2的冪

2的冪是指符合 $2^{n}$ 型式，而 $n$ 也是整數的數，也就是底數為2，指數為整數 $n$ 的冪。

從1到1024（2⁰ 至 2¹⁰）

在有些情形下，會將 $n$ 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

$2^{n}$
$2\uparrow n$
$2[3]n$
2 ^ n
2 ** n
power(2, n)
2的n次冪
2的n次方

與2的冪有關的數字

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（ $3=4-1=2^{2}-1$ ）。
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（ $3=2+1=2^{1}+1$ ）。
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

$2^{n},n\in [0,63],n\in Z$

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

2^2⁰ = 2¹ = 2

2^2¹ =2² = 4

2^2² =2⁴ = 16

2^2³ =2⁸ = 256

2^2⁴ =2¹⁶ = 65,536

2^2⁵ =2³² = 4,294,967,296

2^2⁶ =2⁶⁴ = 18,446,744,073,709,551,616

2^2⁷ =2¹²⁸ = 340,282,366,920,938,463,463,374,607,431,768,211,456

2^2⁸ =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

^ 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.

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