2的冪
2的冪是指符合型式，而也是整數的數，也就是底數為2，指數為整數 n的冪。
在有些情形下，會將限制在正整數及零的範圍內[1]，因此2的冪包括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的冪也很常見。若將2的冪用二進制表示，會是100…000、0.00…001或是1的形式，類似用十進制表示10的冪的情形。
表示方法
[編輯]2 ^ n2 ** npower(2, n)- 2的n次冪
- 2的n次方
![{\displaystyle 2[3]n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a0352feee1240f95d642d45b400afd6f5a9c7b)
與2的冪有關的數字
[編輯]- 比某一個2的冪小1的質數，在數學上稱為梅森質數；例如數字3是最小的梅森質數（）。
- 比某一個2的冪大1的質數，在數學上稱為費馬質數；如數字3也是最小的費馬質數（）。
- 一個以2的冪為分母的分數稱為二進有理數。
- 可以表示為連續正整數和的數稱為禮貌數，2的冪不會是禮貌數。
2的冪列表
[編輯]
![{\displaystyle 2^{n},n\in [0,63],n\in Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a49a0517716c72ac6467102e83acc611fef31b27)
| 20 | = | 1 | 216 | = | 65,536 | 232 | = | 4,294,967,296 | 248 | = | 281,474,976,710,656 | |||
| 21 | = | 2 | 217 | = | 131,072 | 233 | = | 8,589,934,592 | 249 | = | 562,949,953,421,312 | |||
| 22 | = | 4 | 218 | = | 262,144 | 234 | = | 17,179,869,184 | 250 | = | 1,125,899,906,842,624 | |||
| 23 | = | 8 | 219 | = | 524,288 | 235 | = | 34,359,738,368 | 251 | = | 2,251,799,813,685,248 | |||
| 24 | = | 16 | 220 | = | 1,048,576 | 236 | = | 68,719,476,736 | 252 | = | 4,503,599,627,370,496 | |||
| 25 | = | 32 | 221 | = | 2,097,152 | 237 | = | 137,438,953,472 | 253 | = | 9,007,199,254,740,992 | |||
| 26 | = | 64 | 222 | = | 4,194,304 | 238 | = | 274,877,906,944 | 254 | = | 18,014,398,509,481,984 | |||
| 27 | = | 128 | 223 | = | 8,388,608 | 239 | = | 549,755,813,888 | 255 | = | 36,028,797,018,963,968 | |||
| 28 | = | 256 | 224 | = | 16,777,216 | 240 | = | 1,099,511,627,776 | 256 | = | 72,057,594,037,927,936 | |||
| 29 | = | 512 | 225 | = | 33,554,432 | 241 | = | 2,199,023,255,552 | 257 | = | 144,115,188,075,855,872 | |||
| 210 | = | 1,024 | 226 | = | 67,108,864 | 242 | = | 4,398,046,511,104 | 258 | = | 288,230,376,151,711,744 | |||
| 211 | = | 2,048 | 227 | = | 134,217,728 | 243 | = | 8,796,093,022,208 | 259 | = | 576,460,752,303,423,488 | |||
| 212 | = | 4,096 | 228 | = | 268,435,456 | 244 | = | 17,592,186,044,416 | 260 | = | 1,152,921,504,606,846,976 | |||
| 213 | = | 8,192 | 229 | = | 536,870,912 | 245 | = | 35,184,372,088,832 | 261 | = | 2,305,843,009,213,693,952 | |||
| 214 | = | 16,384 | 230 | = | 1,073,741,824 | 246 | = | 70,368,744,177,664 | 262 | = | 4,611,686,018,427,387,904 | |||
| 215 | = | 32,768 | 231 | = | 2,147,483,648 | 247 | = | 140,737,488,355,328 | 263 | = | 9,223,372,036,854,775,808 |
2的2的冪次方列表
[編輯]- 220 = 21 = 2
- 221 =22 = 4
- 222 =24 = 16
- 223 =28 = 256
- 224 =216 = 65,536
- 225 =232 = 4,294,967,296
- 226 =264 = 18,446,744,073,709,551,616
- 227 =2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456
- 228 =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
參考資料
[編輯]- ^ Lipschutz, Seymour. Schaum's Outline of Theory and Problems of Essential Computer Mathematics. New York: McGraw-Hill. 1982: 3. ISBN 0-07-037990-4.
- ^ Sewell, Michael J. Mathematics Masterclasses. Oxford: Oxford University Press. 1997: 78. ISBN 0-19-851494-8.
