Binary and Denary Systems
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Video 1.1.2 ·
6:00 · Converting both ways, and the maximum a byte can hold
1. The Denary System (Base 10)
Denary is the standard numbering system used by humans. It is positional, meaning the value of a digit depends on its place in the number.
Example: In the number 523, the '5' represents 5 hundreds, the '2' represents 2 tens, and the '3' represents 3 units.
Denary Place Values
| 103 | 102 | 101 | 100 |
| 1000 | 100 | 10 | 1 |
2. The Binary System (Base 2)
Binary uses only two digits: 0 and 1. In a computer, these represent the absence or presence of an electrical pulse.
Binary Place Values (The 8-bit Byte)
To convert between systems, always write out your binary grid first:
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
3. Conversion: Binary to Denary
To convert a binary number to denary, simply add the place values where a '1' appears.
Example: Convert 10101000 to Denary
- 128 + 0 + 32 + 0 + 8 + 0 + 0 + 0
- $128 + 32 + 8 = 168$
- Result: 168
4. Conversion: Denary to Binary
There are two main methods. The "Subtraction Method" is often the most intuitive for IGCSE students:
- Start from the left of the binary grid (128).
- Check if the Denary number is greater than or equal to the place value.
- If YES: Write a 1 and subtract the place value from your total.
- If NO: Write a 0 and move to the next place value on the right.
- Repeat until you reach the '1' column.
Example: Convert 75 to Binary
- Is $75 \ge 128$? No (0)
- Is $75 \ge 64$? Yes (1). Remaining: $75 - 64 = 11$
- Is $11 \ge 32$? No (0)
- Is $11 \ge 16$? No (0)
- Is $11 \ge 8$? Yes (1). Remaining: $11 - 8 = 3$
- Is $3 \ge 4$? No (0)
- Is $3 \ge 2$? Yes (1). Remaining: $3 - 2 = 1$
- Is $1 \ge 1$? Yes (1). Remaining: 0
5. Maximum Values
It is useful to remember the capacity of a standard 8-bit byte:
- Smallest value: 00000000 (0)
- Largest value: 11111111 (255)
- Total combinations: $2^8 = 256$ different values.