Number Base Converter Guide: Binary, Decimal, Hex, and Octal

Colour codes in CSS, file permissions on a Linux server, memory addresses in a debugger, and IP subnet masks all use bases other than ten. Once you can move between binary, octal, decimal, and hexadecimal by hand, those notations stop being opaque strings and start being numbers you can reason about.

What a Base Actually Means

The base tells you how many digits are available and what each position is worth. In base ten, the digits are 0 to 9 and positions are powers of ten, so 407 is 4 hundreds, 0 tens, 7 units. In base two the digits are 0 and 1 and positions are powers of two. In base sixteen there are sixteen digits, so letters fill in past 9: A is 10, B is 11, C is 12, D is 13, E is 14, F is 15.

Binary to Decimal

Write the place values above the digits and add the ones that are set. For 10110110, the places from the right are 1, 2, 4, 8, 16, 32, 64, 128. The set bits are at 2, 4, 16, 32, and 128, giving 128 + 32 + 16 + 4 + 2 = 182. An 8-bit byte covers 0 to 255, which is why colour channels and older character codes stop at 255.

Decimal to Binary

Divide repeatedly by two and read the remainders bottom to top. For 156: 156/2 = 78 r0, 78/2 = 39 r0, 39/2 = 19 r1, 19/2 = 9 r1, 9/2 = 4 r1, 4/2 = 2 r0, 2/2 = 1 r0, 1/2 = 0 r1. Reading upward gives 10011100. A quicker route for people who know the powers of two is greedy subtraction: 156 minus 128 is 28, minus 16 is 12, minus 8 is 4, minus 4 is 0, so bits 128, 16, 8, and 4 are set, which is the same 10011100.

Hexadecimal Is Just Binary in Groups of Four

Because sixteen is two to the fourth power, every hex digit maps to exactly four bits. That makes conversion mechanical rather than arithmetic.

• 0 = 0000, 1 = 0001, 2 = 0010, 3 = 0011
• 4 = 0100, 5 = 0101, 6 = 0110, 7 = 0111
• 8 = 1000, 9 = 1001, A = 1010, B = 1011
• C = 1100, D = 1101, E = 1110, F = 1111

So B6 in hex is 1011 0110 in binary, which is the 182 we calculated earlier. Going the other way, group the bits in fours from the right and translate each group. This is why programmers write memory addresses in hex: one glance gives you the bit pattern.

Octal and Unix File Permissions

Octal uses digits 0 to 7 and each digit maps to three bits. Its main surviving use is file permissions, where read is 4, write is 2, and execute is 1. A permission of 755 means the owner has 4+2+1 (read, write, execute), while group and others have 4+1 (read and execute). 644 means the owner can read and write while everyone else can only read. Understanding that each digit is a three-bit field makes chmod values obvious instead of memorised.

Converting Between Any Two Bases

The general approach is to route through decimal: convert the source number to decimal by multiplying each digit by its place value, then convert decimal to the target base by repeated division. Base 12 number 2A5 becomes 2 x 144 + 10 x 12 + 5 = 413 in decimal, and 413 into base 7 is 1136, since 413 = 1 x 343 + 1 x 49 + 3 x 7 + 6. When both bases are powers of two, skip decimal entirely and regroup the bits.

Where These Bases Turn Up Day to Day

Mistakes That Cause Bugs

Reading a hex value as decimal is the classic one: port 0x50 is 80, not 50. Forgetting that a leading 0 in some languages means octal turns 010 into 8. Assuming a fixed width also bites, since 0xFF is one byte but 0x0FF written into a 16-bit field leaves the top byte zero. Prefixes exist to prevent all three, so keep 0b for binary, 0o for octal, and 0x for hex in code and notes.

Convert Bases With a Free Tool

The number base converter on SimpleCounter moves values between binary, octal, decimal, and hexadecimal in one field, showing the equivalent in every base at once. It is useful for checking a colour code, a permission value, or a bit pattern, and it runs entirely in your browser.