Powers of 2

The fundamental numbers of computing and digital technology

The powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...) are the foundation of all modern computing. Each bit in a computer can be 0 or 1, which makes powers of 2 omnipresent in digital technology.

Why are powers of 2 important?

Computers operate using the binary system, where all information is represented using only two digits: 0 and 1. Each position in a binary number represents a power of 2, just as each position in the decimal system represents a power of 10.

For this reason, memory capacities, storage, and many technical parameters are expressed in powers of 2:

1 Kilobyte (KB) 210 = 1,024 bytes
1 Megabyte (MB) 220 = 1,048,576 bytes
1 Gigabyte (GB) 230 = 1,073,741,824 bytes
Colores en pantalla (RGB) 224 = 16,777,216 colores
Direcciones IPv4 máximas 232 = 4,294,967,296
Resoluciones de pantalla 1024×768, 2048×1536...

Complete table of powers of 2

Below is the table with powers of 2 from 20 to 230, along with their numerical value and their use in computing when relevant.

Exponent Value In computing
20 1 1 — base
21 2 bit
22 4
23 8 valores de un nibble bajo
24 16 valores de un nibble
25 32
26 64
27 128 valores ASCII
28 256 valores de un byte
29 512
210 1,024 1 KB (kibibyte)
211 2,048
212 4,096
213 8,192
214 16,384
215 32,768
216 65,536 65.536 — rango entero 16 bits
217 131,072
218 262,144
219 524,288
220 1,048,576 1 MB (mebibyte)
221 2,097,152
222 4,194,304
223 8,388,608
224 16,777,216 16,7 M colores 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 GB (gibibyte)

Mathematical properties

Powers of 2 have fascinating properties that make them unique among natural numbers:

Binary representation Always 10...0 (a 1 followed by zeros)
Only prime Only 2 is prime among all powers of 2
Cumulative sum 20 + 21 + ... + 2n = 2n+1 − 1
Bitwise property n is a power of 2 if n & (n−1) = 0

Each power of 2 in binary has exactly one bit set to 1. For example: 8 = 1000₂, 16 = 10000₂, 32 = 100000₂. This property is why operations with powers of 2 are extremely fast on processors: it is enough to shift bits to the left.

The sum of all powers of 2 up to 2n equals 2n+1 − 1. For example: 1 + 2 + 4 + 8 + 16 = 31 = 25 − 1. These numbers (2n − 1) are known as Mersenne numbers, and when they are prime, they are called Mersenne primes.

Powers of 2 in nature

Exponential growth based on powers of 2 constantly appears in nature and in classic mathematical problems:

Cell division 1 → 2 → 4 → 8 → 16 → 32 cells
Bacterial growth Each bacterium divides into 2, doubling the population
Chess legend 1 grain on the first square, 2 on the second, 4 on the third...

The famous wheat and chessboard legend illustrates the power of exponential growth: if we place 1 grain of wheat on the first square, 2 on the second, 4 on the third, and so on, the 64th square would have 263 = 9,223,372,036,854,775,808 grains. The total would be 264 − 1 = more than 18 quintillion grains, enough to cover the entire surface of the Earth.

The first 20 powers of 2

Click any number to explore all its mathematical properties:

2 4 8 16 32 64 128 256 512 1,024 2,048 4,096 8,192 16,384 32,768 65,536 131,072 262,144 524,288 1,048,576

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