Decimal to Binary Converter

To convert decimal to binary, repeatedly divide the number by 2 and record the remainder at each step. The remainders, read from bottom to top, form the binary number. For example, 13 ÷ 2 gives remainders 1, 0, 1, 1 — so 13 in binary is 1101.

Binary is a base-2 numeral system that uses only two digits: 0 and 1. Each digit (bit) represents a power of 2. It is the fundamental language of computers and digital electronics because electronic circuits can easily represent two states — on (1) and off (0).

The decimal system is a base-10 numeral system, the most commonly used in everyday life. It uses ten symbols (0–9), and each digit position represents a power of 10. For example, 653 = 6×10² + 5×10¹ + 3×10⁰.

The number of bits required is ⌊log₂(n)⌋ + 1 for any positive integer n. For example, the decimal number 255 requires 8 bits (11111111 in binary), while 256 requires 9 bits (100000000).

Grouping binary digits makes long binary strings easier to read. Grouping by 4 bits is called a nibble, and grouping by 8 bits is called a byte. For example, the binary number 11010110 can be grouped as 1101 0110 in nibble notation.

Negative numbers in binary are typically represented using two's complement notation in computing. This converter handles non-negative integers. For negative numbers, the two's complement is computed by inverting all bits of the positive binary value and adding 1.

Binary (base 2), octal (base 8), and hexadecimal (base 16) are all positional numeral systems used in computing. Octal groups binary digits by 3, and hexadecimal groups them by 4, making large binary numbers more compact and readable. All three can exactly represent the same values as decimal.

Computers use binary because transistors — the basic building blocks of processors — have two states: on and off, naturally mapping to 1 and 0. All data, instructions, and programs in a computer are ultimately stored and processed as sequences of binary digits (bits).