Binary Shifts (AQA GCSE Computer Science): Revision Notes
Binary shifts
What are binary shifts?
When we move binary digits to the left or right within a binary number, this process is known as a binary shift. Think of it like sliding all the digits in one direction - this simple movement actually performs mathematical operations on our numbers!
Binary shifts are incredibly useful because they provide a quick and efficient way to multiply or divide binary numbers by powers of 2. Computers use this technique all the time because it's much faster than traditional multiplication and division methods.
Binary shifts for multiplication
How left shifts work
Every time we shift the digits of a binary number one position to the left, we multiply the original number by 2. This happens because we're essentially moving each digit to a higher place value position.
Let's see this in action with the decimal number 40, which equals 101000 in binary:
Worked Example: Left Shift Multiplication
Starting number: 40 (decimal) = 101000 (binary)
Step 1: Shift one position left and add 0 to the right
- Original:
101000 - After 1 left shift:
1010000 - Result: 80 (decimal)
- Calculation: ✓
Step 2: Shift two positions left from original
- Original:
101000 - After 2 left shifts:
10100000 - Result: 160 (decimal)
- Calculation: ✓
When we shift this number one place to the left and add a 0 to the rightmost position, something interesting happens. The original number (which was in the highlighted positions) moves left, effectively doubling its value.
The pattern for multiplication
Key Pattern for Left Shifts:
- 1 left shift = multiply by
- 2 left shifts = multiply by
- 3 left shifts = multiply by
- left shifts = multiply by
This works because each position in binary represents a power of 2, so moving digits left increases their place value accordingly.
Binary shifts for division
How right shifts work
Moving digits to the right has the opposite effect - it divides our number by 2 for each position we shift. This happens because we're moving each digit to a lower place value position.
Starting with our same example of decimal 40 (101000 in binary), let's shift the digits one place to the right:
Worked Example: Right Shift Division
Starting number: 40 (decimal) = 101000 (binary)
Step 1: Shift one position right
- Original:
101000 - After 1 right shift:
10100 - Result: 20 (decimal)
- Calculation: ✓
Step 2: Shift two positions right from original
- Original:
101000 - After 2 right shifts:
1010 - Result: 10 (decimal)
- Calculation: ✓
The pattern for division
Just like with multiplication, there's a clear pattern:
Key Pattern for Right Shifts:
- 1 right shift = divide by
- 2 right shifts = divide by
- 3 right shifts = divide by
- right shifts = divide by
Important note about division
Warning About Right Shifts and Remainders
When we shift right and there are digits that would "fall off" the right end, we lose that information. This means that if our original number isn't evenly divisible by the power of 2 we're dividing by, we'll lose the remainder and only get the whole number part of the answer.
Why binary shifts matter
Binary shifts are important because they demonstrate fundamental computing concepts:
- They're extremely fast operations for computers to perform
- They help us understand how binary place values work
- They show the relationship between binary and powers of 2
- They're used in many computer algorithms and calculations
Understanding binary shifts also helps you better grasp how computers handle numbers internally and why binary is so fundamental to computing.
Remember!
Key Points to Remember:
- Binary shift: Moving all digits in a binary number left or right
- Left shifts multiply: Each position left multiplies the number by 2
- Right shifts divide: Each position right divides the number by 2
- Powers of 2: The number of positions shifted determines which power of 2 you multiply or divide by
- Efficient operation: Binary shifts are much faster than traditional multiplication/division for computers