:such as the
114:square roots
91:
83:arithmetical
50:
44:
1626:Brooks Cole
1592:Number form
1480:conjugation
1393:quaternions
1335:defined on
1017:, then the
907:, with the
846:independent
678:slide rules
612:10×10 = 100
397:subtraction
361:, but in a
327:John Napier
316:John Wallis
259:dimensional
129:real number
98:subtraction
51:number line
1923:Categories
1908:Vitali set
1838:Cantor set
1604:References
1572:Chronology
1442:direct sum
1329:completion
850:set theory
628:1/100 = 10
418:multiplied
359:horizontal
219:isomorphic
184:slide rule
136:inequality
1898:Real line
1500:−
1497:→
1460:⊕
1290:dimension
1279:over the
1201:connected
1172:. (Up to
1155:dimension
951:−
852:known as
822:intervals
789:countable
624:1/10 = 10
211:real line
200:equations
148:functions
140:intervals
106:fractions
87:geometric
1828:0.999...
1753:(1966).
1729:Topology
1727:(1999).
1704:Archived
1546:See also
1425: :
1387: :
1162:boundary
1118:topology
1037:interval
818:nonempty
814:disjoint
655:molecule
647:electron
639:Universe
607:positive
520:interval
425:Division
176:division
158:relates
144:segments
94:addition
75:integers
1656:pp. 265
1534:form a
1478:then a
1437:= 0 }.
1321:measure
1251:or the
781:maximum
630:, etc.
626:, then
618:, then
614:, then
310:History
294:, or a
255:
233:
221:to the
204:graphed
59:numbers
1761:
1739:
1688:pp. 98
1632:
1530:, the
1303:. The
1193:normal
1183:and a
1166:circle
1144:(0, 1)
1015:> 0
870:metric
795:subset
671:galaxy
665:, the
661:, the
643:photon
196:tuples
164:angles
156:circle
116:, and
79:points
67:origin
1360:When
1348:on a
1331:of a
1281:field
820:open
792:dense
770:dense
663:Earth
659:human
649:, an
645:, an
411:added
186:. In
63:ruler
1759:ISBN
1737:ISBN
1630:ISBN
1536:ring
1413:j +
1409:i +
1305:norm
1234:ends
1211:and
1191:and
1087:The
1023:ball
1010:and
993:-by-
977:The
892:ball
868:The
766:<
758:<
669:, a
657:, a
653:, a
651:atom
590:and
290:, a
286:, a
282:, a
174:and
96:and
71:zero
49:, a
1482:on
1379:+ i
1222:of
1153:of
1025:in
1000:If
884:An
854:ZFC
824:in
783:or
764:by
527:ray
380:ray
329:'s
318:'s
231:or
223:set
213:or
45:In
1925::
1735:.
1693:^
1652:.
1433:=
1429:=
1405:+
1401:=
1375:=
1352:.
1311:.
1259:.
1146:.
1058:.
1052:+
1048:,
1044:−
1005:∈
856:.
816:,
776:.
712:.
399:.
386:.
306:.
206:.
190:,
112:,
108:,
1809:e
1802:t
1795:v
1767:.
1745:.
1638:.
1528:n
1517:V
1503:v
1494:v
1484:A
1466:,
1463:V
1457:R
1454:=
1451:A
1435:z
1431:y
1427:x
1423:q
1417:k
1415:z
1411:y
1407:x
1403:w
1399:q
1389:y
1385:z
1381:y
1377:x
1373:z
1362:A
1338:R
1293:1
1285:R
1225:R
1158:1
1135:R
1112:.
1084:.
1056:)
1054:ε
1050:p
1046:ε
1042:p
1040:(
1033:p
1028:R
1021:-
1019:ε
1013:ε
1007:R
1003:p
995:n
991:n
987:n
962:.
958:|
954:y
948:x
944:|
940:=
937:)
934:y
931:,
928:x
925:(
922:d
897:a
890:-
887:ε
876:.
841:R
827:R
729:z
721:y
717:x
592:y
588:x
584:y
580:x
574:x
570:y
566:x
562:y
558:x
554:y
267:R
242:R
228:R
124:π
34:.
20:)
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