Number Base Converter

A number base (or radix) defines how many unique digits are used in a positional numeral system. Base 10 (decimal) uses 0–9. Base 2 (binary) uses only 0 and 1. Base 16 (hexadecimal) uses 0–9 plus A–F.

Binary uses only 0 and 1. Each digit (bit) represents a power of 2. The rightmost bit is 2⁰=1, next is 2¹=2, then 2²=4, etc. Binary 1010 = (1×8) + (0×4) + (1×2) + (0×1) = 10 in decimal.

Hex is compact — one hex digit represents exactly 4 bits (a nibble). Two hex digits represent a full byte. It is much easier to read and write memory addresses, color codes (#FF5733), and machine code in hex than binary.

Two's complement is how computers represent negative integers. To negate a number: flip all bits (one's complement) then add 1. In 8-bit: +5 = 00000101, −5 = 11111010 + 1 = 11111011. This makes addition and subtraction use the same hardware circuit.

Divide the number by 2 repeatedly and record the remainders from bottom to top. Example: 13 ÷ 2 = 6 R1, 6 ÷ 2 = 3 R0, 3 ÷ 2 = 1 R1, 1 ÷ 2 = 0 R1 → read remainders up: 1101.

Octal (base 8) uses digits 0–7. It was used in early computing because 3 bits map to one octal digit. Today it appears in Unix/Linux file permissions (chmod 755 means rwxr-xr-x — each digit is 3 permission bits).

#RRGGBB format: each pair of hex digits represents red, green, blue intensity from 00 (0) to FF (255). #FF0000 = pure red, #00FF00 = pure green, #0000FF = pure blue, #FFFFFF = white, #000000 = black.

IEEE 754 is the standard for representing decimal fractions in binary. A 32-bit float has 1 sign bit, 8 exponent bits, and 23 mantissa bits. It can represent approximately 7 significant decimal digits. Example: 3.14 in binary IEEE 754 is 0 10000000 10010001111010111000011.

Yes. Bases 2 through 36 are supported (using digits 0–9 and letters A–Z). Base 36 (alphanumeric) is used for URL shorteners. Base 60 (sexagesimal) is used for time (60 seconds/minutes) and angles (360°).

JavaScript's Number.MAX_SAFE_INTEGER is 2⁵³ − 1 = 9,007,199,254,740,991. Beyond this, integer precision is lost. For larger numbers, BigInt must be used. This calculator handles values up to MAX_SAFE_INTEGER accurately.

The most practical approach is to convert to decimal as an intermediate step. However, binary and hex have a shortcut: since 16 = 2⁴, each hex digit maps directly to exactly 4 binary digits. Similarly, octal digits map to 3 binary digits each.

A nibble is 4 bits (one hex digit, values 0–15). A byte is 8 bits (two hex digits, values 0–255). A word is typically 16 bits (4 hex digits, 0–65535). These are fundamental units in computer architecture.

Zero-based indexing aligns with binary addressing. The first address is 0 (all bits zero). An array of N elements occupies addresses 0 through N−1, allowing the size calculation as last − first = N−1 − 0 + 1 = N, consistent with binary arithmetic.

BCD encodes each decimal digit separately as 4 bits. The decimal number 59 in BCD is 0101 1001 (5 = 0101, 9 = 1001). BCD is used in financial applications and digital displays where exact decimal representation is needed without floating-point rounding.

Three main methods exist: sign-magnitude (leftmost bit = sign), one's complement (flip all bits), and two's complement (flip all bits + 1). Modern computers universally use two's complement because it avoids having two representations of zero and simplifies arithmetic circuits.