Powers of 2

Powers of 2 are the numbers you get by doubling, starting from 1: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1,024. 2 to the power of 5, written 2⁵, means five 2s multiplied together: 2 × 2 × 2 × 2 × 2 = 32. You keep meeting these numbers in computing because every extra bit doubles how many values a computer can store, which is why 256, 1,024 and 65,536 turn up so often.

Key takeaways

	A power of 2 is 2 multiplied by itself. Each step doubles the last: 1, 2, 4, 8, 16.
	n bits give 2 to the power of n patterns, which is why 256 and 1,024 keep turning up.
	In binary every power of 2 is a 1 followed by zeros, and one less is a row of 1s: 255 is 11111111.
	1 KiB is 1,024 bytes, but drive makers count 1 KB as 1,000, so a 1 TB drive shows about 931 GB.
	2 to the 10th is 1,024, close to 1,000, so every 10 steps in the exponent adds about three zeros.

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The values you will meet most often:

Power	Value	Power	Value
20	1	27	128
21	2	28	256
22	4	29	512
23	8	210	1,024
24	16	212	4,096
25	32	216	65,536
26	64	220	1,048,576

Work out any power of 2

Type an exponent and the box shows the exact value, how many digits it has and what it looks like in binary. It handles whole numbers from -64 up to 1,000.

Powers of 2 chart from 0 to 64

The full list up to 2⁶⁴, with a note wherever the number shows up in computing.

Power	Value	Where you meet it
20	1	any number to the power of 0 is 1
21	2	one bit holds 2 values
22	4
23	8	one octal digit
24	16	one hex digit
25	32
26	64
27	128	ASCII has 128 codes
28	256	values in one byte
29	512
210	1,024	1 KiB in bytes
211	2,048
212	4,096
213	8,192
214	16,384
215	32,768	32,767 is the top of a signed 16-bit number
216	65,536	values in 16 bits
217	131,072
218	262,144
219	524,288
220	1,048,576	1 MiB in bytes
221	2,097,152
222	4,194,304
223	8,388,608
224	16,777,216	colors in 24-bit RGB
225	33,554,432
226	67,108,864
227	134,217,728
228	268,435,456
229	536,870,912
230	1,073,741,824	1 GiB in bytes
231	2,147,483,648	2,147,483,647 is the top of a signed 32-bit number
232	4,294,967,296	IPv4 addresses
233	8,589,934,592
234	17,179,869,184
235	34,359,738,368
236	68,719,476,736
237	137,438,953,472
238	274,877,906,944
239	549,755,813,888
240	1,099,511,627,776	1 TiB in bytes
241	2,199,023,255,552
242	4,398,046,511,104
243	8,796,093,022,208
244	17,592,186,044,416
245	35,184,372,088,832
246	70,368,744,177,664
247	140,737,488,355,328
248	281,474,976,710,656
249	562,949,953,421,312
250	1,125,899,906,842,624	1 PiB in bytes
251	2,251,799,813,685,248
252	4,503,599,627,370,496
253	9,007,199,254,740,992	JavaScript counts every whole number exactly up to here
254	18,014,398,509,481,984
255	36,028,797,018,963,968
256	72,057,594,037,927,936
257	144,115,188,075,855,872
258	288,230,376,151,711,744
259	576,460,752,303,423,488
260	1,152,921,504,606,846,976	1 EiB in bytes
261	2,305,843,009,213,693,952
262	4,611,686,018,427,387,904
263	9,223,372,036,854,775,808	top of a signed 64-bit number is this minus 1
264	18,446,744,073,709,551,616	values in 64 bits

Why powers of 2 matter in computing

One bit has two states. Two bits have four patterns, three bits have eight, and in general n bits have 2ⁿ patterns. That one fact explains most of the "odd" numbers in computing.

• A byte is 8 bits, so it holds 2⁸ = 256 values, from 0 to 255. That is why color channels and IP address parts stop at 255.
• A 24-bit color uses 8 bits each for red, green and blue, which gives 2²⁴ = 16,777,216 colors.
• IPv4 addresses are 32 bits long, so there can be at most 2³² = 4,294,967,296 of them. The world ran short, which is why IPv6 uses 128 bits.
• A signed 32-bit number tops out at 2³¹ - 1 = 2,147,483,647. YouTube changed its view counter in 2014 after Gangnam Style got close to that number.
• Many systems count time as seconds since 1 January 1970 in a signed 32-bit number. It runs out at 03:14:07 UTC on 19 January 2038, a date known as the Year 2038 problem.

Powers of 2 in binary

In binary, every power of 2 is a 1 followed by zeros. 2³ = 8 is 1000 and 2⁶ = 64 is 1000000. The exponent tells you how many zeros come after the 1, the same way 10³ = 1000 in decimal.

Take one away and you get a row of 1s. 2⁸ - 1 = 255 is 11111111, and 2⁴ - 1 = 15 is 1111. This is also why the place values in a byte are 128, 64, 32, 16, 8, 4, 2 and 1: each one is a power of 2. Our guide on how to read binary uses those place values to decode letters, and the binary numbers chart shows every value up to 1023.

Powers of 2 and file sizes

Storage is where powers of 2 cause the most confusion. Memory chips are built in powers of 2, so 1 KiB (kibibyte, one of the IEC binary prefixes) is 2¹⁰ = 1,024 bytes, 1 MiB is 2²⁰ bytes and 1 GiB is 2³⁰ = 1,073,741,824 bytes. Drive makers use powers of 10 instead, so their 1 GB is exactly 1,000,000,000 bytes.

Binary unit	Bytes	Decimal unit	Bytes
1 KiB = 210	1,024	1 KB = 103	1,000
1 MiB = 220	1,048,576	1 MB = 106	1,000,000
1 GiB = 230	1,073,741,824	1 GB = 109	1,000,000,000
1 TiB = 240	1,099,511,627,776	1 TB = 1012	1,000,000,000,000

Windows counts in binary units but labels them KB, MB and GB, so a 1 TB drive shows up as about 931 GB. Nothing is missing: 1,000,000,000,000 divided by 1,073,741,824 is 931.3, so it is the same number of bytes counted in bigger units. The GiB to GB converter and the byte converter switch between the two systems.

How to tell if a number is a power of 2

Write the number in binary. If it has exactly one 1, it is a power of 2. 64 is 1000000, so it is. 96 is 1100000, with two 1s, so it is not.

Programmers use a shortcut based on the same idea. Subtracting 1 from a power of 2 flips its single 1 to 0 and every 0 after it to 1, so the two numbers share no 1 bits. In code that check is n > 0 and (n & (n - 1)) == 0. The n > 0 part is there because 0 would also give 0 and wrongly pass. For 64 the AND compares 1000000 with 0111111 and gets 0. The binary calculator lets you try the AND step yourself.

2 to the power of 0, and negative powers

2⁰ = 1. Walk down the list and each step halves the number: 2³ = 8, 2² = 4, 2¹ = 2. The next step, 2⁰, is 2 halved, which is 1. Keep halving and you reach the negative powers.

Power	Fraction	Decimal	Binary
2-1	1/2	0.5	0.1
2-2	1/4	0.25	0.01
2-3	1/8	0.125	0.001
2-4	1/16	0.0625	0.0001

Computers store fractions as sums of these values. A number like 0.1 cannot be written as an exact sum of them, so it is stored as a close approximation, as Python's floating-point tutorial explains. That is why 0.1 + 0.2 shows up as 0.30000000000000004 in many programming languages. The IEEE 754 converter shows the exact bits a computer keeps for any decimal.

A quick way to estimate big powers of 2

2¹⁰ = 1,024 is close to 1,000. So every 10 steps in the exponent adds roughly three zeros: 2²⁰ is about a million, 2³⁰ about a billion and 2⁴⁰ about a trillion. To estimate 2³⁵, split it into 2³⁰ × 2⁵, about a billion times 32, which gives about 32 billion. The exact answer is 34,359,738,368.

Questions people ask

What is 2 to the power of 10?

2¹⁰ = 1,024. It is the number of bytes in a kibibyte and the reason a "kilobyte" in Windows is 1,024 bytes.

What is 2 to the power of 0?

2⁰ = 1. Any number other than 0 raised to the power of 0 equals 1.

Is 1 a power of 2?

Yes. 1 = 2⁰. In binary it is a single 1 with no zeros after it.

Is 0 a power of 2?

No. You can halve 2 as many times as you like and the result gets smaller, but it never reaches 0.

What is the largest power of 2 that fits in 64 bits?

2⁶³ = 9,223,372,036,854,775,808. An unsigned 64-bit number goes up to 2⁶⁴ - 1, which is all 64 bits set to 1.

Why do computers use powers of 2?

Each bit doubles the number of values a computer can store. Memory addresses, data types and file sizes are all built from whole numbers of bits, so their limits land on powers of 2.

About the authors

Written byZachary PainterTechnical writer at GitLab

Zachary Painter is a technical writer at GitLab, where he writes developer documentation and UI text. He has written API documentation for REST and GraphQL APIs and reference docs for Kubernetes, Docker and command-line tools, earlier as a technical writer at Pomerium and a senior technical content writer at Stream. He holds a BA in English and German studies from the University of North Carolina at Greensboro. On binarytranslator.ai he writes guides and the how-to sections on tool pages.

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Reviewed bySam SiewertProfessor of computer science, California State University, Chico

Sam Siewert is the O'Connell Endowed Professor of computer science at California State University, Chico, where he teaches numeric and parallel computing, computer vision and machine learning. He has taught real-time embedded systems at the University of Colorado Boulder since 2000 and co-founded its Embedded Systems Engineering program. Both fields depend on how computers store numbers in binary, from fixed-width integers to floating point. He earned his PhD and MS in computer science at the University of Colorado Boulder and is a senior member of IEEE. On binarytranslator.ai he reviews the math behind the converters and the number system guides.

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