Binary Number System: What is it? (Definition & Examples)

What is the Binary Number System?

In digital electronics, the binary number system provides a convenient way to represent data with two digits. Decimal is a useful comparison because both systems use place value, but binary uses powers of 2 instead of powers of 10.

The decimal, or base-10, system uses the ten digits 0 through 9. A single decimal digit can therefore represent any value from zero to nine.

For larger values, decimal combines digits in positions. Each position has ten times the weight of the position to its right. For example, 625 means six hundreds plus two tens plus five ones.

Binary follows the same positional idea but uses only 0 and 1. Each position has twice the weight of the position to its right. The linked guide explains the conversion of binary to decimal numbers in more detail.

History of Binary Number System

Around 200 BCE, the Indian prosodist Pingala described procedures for arranging and counting patterns of light and heavy syllables. These two-state patterns parallel binary enumeration, but they were not the modern positional notation that uses the digits 0 and 1.

Gottfried Wilhelm Leibniz described binary arithmetic in an unpublished 1679 manuscript and presented a published account in 1703.
Any nonnegative whole number can be written in binary. Digital systems use bits because two distinct physical state ranges can be labelled 0 and 1.

The table shows every two-digit decimal string from 00 through 99. Its two places are decimal digit positions, not bits. The left-hand digit has the greater place value, while the right-hand digit has the smaller place value.

From 00 to 09, the right-hand digit runs through all ten possible values. It then returns to 0 as the left-hand digit increases by one. The same pattern repeats for 10 to 19, 20 to 29, and each following row.

Two decimal positions provide 100 fixed-width strings, from 00 through 99, because 100 equals 10². The general count is (number of basic digits)^{number of bits}. With three decimal positions, 10³ gives 1,000 strings from 000 through 999. With n decimal positions, the count is 10ⁿ.

The binary system applies this place-value method with two digits: 0 and 1. Each binary digit is called a bit.

In a binary integer, the rightmost position has a weight of 1. Moving one position to the left doubles the weight to 2, then 4, 8, and so on. The rightmost bit has the smallest place value, while the leftmost occupied bit has the greatest place value.

A two-bit unsigned field has 2², or four, patterns: 00, 01, 10, and 11. The general count remains (number of basic digits)^{number of bits}. Therefore, n bits provide 2 to the power of n fixed-width patterns and represent unsigned values from 0 through 2 to the power of n minus 1. Longer finite strings can represent larger nonnegative integers.

Binary Number System Definition

The binary number system is a positional number system with base, or radix, 2. Its two digits are 0 and 1.

Consider the decimal number 625.

Number the digit positions from right to left, starting at 0. The digit 5 is at position 0, the digit 2 is at position 1, and the digit 6 is at position 2. Therefore, 625 equals 6 × 100 + 2 × 10 + 5 × 1. In general, a decimal number can be expressed as:
∑digit × 10^{corresponding position or bit}
Here, 10 is the number of digits in the decimal system, from 0 through 9.
The same positional rule works in any base when 10 is replaced by the number of digits available in that system. The general expression is:
∑digit × (number of digits used in that system)^{corresponding position or bit}
Binary has two available digits. A binary value can therefore be expressed as:
∑digit × (2)^{corresponding position or bit}

For the binary number 101010, the place-value calculation gives 32 + 8 + 2, so its decimal value is 42:

The table below applies the same calculation to several binary values:

The animation below shows how successive binary numbers are counted.