Converting Between Binary and Decimal Revision Notes for AQA GCSE Computer Science

Converting between binary and decimal

Understanding number base conversions

When working with computers, you'll often need to switch between binary (base 2) and decimal (base 10) number systems. Binary uses only 0s and 1s, while decimal uses digits 0-9. The key to converting between these systems is understanding place values - each position in a binary number represents a different power of 2.

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Understanding place values is fundamental to all number base conversions. Each position in a binary number has a specific value based on powers of 2, which makes the conversion process systematic and predictable.

The binary place value table

The foundation for all binary-decimal conversions is the place value table. Each column represents a power of 2, starting from 1 on the right and doubling as you move left:

128	64	32	16	8	4	2	1

This table shows the place values for an 8-bit binary number. Each position can contain either a 0 or a 1, and when there's a 1 in a position, that place value contributes to the total decimal number.

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Remember the Pattern: Each place value in binary is exactly double the previous one. Starting from the right: $1, 2, 4, 8, 16, 32, 64, 128...$

This doubling pattern follows the mathematical formula: $2^n$ where $n$ is the position number (counting from 0 on the right).

Converting binary to decimal

Converting from binary to decimal is straightforward - you simply add up the place values where there's a 1 in the binary number.

Method for binary to decimal conversion

Write the binary number in the place value table
Fill the table from right to left, ensuring the rightmost digit goes in the '1' column
Add up all the place values that have a '1' underneath them
The sum is your decimal answer

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Worked Example: Converting 100111₂ to decimal

Let's convert the binary number 100111 to decimal using our method:

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Step-by-step solution:

Place the binary digits in the table: 100111
Identify where the 1s appear: positions 32, 4, 2, and 1
Add these place values together: $32 + 4 + 2 + 1 = 39$
Answer: 100111₂ = 39₁₀

Converting decimal to binary

Converting decimal numbers to binary requires a different approach using subtraction. You work from the largest place value down to the smallest, deciding whether each place value fits into your remaining decimal number.

Method for decimal to binary conversion

Start with your decimal number
Begin with the largest place value (128 for 8-bit numbers)
Ask: "Is this place value smaller than or equal to my remaining number?"
If YES: Put a 1 in that column and subtract the place value from your number
If NO: Put a 0 in that column and move to the next smaller place value
Repeat until you reach the 1s column

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Worked Example: Converting 142₁₀ to binary

Let's convert the decimal number 142 to binary step by step:

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Step-by-step process:

128 column: Is $128 \leq 142$ ? YES → Put 1, subtract: $142 - 128 = 14$
64 column: Is $64 \leq 14$ ? NO → Put 0
32 column: Is $32 \leq 14$ ? NO → Put 0
16 column: Is $16 \leq 14$ ? NO → Put 0
8 column: Is $8 \leq 14$ ? YES → Put 1, subtract: $14 - 8 = 6$
4 column: Is $4 \leq 6$ ? YES → Put 1, subtract: $6 - 4 = 2$
2 column: Is $2 \leq 2$ ? YES → Put 1, subtract: $2 - 2 = 0$
1 column: Is $1 \leq 0$ ? NO → Put 0

Result: 142₁₀ = 10001110₂

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Worked Example: Converting 83₁₀ to binary

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Following the same method for 83:

$64 \leq 83$ ? YES → 1 (remaining: $83 - 64 = 19$ )
$32 \leq 19$ ? NO → 0
$16 \leq 19$ ? YES → 1 (remaining: $19 - 16 = 3$ )
$8 \leq 3$ ? NO → 0
$4 \leq 3$ ? NO → 0
$2 \leq 3$ ? YES → 1 (remaining: $3 - 2 = 1$ )
$1 \leq 1$ ? YES → 1 (remaining: $1 - 1 = 0$ )

Result: 83₁₀ = 1010011₂ (or 01010011₂ with leading zero)

Key exam tips

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Critical Exam Strategies

For binary to decimal conversion:

Always use the place value table to avoid mistakes
Remember to align your binary number from the right (starting with the 1s column)
Simply add up the place values where you see a 1

For decimal to binary conversion:

Work systematically from left to right (largest to smallest place value)
Always show your working by displaying each subtraction step
Double-check your answer by converting back to decimal
Remember that you can only use each place value once

Important exam note: In GCSE exams, you'll typically work with binary numbers up to 8 bits (digits), meaning the largest decimal number you'll convert is 255 (11111111₂ in binary).

Practice makes perfect

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Regular practice with conversion exercises will help you become faster and more confident. Try converting both ways - start with smaller numbers and gradually work up to larger ones as you get more comfortable with the process.

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Key Points to Remember:

Binary place values double each time: $1, 2, 4, 8, 16, 32, 64, 128...$
Binary to decimal: Add up place values where there's a 1
Decimal to binary: Use subtraction method, working from largest to smallest place value
Always show your working in exams - use the place value table and write out your calculations
Check your answers by converting back the other way to verify correctness

Converting Between Binary and Decimal (AQA GCSE Computer Science): Revision Notes