## How to convert base

### Base-N (non-decimal) to decimal

Multiply $N^0, N^1, ... N^n$ from the bottom of the base-N and add up.

e.g. When converting the binary number 1101100 to decimal number.

$0 \times 2^0+0 \times 2^1 + 1 \times 2^2 + 1 \times 2^3 + 0 \times 2^4 + 1 \times 2^5 + 1 \times 2^6 + 0 = 108$e.g. When converting hexadecimal 6C to decimal (C = 12).

$12 \times 16^0 + 6 \times 16^1 = 108$### Decimal to base-N (non-decimal)

- Divide the decimal number until the quotient is less than N.
- Arrange the remainders in order from the end, with the last quotient at the top.

e.g. When converting decimal 108 to binary.

108 / 2 = 54, remainder of 0

54 / 2 = 27, remainder of 0

27 / 2 = 13, remainder of 1

13 / 2 = 6, remainder of 1

6 / 2 = 3, remainder of 0

3 / 2 = 1, remainder of 1

-> 1101100

In other words, the algorithm that decomposes into the following format.

$0 + 0 \times 2 + 1 \times 2^2 + 1 \times 2^3 + 0 \times 2^4 + 1 \times 2^5 + 1 \times 2^6 + 0$e.g. When converting decimal 108 to hexadecimal.

108 / 16 = 6, remainder of 12 (C)

-> 6C

## Cheat sheet

Dec (10) | Bin (2) | Qua (4) | Oct (8) | Hex (16) |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 1 |

2 | 10 | 2 | 2 | 2 |

3 | 11 | 3 | 3 | 3 |

4 | 100 | 10 | 4 | 4 |

5 | 101 | 11 | 5 | 5 |

6 | 110 | 12 | 6 | 6 |

7 | 111 | 13 | 7 | 7 |

8 | 1000 | 20 | 10 | 8 |

9 | 1001 | 21 | 11 | 9 |

10 | 1010 | 22 | 12 | A |

11 | 1011 | 23 | 13 | B |

12 | 1100 | 30 | 14 | C |

13 | 1101 | 31 | 15 | D |

14 | 1110 | 32 | 16 | E |

15 | 1111 | 33 | 17 | F |

16 | 10000 | 100 | 20 | 10 |