# Definition

Put definition of generic base n

# Popular Bases

## Binary

Binary, or base 2, is the most commonly used number system in computers. The base 2 system only uses the digits, 0 and 1. Each digit in binary represents an extra power of two starting from the rightmost digit. For example the binary number **1101** can be expressed as **(1*2 ^{3}) + (1*2^{2}) + (0*2^{1}) + (1*2^{0})**.

### Converting from Decimal to Binary

The fastest and most efficient way to convert from decimal to binary is by dividing by 2 and keeping track of the remainder. We stop when the quotient = 0. Then read the numbers stating from the bottom to get the binary number.

For example, to convert 156 from decimal to binary we do:

156 | / 2 = | 78 | r = | 0 |

78 | / 2 = | 39 | r = | 0 |

39 | / 2 = | 19 | r = | 1 |

19 | / 2 = | 9 | r = | 1 |

9 | / 2 = | 4 | r = | 1 |

4 | / 2 = | 2 | r = | 0 |

2 | / 2 = | 1 | r = | 0 |

1 | / 2 = | 0 | r = | 1 |

Thus by reading the numbers starting from the bottom we get, **10011100**

## Hexadecimal

Hexadecimal or base 16 is commonly used because of the less space it takes up. A hexadecimal number consists of the digits 0-9 and A-F to represent the values 10-15. This works in the same way as binary but instead of incrementing in powers of 2 it increments in powers of 16.

### Example

The hexadecimal number **A5B** can be represented as **(10*16 ^{2}) + (5*16^{1}) + (11*16^{0})** which is 2651 in decimal.

## Converting from Binary to Hexadecimal

You can group 4 digits in binary as 1 digit in hexadecimal because 16 is 2^^4.

### Example

To convert **100111000100** to hexadecimal we group in fours starting from the right to convert to hexadecimal.