# Convert binary to decimal

Please provide values below to convert binary to decimal, or *vice versa*.

### Binary

**Definition:** The binary numeral system is a base-2 numeral system that typically only uses two symbols: 0 and 1. Thus, it has a radix, or a base number of unique digits of two. Each digit in binary is referred to as a bit.

It is a system that uses positional notation in which the same symbol is used for different orders of magnitude, where each "place" represents a different value dependent on whichever base is being used; in the case of binary, the base is 2.

In the binary number 101, the first "1" on the left is in the 2^{2} place, the "0" is in the 2^{1} place, and the second "1" is in the 2^{0} place. If this were converted to decimal:

101 = 1 × 2^{2} + 0 × 2^{1} + 1 × 2^{0} = 4 + 0 + 1 = 5

**History/origin:** There is evidence of systems related to binary numbers in a number of different cultures including that of ancient Egypt, China, and India. However, the modern binary number system was studied and developed by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz in the 16th and 17th centuries.

**Current use:** The binary system is widely used in almost all modern computers or computer-based devices. Because of this, it is sometimes referred to as the "language of computers." Its widespread use can be attributed to the ease with which it can be implemented in a compact, reliable manner using 0s and 1s to represent states such as on or off, open or closed, etc.

### Decimal

**Definition:** The decimal numeral system is a base-10 numeral system, also known as the Arabic number system, and is the standard system used to represent integer and non-integer numbers, using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is a system that uses positional notation, where the same symbol is used in different positions, and the magnitude is determined by which "place" the symbol holds. For example, the number 111:

111 = 1 × 10^{2} + 1 × 10^{1} + 1 × 10^{0} = 100 + 10 + 1 = 111

As can be seen, even though each symbol (the "1") is the same in each position, they all have different magnitudes. Decimal fractions can also be represented by using a decimal point (".").

**History/origin:** Numerals based on ten have been used by many cultures since ancient times including the Indus Valley Civilization, ancient Egyptians, the Bronze Age cultures of Greece, the classical Greeks, and the Romans, among others. Some believe that this is linked to the human hand usually having ten digits.

The positional decimal system in use today has roots as early as around the year 500, in Hindu mathematics during the Gupta period. The earliest known evidence of the Hindu-Arabic numerals being used in Europe was found in the Codex Vigilanus, a compilation of historical documents written in the year 976. The numerals that people today are accustomed to were a result of early typesetting in the late 15^{th} to earthly 16^{th} century.

**Current use:** The decimal numeral system is the most common system used around the world for the symbolic representation of numbers. It is used ubiquitously for everyday applications, mathematics, and within many other contexts.