The Binary Number System

A number system is a way of writing numbers. In everyday life we use the decimal system: it has ten digits (0–9), and each next place is ten times larger. The binary system has only two digits, 0 and 1, because the electronic circuits of a computer have two states: there is a signal or there is not.

To convert a binary number to decimal, multiply each digit by the matching power of two and add. From right to left the powers are 1, 2, 4, 8, 16, … For example, 1101₂ = 1·8 + 1·4 + 0·2 + 1·1 = 13. To convert a decimal number to binary, divide it by 2 repeatedly, writing down the remainders, and then read the remainders in reverse order.

Example: let us convert 13 to binary. 13 : 2 = 6 (remainder 1), 6 : 2 = 3 (remainder 0), 3 : 2 = 1 (remainder 1), 1 : 2 = 0 (remainder 1). Reading the remainders from bottom to top gives 1101. Check: 8 + 4 + 1 = 13, correct. This is how a computer stores text, images and sound as sequences of zeros and ones.

Key takeaways

• The binary system uses only the digits 0 and 1.
• Binary to decimal: multiply the digits by powers of two and add.
• Decimal to binary: divide by 2 and read the remainders in reverse.
• A computer stores all information in binary code.