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
Lesson 3. Binary Arithmetic
Lesson Notes
Addition - Column Method
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 |
Binary Addition - THE RULES!!
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 |
Overflow Error
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 |
Adding 3 Binary Numbers
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.
Logical Binary Shifts
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 |