Binary to Decimal Calculator

Each binary digit (bit) represents a power of 2, starting from 2^0 (= 1) on the right. Multiply each bit by its corresponding power of 2 and sum the results. For example, 1010 in binary: 0 x 2^0 = 0, 1 x 2^1 = 2, 0 x 2^2 = 0, 1 x 2^3 = 8. Total = 0 + 2 + 0 + 8 = 10 in decimal.

Two's complement is the most common way to represent negative integers in binary. In an 8-bit signed integer, the leftmost bit is the sign bit: 0 means positive, 1 means negative. The range of an 8-bit signed integer is -128 to +127. To find the 2's complement of a binary number: invert all bits (flip 0s and 1s), then add 1. For example, 11111111 in 8-bit 2's complement = -(00000001) = -1.

An 8-bit unsigned integer can hold values from 0 (00000000) to 255 (11111111). There are 2^8 = 256 possible values. An 8-bit signed integer (using 2's complement) ranges from -128 (10000000) to +127 (01111111), also 256 values. Unsigned integers are used when negative values are not needed; signed integers are used when both positive and negative values are required.

Binary, octal, and hex are all related to powers of 2. One octal digit represents exactly 3 binary bits (2^3 = 8). One hex digit represents exactly 4 binary bits (2^4 = 16). This makes conversion between binary and these bases very easy without going through decimal: group binary digits in threes for octal, or fours for hex, and convert each group independently.

All digital computers operate on binary at the hardware level. Every instruction, number, character, image, and video is ultimately stored as sequences of 0s and 1s. Modern CPUs process 64 bits at once (64-bit architecture). Binary arithmetic is the foundation of all computer arithmetic, logic gates, memory addressing, and networking protocols. Programming languages abstract this away, but understanding binary helps with low-level programming, debugging, and networking.