2的幂

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從1到1024(20 至 210

2的幂是指符合型式,而也是整數的數,也就是底數2指數為整數 n

在有些情形下,會將限制在正整數及零的範圍內[1],因此2的幂包括1、2以及2自乘多次的乘積[2]

因為2是二進制的底數,因此在常出現二進制的電腦科學中,2的幂也很常見。若將2的幂用二進制表示,會是100…000、0.00…001或是1的形式,類似用十進制表示10的幂英语Power of 10的情形。

表示方法[编辑]

  • 2 ^ n
  • 2 ** n
  • power(2, n)
  • 2的n次幂
  • 2的n次方

與2的冪有關的數字[编辑]

  • 比某一个21素数数学上被称为梅森素数;例如数字3是最小的梅森素数()。
  • 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()。
  • 一个以2的幂为分母分数被称为二进有理数
  • 可以表示为连续正整数被称为礼貌数,这些数则是完全不含2的幂的数。

2的幂的列表(部分)[编辑]

2的正指数幂[编辑]

20 = 1 212 = 4,096 224 = 16,777,216 236 = 68,719,476,736 248 = 281,474,976,710,656 260 = 1,152,921,504,606,846,976 272 = 4,722,366,482,869,645,213,696 284 = 19,342,813,113,834,066,795,298,816 296 = 79,228,162,514,264,337,593,540,590,336
21 = 2 213 = 8,192 225 = 33,554,432 237 = 137,438,953,472 249 = 562,949,953,421,312 261 = 2,305,843,009,213,693,952 273 = 9,444,732,965,739,290,427,392 285 = 38.685,626,227,668,133,590,597,632 297 = 158,456,325,028,528,675,187,087,900,672
22 = 4 214 = 16,384 226 = 67,108,864 238 = 274,877,906,944 250 = 1,125,899,906,842,624 262 = 4,611,686,018,427,387,904 274 = 18,889,465,931,478,580,854,784 286 = 77,371,252,455,336,267,181,195,264 298 = 316,912,650,057,057,350,374,175,081,344
23 = 8 215 = 32,768 227 = 134,217,728 239 = 549,755,813,888 251 = 2,251,799,813,685,248 263 9,223,372,036,854,775,808 275 = 37,778,931,862,957,161,709,568 287 = 154,742,504,910,672,534,362,390,528 299 = 633,825,300,114,114,700,748,351,602,688
24 = 16 216 = 65,536 228 = 268,435,456 240 = 1,099,511,627,776 252 = 4,503,599,627,370,496 264 = 18,446,744,073,709,551,616 276 = 75,557,863,725,914,323,419,136 288 = 309,485,009,821,345,068,724,781,056 2100 = 1,267,650,600,228,229,401,496,703,205,376
25 = 32 217 = 131,072 229 = 536,870,912 241 = 2,199,023,255,552 253 = 9,007,199,254,740,992 265 = 36,893,488,147,419,103,232 277 = 151,115,727,451,828,646,838,272 289 = 618,970,019,642,690,137,449,562,112 2101 = 2,535,351,200,456,458,802,993,306,410,752
26 = 64 218 = 262,144 230 = 1,073,741,824 242 = 4,398,046,511,104 254 = 18,014,398,509,481,984 266 = 73,786,976,294,838,206,464 278 = 302,231,454,903,657,293,676,544 290 = 1,237,940,039,285,380,274,899,124,224
27 = 128 219 = 524,288 231 = 2,147,483,648 243 = 8,796,093,022,208 255 = 36,028,797,018,963,968 267 = 147,573,952,589,676,412,928 279 = 604,462,909,817,314,587,353,088 291 = 2,475,880,078,570,760,549,798,248,448
28 = 256 220 = 1,048,576 232 = 4,294,967,296 244 = 17,592,186,044,416 256 = 72,057,594,037,927,936 268 = 295,147,905,179,352,825,856 280 = 1,208,925,819,614,629,174,706,176 292 = 4,951,760,157,141,521,099,596,496,896
29 = 512 221 = 2,097,152 233 = 8,589,934,592 245 = 35,184,372,088,832 257 = 144,115,188,075,855,872 269 = 590,295,810,358,705,651,712 281 = 2,417,851,639,229,258,349,412,352 293 = 9,903,520,314,283,042,199192,993,792
210 = 1,024 222 = 4,194,304 234 = 17,179,869,184 246 = 70,368,744,177,664 258 = 288,230,376,151,711,744 270 = 1,180,591,620,717,411,303,424 282 = 4,835,703,278,458,516,698,824,704 294 19,807,040,628,566,084,398,385,987,584
211 = 2,048 223 = 8,388,608 235 = 34,359,738,368 247 = 140,737,488,355,328 259 = 576,460,752,303,423,488 271 = 2,361,183,241,434,822,606,848 283 = 9,671,406,556,917,033,397,649,408 295 39,614,081,257,132,168,796,771,975,168
21 = 2
22 = 4
24 = 16
28 = 256
216 = 65,536
232 = 4,294,967,296
264 = 18,446,744,073,709,551,616
2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456
2256 = 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[编辑]

2的首60个负指数幂

2-1 = 0.5

2-2 = 0.25

2-3 = 0.125

2-4 = 0.0625

2-5 = 0.03125

2-6 = 0.015625

2-7 = 0.0078125

2-8 = 0.00390625

2-9 = 0.001953125

2-10 = 0.0009765625

2-11 = 0.00048828125

2-12 = 0.000244140625

2-13 = 0.0001220703125

2-14 = 0.00006103515625

2-15 = 0.000030517578125

2-16 = 0.0000152587890625

2-17 = 0.00000762939453125

2-18 = 0.000003814697265625

2-19 = 0.0000019073486328125

2-20 = 0.00000095367431640625

2-21 = 0.000000476837158203125

2-22 = 0.0000002384185791015625

2-23 = 0.00000011920928955078125

2-24 = 0.000000059604644775390625

2-25 = 0.0000000298023223876953125

2-26 = 0.00000001490116119384765625

2-27 = 0.000000007450580596923828125

2-28 = 0.0000000037252902984619140625

2-29 = 0.00000000186264514923095703125

2-30 = 0.000000000931322574615478515625

2-31 = 0.0000000004656612873077392578125

2-32 = 0.00000000023283064365386962890625

2-33 = 0.000000000116415321826934814453125

2-34 = 0.0000000000582076609134674072265625

2-35 = 0.00000000002910383045673370361328125

2-36 = 0.000000000014551915228366851806640625

2-37 = 0.0000000000072759576141834259033203125

2-38 = 0.00000000000363797880709171295166015625

2-39 = 0.000000000001818989403545856475830078125

2-40 = 0.0000000000009094947017729282379150390625

2-41 = 0.00000000000045474735088646411895751953125

2-42 = 0.000000000000227373675443232059478759765625

2-43 = 0.0000000000001136868377216160297393798828125

2-44 = 0.00000000000005684341886080801486968994140625

2-45 = 0.000000000000028421709430404007434844970703125

2-46 = 0.0000000000000142108547152020037174224853515625

2-47 = 0.00000000000000710542735760100185871124267578125

2-48 = 0.000000000000003552713678800500929355621337890625

2-49 = 0.0000000000000017763568394002504646778106689453125

2-50 = 0.00000000000000088817841970012523233890533447265625

2-51 = 0.000000000000000444089209850062616169452667236328125

2-52 = 0.0000000000000002220446049250313080847263336181640625

2-53 = 0.00000000000000011102230246251565404236316680908203125

2-54 = 0.000000000000000055511151231257827021181583404541015625

2-55 = 0.0000000000000000277555756156289135105907917022705078125

2-56 = 0.00000000000000001387778780781445675529539585113525390625

2-57 = 0.000000000000000006938893903907228377647697925567626953125

2-58 = 0.0000000000000000034694469519536141888238489627838134765625

2-59 = 0.00000000000000000173472347597680709441192448139190673828125

2-60 = 0.000000000000000000867361737988403547205962240695953369140625

參考資料[编辑]

  1. ^ Lipschutz, Seymour. Schaum's Outline of Theory and Problems of Essential Computer Mathematics. New York: McGraw-Hill. 1982: 3. ISBN 0-07-037990-4. 
  2. ^ Sewell, Michael J. Mathematics Masterclasses. Oxford: Oxford University Press. 1997: 78. ISBN 0-19-851494-8. 

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