Computer Science › Math for Programmers

Binary and Hexadecimal

Base-2 and base-16 number systems.

Also known as: binary, hexadecimal, hex, base 2, base 16

Everyday numbers use base 10: ten digits (0-9), where each place is worth ten times the one to its right. Computers use binary (base 2), with just 0 and 1, because that matches on and off in hardware. Hexadecimal (base 16) is a compact way to write binary.

Binary

Each place is a power of two: 1, 2, 4, 8, 16…

1011 in binary = 1×8 + 0×4 + 1×2 + 1×1 = 11

bin(11)            # '0b1011'
int("1011", 2)     # 11
0b1011             # 11 written as a literal

Hexadecimal

Uses 0-9 and A-F (A=10 … F=15). One hex digit is exactly four bits, so a byte is two hex digits.

11111111 (binary) = FF (hex) = 255 (decimal)
0xFF               # hex literal in many languages

hex(255)           # '0xff'
int("ff", 16)      # 255

Where you meet them

• Colors: #ff8800 is three bytes: red 255, green 136, blue 0.
• Memory addresses and hashes: written in hex (3f9c2a1).
• Permissions: file modes like 755 (octal, base 8).
• Bit flags and masks (bitwise operations).
• Debugging bytes: hex dumps show raw data (bits and bytes, ASCII).
• IPv6 addresses and MAC addresses.

Handy facts

• n bits can represent 2ⁿ values: 8 bits give 256 (0-255).
• Powers of two (1, 2, 4, 8, 16, 32, 64, 128, 256, 1024) are worth memorizing.
• A prefix such as 0x (hex) or 0b (binary) says which base a literal is in.