## Lesson Objective

- Be able to add together up to three binary numbers.
- Apply binary shifts to multiply and divide numbers.

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KS3, GCSE, A-Level Computing Resources

- Be able to add together up to three binary numbers.
- Apply binary shifts to multiply and divide numbers.

Work right to left:

- Add the Units
- If Over 9 = 0 Carry Tens
- Add Tens

Here are examples of Base 10 numbers:

1 | 1 | ||||||

1 | 5 | 2 | 6 | ||||

+ | 1 | 7 | + | 1 | 9 | ||

3 | 2 | 4 | 5 |

Work right to left and apply these simple rules:

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 Carry 1
- 1 + 1 + 1 = 1 Carry 1

Here are examples of two 8 bit numbers being added together:

1 | 1 | 1 | |||||||

0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 14 | |

+ | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 162 |

1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 176 |

...

1 | 1 | 1 | 1 | ||||||

0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 71 | |

+ | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 79 |

1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 150 |

When and extra bit is created to represent a number.

Here is an example of an overflow error:

1 | 1 | 1 | 1 | 1 | |||||

1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 204 | |

+ | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 157 |

1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 361 |

**Method 1.**

Add the first two, then add the third to the result.

We will carry out the addition 1011 + 0111 + 101

We can can see the 1011 (11) + 0111 (7) + 101 (5) = 23

1 | 1 | 1 | 1 | ||||

1 | 0 | 1 | 1 | 11 | |||

+ | 0 | 1 | 1 | 1 | 7 | ||

1 | 0 | 0 | 1 | 0 | |||

+ | 1 | 0 | 1 | 5 | |||

1 | 0 | 1 | 1 | 1 | 23 |

**Method 2.**

Left Shift = Multiply. Each shift is the number multiplied by a power of 2

0 Shift | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | Original |

1 Shift | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | *2 |

2 Shift | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | *4 |

3 Shift | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | *8 |

Right Shift = Divide. Each shift is the number division by a power of 2

0 Shift | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | Original |

1 Shift | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | /2 |

2 Shift | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | /4 |

3 Shift | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | /8 |

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