**CONVERSION OF OTHER BASES TO DENARY SYSTEM**

Conversion of other bases to denary system. To convert numbers in the other bases to denary system, expand the given number in power of its bases.

Examples:

- Convert 354
_{16}to denary

354_{16} = 3 x 16^{2} + 5 X 16^{1} + 4 x 16^{}

= 3 x 256 + 5 x 16 + 4 x 1

= 768 + 80 + 4

852_{ten}

- Convert 255
_{eight}to base ten

255_{8} = 2 x 8^{2 }+ 5 x 8^{1} + 5 x 8^{}

= 2 x 64 + 5 x 8 + 5 x 1

= 128 + 40 + 5

= 173_{ten}

- Convert 1011001
_{2}to base ten

1011001_{two} = 1 x 2^{6} + 0 x 2^{5} + 1 x 2^{4} + 1 x 2^{3} + 0 x 2^{2} + 0 x 2^{1} + 1 x 2^{}

= 1 x 64 + 0 x 32 + 1 x 16 + 1 x 8 + 0 X 4 + 0 x 2 + 1 x 1

= 64 + 0 + 16 + 8 + 0 + 0 +1

= 89_{ten}

- Convert 3A7
_{16}to base ten

3A7_{16} = 3 x 16^{2} + 10 x 16^{1} + 7 x 16^{}

^{ }= 3 x 256 + 10 x 16 + 7 x 1

= 768 + 160 + 7

= 935_{ten}

**ADDITION, SUBTRACTION AND MULTIPLICATION OF BINARY NUMBERS **

There are four rule guiding binary addition

0 + 0 = 0

0 + 1 = 1 (Always remember you are working with binary digit not decimal)

1 + 0 = 1

1 + 1 = 10

- Add 101
_{2}+ 1000_{2} - Add 1001101
_{2}+ 111001_{2}

- 101
_{2}

+ 1000_{2}

1101_{2}

- 1001101
_{2}

+ 111001_{2}

10000110_{2}

**Binary Subtraction**

0 – 0 = 0; 0 – 1 = 1; 1 – 0 = 1; 1 – 1 = 0

(1). Subtract 101_{2} from 1001_{2 } (2). 10001_{2} – 1111_{2}

- 1001
_{2}

– 101_{2}

100_{2}

- 10001

– 1111

0010_{2}

**Binary Multiplication **

The rules for binary multiplication are: 0 x0 = 0; 1 x 0 = 0; 0 x 1 = 0; 1 x 1 = 1

(1).1101_{2} x 110_{2} (2). 101_{2} x 10_{2}

- 1101
_{2}

x 110_{2}

0000_{2}

+ 1101_{2}

1101_{2}

1001110_{2}

- 101
_{2}

x 10_{2}

000_{2}

+ 101_{2}

1010_{2}

Decimal |
Binary |
Octal |
Hexadecimal |

0000 | |||

1 | 0001 | 1 | 1 |

2 | 0010 | 2 | 2 |

3 | 0011 | 3 | 3 |

4 | 0100 | 4 | 4 |

5 | 0101 | 5 | 5 |

6 | 0110 | 6 | 6 |

7 | 0111 | 7 | 7 |

8 | 1000 | 10 | 8 |

9 | 1001 | 11 | 9 |

10 | 1010 | 12 | A |

11 | 1011 | 13 | B |

12 | 1100 | 14 | C |

13 | 1101 | 15 | D |

14 | 1110 | 16 | E |

15 | 1111 | 17 | F |

**EVALUATION**

- Define number base
- Convert the following to base 10

- 10011
_{two} - 317
_{eight}

- Evaluate the following:

- 101
_{2}x 101_{2} - 111001
_{2}+ 1001_{2} - 10111
_{2}– 100_{2}

See also

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