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Order of magnitude

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3061: 43: 1831:, which has a logarithm (in base 10) of 6.602, has 7 as its nearest order of magnitude, because "nearest" implies rounding rather than truncation. For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten (about 3.162). For example, the nearest order of magnitude for 127:" (i.e., factors of ten). For example, there is one order of magnitude between 2 and 20, and two orders of magnitude between 2 and 200. Each division or multiplication by 10 is called an order of magnitude. This phrasing helps quickly express the difference in scale between 2 and 2,000,000: they differ by 6 orders of magnitude. 137:
Below are examples of different methods of partitioning the real numbers into specific "orders of magnitude" for various purposes. There is not one single accepted way of doing this, and different partitions may be easier to compute but less useful for approximation, or better for approximation but
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has a logarithm (in base 10) of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between 10 and 10. In a similar example, with the phrase "seven-figure income", the order of magnitude is the number of figures minus one, so it is very easily determined
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are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits minus one in the base-10 representation of the value. Similarly, if the
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For example, 1 and 1.02 are within an order of magnitude. So are 1 and 2, 1 and 3, or 1 and 0.98. However, 1 and 15 are not within an order of magnitude, since their ratio is 15/1 = 15 > 10. The reciprocal ratio, 1/15, is less than 0.1, so the same result is obtained.
2043:. It can be seen that the order of magnitude is included in the number name in this example, because bi- means 2, tri- means 3, etc. (these make sense in the long scale only), and the suffix -illion tells that the base is 2032:
of the world use a larger base to better envision the size of the number, and have created names for the powers of this larger base. The table shows what number the order of magnitude aim at for base 10 and for base
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The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the
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Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are within about a factor of 10 of each other.
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Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number. To work out the order of magnitude of a number
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to the nearest power of ten. For example, an order-of-magnitude estimate for a variable between about 3 billion and 30 billion (such as the human
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Two quantities A and B which are within about a factor of 10 of each other are then said to be "of the same order of magnitude," written A∼B.
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Physicists and engineers use the phrase "order of magnitude" to refer to the smallest power of ten needed to represent a quantity.
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greater than the previous level. Thus, a level being 5 magnitudes brighter than another indicates that it is a factor of
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format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.
1821:. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus 2897: 2842: 2792: 2778: 2768: 2426: 2400: 2365: 2325: 131: 1940: 926:
Orders of magnitude are used to make approximate comparisons. If numbers differ by one order of magnitude,
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of a value relative to some contextually understood reference value, usually 10, interpreted as the
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Other orders of magnitude may be calculated using bases other than integers. In the field of
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An order-of-magnitude estimate of a variable, whose precise value is unknown, is an estimate
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without a calculator to be 6. An order of magnitude is an approximate position on a
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represents the order of magnitude of the number. The order of magnitude can be any
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Cosmos – an Illustrated Dimensional Journey from microcosmos to macrocosmos
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reference value is one of some powers of 2 since computers store data in a
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of the logarithm and the representative of values of magnitude one.
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This series of magnitudes forms a logarithmic scale with a base of
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is 9. An order-of-magnitude estimate is sometimes also called a
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with base 1024 were invented for use in electronic technology.
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Examples of numbers of different magnitudes can be found at
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times brighter: that is, two base 10 orders of magnitude.
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magnitudes, they are names of "magnitudes", that is the
2260: 166:, the number is first expressed in the following form: 2435:, an algebraic generalization of "order of magnitude" 2020: 1994: 1943: 1908: 814: 773: 751: 725: 693: 670: 640: 613: 586: 545: 504: 387: 346: 324: 294: 262: 217: 175: 152: 141: 2263:units in the table at right are used together with 119:in order of magnitude can be measured on a base-10 2009: 1977: 1929: 1854: 820: 798: 757: 731: 711: 676: 652: 626: 599: 572: 531: 393: 371: 330: 300: 280: 248: 200: 158: 1885: 1865:An order of magnitude is an approximation of the 1839:is 8, whereas the nearest order of magnitude for 3077: 2057:than in the first chapter) are not names of the 1898:in which each increasing level is brighter by a 1766:Calculating the order of magnitude by truncation 1800: 2709: 2291:Mathematical operators and symbols in Unicode 281:{\displaystyle 0.316\lesssim a\lesssim 3.16} 2716: 2702: 1930:{\displaystyle {\sqrt{100}}\approx 2.512} 87:Learn how and when to remove this message 106:of numbers in relation to one another. 2723: 2562: 664:within the range of possible values of 14: 3078: 2587: 1978:{\displaystyle ({\sqrt{100}})^{5}=100} 2697: 2423:Unicode symbols for CJK Compatibility 934:ten times different in quantity than 687:Some use a simpler definition where 36: 1778:part of the logarithm, obtained by 32:Order of magnitude (disambiguation) 27:Scale of numbers with a fixed ratio 24: 2671:– from Digital Nature Agency 2635: 2516:. British Broadcasting Corporation 2485: 2311:Orders of magnitude (acceleration) 2021:Base 1,000,000 orders of magnitude 942:over time, most recently in 2022. 607:, meaning that a value of exactly 142:Calculating the Order of Magnitude 25: 3107: 3096:Logarithmic scales of measurement 2652: 2391:Orders of magnitude (temperature) 2351:Orders of magnitude (illuminance) 102:is a concept used to discuss the 3060: 3059: 2991:Earth's location in the Universe 2919:Back-of-the-envelope calculation 799:{\displaystyle N=a\times 10^{b}} 372:{\displaystyle N=a\times 10^{b}} 201:{\displaystyle N=a\times 10^{b}} 41: 2924:Best-selling electronic devices 2381:Orders of magnitude (radiation) 2346:Orders of magnitude (frequency) 1855:Non-decimal orders of magnitude 2581: 2556: 2528: 2506: 2479: 2451: 2376:Orders of magnitude (pressure) 2321:Orders of magnitude (bit rate) 1960: 1944: 1886:Irrational orders of magnitude 712:{\displaystyle 0.5\leq a<5} 13: 1: 2665:10 meters to universe size 10 2444: 2401:Orders of magnitude (voltage) 2366:Orders of magnitude (numbers) 2326:Orders of magnitude (current) 132:Orders of magnitude (numbers) 2986:Astronomical system of units 2659:The Scale of the Universe 2 2406:Orders of magnitude (volume) 2356:Orders of magnitude (length) 2336:Orders of magnitude (energy) 2010:{\displaystyle {\sqrt{100}}} 7: 2689:What is Order of Magnitude? 2681:at 10 meters and ends with 2646:The Measure of the Universe 2386:Orders of magnitude (speed) 2371:Orders of magnitude (power) 2341:Orders of magnitude (force) 2274: 2253: 2250: 2247: 2242: 2234: 2231: 2228: 2223: 2215: 2212: 2189: 2184: 2176: 2173: 2156: 2151: 2143: 2140: 2129: 2124: 1801:Order-of-magnitude estimate 1782:. For example, the number 1697: 1651: 1608: 1568: 1531: 1497: 1466: 1438: 1413: 1394: 1377: 1360: 1340: 1323: 1306: 1289: 1267: 1235: 1207: 1176: 1142: 1105: 1065: 1022: 976: 138:more difficult to compute. 67:the claims made and adding 10: 3112: 2602:10.1038/d41586-022-03747-9 2588:Gibney, Elizabeth (2022). 2396:Orders of magnitude (time) 2361:Orders of magnitude (mass) 2331:Orders of magnitude (data) 2316:Orders of magnitude (area) 1858: 1849:zeroth order approximation 573:{\displaystyle 10^{b+1/2}} 532:{\displaystyle 10^{b-1/2}} 29: 3055: 3037:The Scale of the Universe 2978: 2911: 2731: 2269:The IEC standard prefixes 1875:Logarithmic distributions 660:) represents a geometric 2439:Scale (analytical tool) 921: 3091:Elementary mathematics 3005:To the Moon and Beyond 2873:Specific heat capacity 2661:Interactive tool from 2301:Names of small numbers 2296:Names of large numbers 2011: 1979: 1931: 822: 800: 759: 733: 713: 678: 654: 628: 627:{\displaystyle 10^{b}} 601: 600:{\displaystyle 10^{b}} 574: 533: 395: 373: 332: 302: 282: 250: 202: 160: 3023:(1968 and 1977 films) 2514:"Orders of magnitude" 2488:"Orders of Magnitude" 2012: 1980: 1932: 823: 801: 760: 734: 714: 679: 655: 629: 602: 575: 534: 396: 374: 333: 303: 283: 251: 203: 161: 2536:"Order of Magnitude" 2459:"Order of Magnitude" 1992: 1941: 1906: 812: 771: 749: 723: 691: 668: 638: 611: 584: 543: 502: 385: 344: 322: 292: 260: 215: 173: 150: 30:For other uses, see 3086:Orders of magnitude 2725:Orders of magnitude 2683:subatomic particles 2433:Valuation (algebra) 2418:Scientific notation 808:Order of magnitude 653:{\displaystyle a=1} 381:Order of magnitude 256:, or approximately 18:Orders of magnitude 3031:(1996 documentary) 2960:Metric (SI) prefix 2086:Order of magnitude 2007: 1975: 1927: 818: 796: 755: 729: 709: 674: 650: 624: 597: 570: 529: 391: 369: 328: 298: 278: 246: 198: 156: 100:Order of magnitude 52:possibly contains 3073: 3072: 2970:Microscopic scale 2965:Macroscopic scale 2540:Wolfram MathWorld 2494:on 22 August 2018 2463:Wolfram MathWorld 2258: 2257: 2005: 1957: 1919: 1861:Logarithmic scale 1795:logarithmic scale 1774:, usually as the 1763: 1762: 919: 918: 821:{\displaystyle b} 758:{\displaystyle N} 732:{\displaystyle b} 677:{\displaystyle a} 492: 491: 394:{\displaystyle b} 331:{\displaystyle N} 301:{\displaystyle b} 244: 228: 227: 159:{\displaystyle N} 121:logarithmic scale 97: 96: 89: 54:original research 16:(Redirected from 3103: 3063: 3062: 2744:Angular momentum 2718: 2711: 2704: 2695: 2694: 2629: 2628: 2626: 2624: 2585: 2579: 2578: 2576: 2575: 2560: 2554: 2553: 2548: 2546: 2532: 2526: 2525: 2523: 2521: 2510: 2504: 2503: 2501: 2499: 2490:. 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Then, 2979:Related 2938:decades 2898:Voltage 2843:Numbers 2793:Entropy 2779:Density 2769:Current 2648:(1983). 2286:Decibel 2141:million 2063:numbers 2027:decimal 1819:billion 1813:of the 1807:rounded 1776:integer 1338:  745:Number 634:(i.e., 318:Number 310:integer 125:decades 59:Please 3048:(2012) 3040:(2010) 2903:Volume 2888:Torque 2813:Length 2784:Energy 2759:Charge 2616:  2608:  2594:Nature 2244:100000 2099:Is log 1900:factor 1880:binary 1318:tenth 1290:0.001 1259:micro- 912:1 × 10 846:1 × 10 835:2 × 10 485:1 × 10 419:1 × 10 408:2 × 10 211:where 2948:100th 2878:Speed 2848:Power 2798:Force 2614:S2CID 2498:9 May 2225:10000 2080:etc. 1925:2.512 1815:Earth 1751:Power 1358:(da) 1356:deca- 1307:0.01 1269:0.000 1237:0.000 1209:0.000 1178:0.000 1144:0.000 1107:0.000 1067:0.000 1024:0.000 978:0.000 958:Power 932:about 264:0.316 104:scale 2943:10th 2936:and 2893:Time 2833:Mass 2774:Data 2749:Area 2625:2022 2606:PMID 2547:2017 2522:2024 2500:2013 2470:2024 2186:1000 1871:base 1396:1000 1378:100 1352:ten 1335:one 1324:0.1 1225:−12 1197:−15 1166:−18 1132:−21 1095:−24 1055:−27 1012:−30 922:Uses 909:1000 704:< 539:and 494:The 482:1000 276:3.16 237:< 123:in " 2598:doi 2208:000 2205:000 2202:000 2199:000 2196:000 2193:000 2169:000 2166:000 2163:000 2160:000 2153:100 2136:000 2133:000 2107:000 2104:000 2091:log 2089:Is 2077:000 2074:000 2071:000 2068:000 2050:000 2047:000 2040:000 2037:000 1998:100 1973:100 1950:100 1912:100 1902:of 1841:3.7 1833:1.7 1828:000 1825:000 1789:000 1786:000 1736:30 1733:10 1728:000 1725:000 1722:000 1719:000 1716:000 1713:000 1710:000 1707:000 1704:000 1701:000 1687:27 1684:10 1679:000 1676:000 1673:000 1670:000 1667:000 1664:000 1661:000 1658:000 1655:000 1641:24 1638:10 1633:000 1630:000 1627:000 1624:000 1621:000 1618:000 1615:000 1612:000 1598:21 1595:10 1590:000 1587:000 1584:000 1581:000 1578:000 1575:000 1572:000 1558:18 1555:10 1550:000 1547:000 1544:000 1541:000 1538:000 1535:000 1521:15 1518:10 1513:000 1510:000 1507:000 1504:000 1501:000 1487:12 1484:10 1479:000 1476:000 1473:000 1470:000 1453:10 1448:000 1445:000 1442:000 1425:10 1420:000 1417:000 1400:10 1381:10 1364:10 1361:10 1344:10 1330:−1 1327:10 1313:−2 1310:10 1296:−3 1293:10 1279:−6 1276:10 1271:001 1250:−9 1247:10 1242:001 1239:000 1222:10 1217:001 1214:000 1211:000 1194:10 1189:001 1186:000 1183:000 1180:000 1163:10 1158:001 1155:000 1152:000 1149:000 1146:000 1129:10 1124:001 1121:000 1118:000 1115:000 1112:000 1109:000 1092:10 1087:001 1084:000 1081:000 1078:000 1075:000 1072:000 1069:000 1052:10 1047:001 1044:000 1041:000 1038:000 1035:000 1032:000 1029:000 1026:000 1009:10 1004:001 1001:000 998:000 995:000 992:000 989:000 986:000 983:000 980:000 930:is 898:999 838:−1 832:0.2 695:0.5 580:is 498:of 471:999 411:−1 405:0.2 63:by 3082:: 2644:, 2612:. 2604:. 2596:. 2592:. 2567:. 2549:. 2538:. 2472:. 2461:. 2427:SI 2261:SI 2126:10 2111:of 2096:of 2093:10 2017:. 1851:. 1845:10 1837:10 1797:. 1456:9 1428:6 1403:3 1384:2 1367:1 1347:0 1341:1 1265:) 915:3 904:3 893:1 887:32 882:1 876:31 871:1 860:1 849:0 788:10 684:. 616:10 589:10 548:10 507:10 488:3 477:3 466:2 460:32 455:1 449:31 444:1 433:1 422:0 361:10 242:10 225:10 190:10 134:. 2717:e 2710:t 2703:v 2627:. 2600:: 2577:. 2524:. 2502:. 2240:5 2221:4 2191:1 2182:3 2158:1 2149:2 2131:1 2122:1 2102:1 2066:1 2045:1 2035:1 2002:5 1970:= 1965:5 1961:) 1954:5 1945:( 1916:5 1843:× 1835:× 1823:4 1784:4 1699:1 1653:1 1610:1 1570:1 1533:1 1499:1 1468:1 1440:1 1415:1 1263:μ 1261:( 936:y 928:x 865:6 854:5 843:1 816:b 792:b 781:a 778:= 775:N 753:N 727:b 707:5 701:a 672:a 648:1 645:= 642:a 620:b 593:b 566:2 562:/ 558:1 555:+ 552:b 525:2 521:/ 517:1 511:b 438:6 427:5 416:1 389:b 365:b 354:a 351:= 348:N 326:N 296:b 270:a 234:a 221:1 194:b 183:a 180:= 177:N 154:N 90:) 84:( 79:) 75:( 57:. 34:. 20:)

Index

Orders of magnitude
Order of magnitude (disambiguation)
original research
improve it
verifying
inline citations
Learn how and when to remove this message
Differences
logarithmic scale
decades
Orders of magnitude (numbers)
integer
geometric mean
SI prefixes
Power
micro-
μ
deca-
Power
common logarithm
integer
truncation
logarithmic scale
rounded
population
Earth
billion
zeroth order approximation
Logarithmic scale
logarithm

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