Welcome to the World of Binary Arithmetic!
Ever wondered how a computer actually "thinks" when it calculates your score in a game or works out a math problem? Since computers are made of tiny switches that are either on (1) or off (0), they have to do all their math using only those two numbers. This is called Binary Arithmetic.
In this guide, we are going to learn how to add binary numbers just like a computer does and how to use "binary shifts" to multiply and divide at lightning speed. Don't worry if it seems tricky at first—once you learn the four simple rules of addition, you'll be a human calculator!
Part 1: Binary Addition
Binary addition is very similar to the addition you learned in primary school. You line up the numbers in columns and work from right to left. The only difference is that you can't have a "2" or a "3"—you have to "carry" into the next column instead.
The Four Golden Rules
When you add a column of two binary digits, there are only four possible outcomes:
1. \( 0 + 0 = 0 \)
2. \( 0 + 1 = 1 \)
3. \( 1 + 0 = 1 \)
4. \( 1 + 1 = 10 \) (This is 0 in the current column, and you carry the 1 to the next column on the left).
Adding Three Numbers (The Pro Level)
Your AQA syllabus says you might need to add up to three binary numbers at once. This means you might find three 1s in a single column!
• If the column adds up to 3 (three 1s), the rule is: Write down a 1 and carry a 1 to the next column. This is because 3 in decimal is 11 in binary.
Quick Review Box:
• Total 0: Write 0, carry 0
• Total 1: Write 1, carry 0
• Total 2: Write 0, carry 1
• Total 3: Write 1, carry 1
Step-by-Step Example
Let's add 0110 (6) and 0111 (7):
Step 1 (Right column): \( 0 + 1 = 1 \). Write 1.
Step 2 (Next column): \( 1 + 1 = 10 \). Write 0, carry the 1.
Step 3 (Next column): \( 1 + 1 + \) (the carried 1) \( = 11 \). Write 1, carry the 1.
Step 4 (Left column): \( 0 + 0 + \) (the carried 1) \( = 1 \). Write 1.
Final Answer: 1101 (which is 13 in decimal!)
Common Mistake: Forgetting to add the "carry" digit. Always write your carries clearly at the top of the next column so you don't miss them!
Key Takeaway: Binary addition is just "counting 1s" in a column. If you have two 1s, it's a carry. If you have three 1s, it's a 1 and a carry.
Part 2: Binary Shifts
A Binary Shift is a clever trick used to multiply or divide a number by powers of 2 (2, 4, 8, 16, etc.). It’s much faster for a computer to "shift" digits left or right than to do a long multiplication sum.
Left Shift: Multiplication
When you perform a left shift, every bit in the binary number moves to the left. For every place you shift, the number doubles.
• Shift Left 1 place: Multiply by \( 2^1 \) (Multiply by 2)
• Shift Left 2 places: Multiply by \( 2^2 \) (Multiply by 4)
• Shift Left 3 places: Multiply by \( 2^3 \) (Multiply by 8)
Example: Take the binary number 0000 0101 (which is 5).
Shift it left 2 places: The "101" moves two spots to the left, and we fill the empty spots on the right with 0s.
Result: 0001 0100 (which is 20! Notice that \( 5 \times 4 = 20 \)).
Right Shift: Division
A right shift moves every bit to the right. This divides the number by 2 for every place shifted.
• Shift Right 1 place: Divide by 2
• Shift Right 2 places: Divide by 4
Example: Take 0000 1000 (which is 8).
Shift it right 1 place: Move everything one spot to the right.
Result: 0000 0100 (which is 4! Notice that \( 8 \div 2 = 4 \)).
Did you know? Binary shifts are "Logical Shifts." This means that when bits move out of the 8-bit boundary, they just disappear! Also, any empty gaps created are always filled with a 0.
Part 3: Real-World Use of Shifts
Why do we bother with shifts instead of normal math? In computer science, efficiency is everything.
1. Speed: Shifting bits is one of the fastest operations a CPU can perform. If a programmer needs to multiply by 8, they will tell the computer to "Shift Left 3" because it's quicker than "Multiply by 8."
2. Graphics: When computers calculate colors or move pixels on a screen, they use binary shifts to process millions of calculations per second.
Encouragement: If shifts feel confusing, think of it like multiplying a decimal number by 10. You just move the digits left and add a zero. Binary shifts are exactly the same, but for the number 2!
Key Takeaway: Left = Larger (Multiplication). Right = Reducer (Division). Each jump is a power of 2.
Quick Summary for Revision
Binary Addition Rules:
• \( 0+0 = 0 \)
• \( 1+0 = 1 \)
• \( 1+1 = 0 \) carry 1
• \( 1+1+1 = 1 \) carry 1
Binary Shift Rules:
• Left Shift \( n \) places: Multiply by \( 2^n \)
• Right Shift \( n \) places: Divide by \( 2^n \)
• 8-bit Limit: In your exam, you will usually work with 8 bits. If a 1 is shifted off the end, it is lost.
Final Tip: Always double-check your work by converting your binary answer back to decimal. If your decimal addition matches your binary addition, you've got it right!