2的幂

时间:2020-03-26 18:18:01

从1到1024(2 至 2)

2的幂是指符合 2 n {displaystyle 2^{n}} 型式,而 n {displaystyle n} 也是整数的数,也就是底数为2,指数为整数 n的幂。

在有些情形下,会将 n {displaystyle n} 限制在正整数及零的范围内[1],因此2的幂包括1、2以及2自乘多次的乘积。

因为2是二进制的底数,因此在常出现二进制的计算机科学中,2的幂也很常见。若将2的幂用二进制表示,会是100…000、0.00…001或是1的形式,类似用十进制表示10的幂英语Power of 10的情形。

表示方法

  • 2 n {displaystyle 2^{n}}
  • 2 n {displaystyle 2uparrow n}
  • 2 [ 3 ] n {displaystyle 2[3]n}
  • 2 ^ n
  • 2 ** n
  • power(2, n)
  • 2的n次幂
  • 2的n次方

与2的幂有关的数字

  • 比某一个2的幂小1的素数在数学上被称为梅森素数;例如数字3是最小的梅森素数( 3 = 4 1 = 2 2 1 {displaystyle 3=4-1=2^{2}-1} )。
  • 比某一个2的幂大1的素数在数学上被称为费马素数;如数字3也是最小的费马素数( 3 = 2 + 1 = 2 1 + 1 {displaystyle 3=2+1=2^{1}+1} )。
  • 一个以2的幂为分母的分数被称为二进有理数。
  • 可以表示为连续正整数的和的数被称为礼貌数,这些数则是完全不含2的幂的数。

2的幂的列表(部分)

2的正指数幂

2 n , n [ 0 , 101 ] , n Z {displaystyle 2^{n},nin [0,101],nin Z}

2=12=4,0962=16,777,2162=68,719,476,7362=281,474,976,710,6562=1,152,921,504,606,846,9762=4,722,366,482,869,645,213,6962=19,342,813,113,834,066,795,298,8162=79,228,162,514,264,337,593,540,590,336
2=22=8,1922=33,554,4322=137,438,953,4722=562,949,953,421,3122=2,305,843,009,213,693,9522=9,444,732,965,739,290,427,3922=38.685,626,227,668,133,590,597,6322=158,456,325,028,528,675,187,087,900,672
2=42=16,3842=67,108,8642=274,877,906,9442=1,125,899,906,842,6242=4,611,686,018,427,387,9042=18,889,465,931,478,580,854,7842=77,371,252,455,336,267,181,195,2642=316,912,650,057,057,350,374,175,081,344
2=82=32,7682=134,217,7282=549,755,813,8882=2,251,799,813,685,24829,223,372,036,854,775,8082=37,778,931,862,957,161,709,5682=154,742,504,910,672,534,362,390,5282=633,825,300,114,114,700,748,351,602,688
2=162=65,5362=268,435,4562=1,099,511,627,7762=4,503,599,627,370,4962=18,446,744,073,709,551,6162=75,557,863,725,914,323,419,1362=309,485,009,821,345,068,724,781,0562=1,267,650,600,228,229,401,496,703,205,376
2=322=131,0722=536,870,9122=2,199,023,255,5522=9,007,199,254,740,9922=36,893,488,147,419,103,2322=151,115,727,451,828,646,838,2722=618,970,019,642,690,137,449,562,1122=2,535,351,200,456,458,802,993,306,410,752
2=642=262,1442=1,073,741,8242=4,398,046,511,1042=18,014,398,509,481,9842=73,786,976,294,838,206,4642=302,231,454,903,657,293,676,5442=1,237,940,039,285,380,274,899,124,224
2=1282=524,2882=2,147,483,6482=8,796,093,022,2082=36,028,797,018,963,9682=147,573,952,589,676,412,9282=604,462,909,817,314,587,353,0882=2,475,880,078,570,760,549,798,248,448
2=2562=1,048,5762=4,294,967,2962=17,592,186,044,4162=72,057,594,037,927,9362=295,147,905,179,352,825,8562=1,208,925,819,614,629,174,706,1762=4,951,760,157,141,521,099,596,496,896
2=5122=2,097,1522=8,589,934,5922=35,184,372,088,8322=144,115,188,075,855,8722=590,295,810,358,705,651,7122=2,417,851,639,229,258,349,412,3522=9,903,520,314,283,042,199192,993,792
2=1,0242=4,194,3042=17,179,869,1842=70,368,744,177,6642=288,230,376,151,711,7442=1,180,591,620,717,411,303,4242=4,835,703,278,458,516,698,824,704219,807,040,628,566,084,398,385,987,584
2=2,0482=8,388,6082=34,359,738,3682=140,737,488,355,3282=576,460,752,303,423,4882=2,361,183,241,434,822,606,8482=9,671,406,556,917,033,397,649,408239,614,081,257,132,168,796,771,975,168
2 = 2
2 = 4
2 = 16
2 = 256
2 = 65,536
2 = 4,294,967,296
2 = 18,446,744,073,709,551,616
2 = 340,282,366,920,938,463,463,374,607,431,768,211,456
2 = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936

2的负指数幂

2的首60个负指数幂

2 = 0.5

2 = 0.25

2 = 0.125

2 = 0.0625

2 = 0.03125

2 = 0.015625

2 = 0.0078125

2 = 0.00390625

2 = 0.001953125

2 = 0.0009765625

2 = 0.00048828125

2 = 0.000244140625

2 = 0.0001220703125

2 = 0.00006103515625

2 = 0.000030517578125

2 = 0.0000152587890625

2 = 0.00000762939453125

2 = 0.000003814697265625

2 = 0.0000019073486328125

2 = 0.00000095367431640625

2 = 0.000000476837158203125

2 = 0.0000002384185791015625

2 = 0.00000011920928955078125

2 = 0.000000059604644775390625

2 = 0.0000000298023223876953125

2 = 0.00000001490116119384765625

2 = 0.000000007450580596923828125

2 = 0.0000000037252902984619140625

2 = 0.00000000186264514923095703125

2 = 0.000000000931322574615478515625

2 = 0.0000000004656612873077392578125

2 = 0.00000000023283064365386962890625

2 = 0.000000000116415321826934814453125

2 = 0.0000000000582076609134674072265625

2 = 0.00000000002910383045673370361328125

2 = 0.000000000014551915228366851806640625

2 = 0.0000000000072759576141834259033203125

2 = 0.00000000000363797880709171295166015625

2 = 0.000000000001818989403545856475830078125

2 = 0.0000000000009094947017729282379150390625

2 = 0.00000000000045474735088646411895751953125

2 = 0.000000000000227373675443232059478759765625

2 = 0.0000000000001136868377216160297393798828125

2 = 0.00000000000005684341886080801486968994140625

2 = 0.000000000000028421709430404007434844970703125

2 = 0.0000000000000142108547152020037174224853515625

2 = 0.00000000000000710542735760100185871124267578125

2 = 0.000000000000003552713678800500929355621337890625

2 = 0.0000000000000017763568394002504646778106689453125

2 = 0.00000000000000088817841970012523233890533447265625

2 = 0.000000000000000444089209850062616169452667236328125

2 = 0.0000000000000002220446049250313080847263336181640625

2 = 0.00000000000000011102230246251565404236316680908203125

2 = 0.000000000000000055511151231257827021181583404541015625

2 = 0.0000000000000000277555756156289135105907917022705078125

2 = 0.00000000000000001387778780781445675529539585113525390625

2 = 0.000000000000000006938893903907228377647697925567626953125

2 = 0.0000000000000000034694469519536141888238489627838134765625

2 = 0.00000000000000000173472347597680709441192448139190673828125

2 = 0.000000000000000000867361737988403547205962240695953369140625

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