Binary Shifts
What is a binary shift?
- A binary shift is how a computer system performs basic multiplication and division
- Binary digits are moved left or right a set number of times
- A left shift multiplies a binary number by 2 (x2)
- A right shift divides a binary number by 2 (/2)
- A shift can move more than one place at a time, the principle remains the same
- A left shift of 2 places would multiply the original binary number by 4 (x4)
How do you perform a left shift of 1?
- Here is the binary representation of the denary number 40
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
- To perform a left binary shift of 1, we move each bit 1 place to the left
- The digit in the 128 column will move left causing an overflow error
- The 1 column becomes empty so is filled with a 0
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | = 40 | |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | = 80 |
- The original binary representation of denary 40 (32+8) has multiplied by 2 and became 80 (64+16)
How do you perform a left shift of 2?
- Here is the binary representation of the denary number 28
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
- To perform a left binary shift of 2, we move each bit 2 place to the left
- The digit in the 128 and 64 column will move left causing an overflow error
- The 1 and 2 column become empty so are filled with a 0
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |
| 0 | 1 | 1 | 1 | 0 | 0 | = 28 | ||
| 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | = 112 |
- The original binary representation of denary 28 (16+8+4) has multiplied by 4 and became 112 (64+32+16)
How do you perform a right shift of 1?
- Here is the binary representation of the denary number 40
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
- To perform a right binary shift of 1, we move each bit 1 place to the right
- The digit in the 1 column will move right causing an underflow error
- The 128 column becomes empty so is filled with a 0
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | = 40 | |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | = 20 |
- The original binary representation of denary 40 (32+8) has divided by 2 and became 20 (16+4)
How do you perform a right shift of 2?
- Here is the binary representation of the denary number 200
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |
- To perform a right binary shift of 2, we move each bit 2 places to the right
- The digits in the 1 and 2 columns will move right causing an underflow error
- The 128 and 64 columns become empty so are filled with a 0
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | |
| 1 | 1 | 0 | 0 | 1 | 0 | = 200 | ||
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | = 50 |
- The original binary representation of denary 200 (128+64+8) has divided by 4 and became 50 (32+16+2)
Worked example
1. Perform a binary shift of 2 places left on the binary number 00001110 [1]
2. Explain the effect of performing a 2 place shift to the left on the binary number 00001110 [2]
Answers
Q1
- Cross out the first 2 digits from the left
- Write down the binary digits left and add 2 zeros to the end
Q2
- Multiplies the number by 4
- Overflow errors can cause loss of precision
