Cambridge IGCSE Computer Science (0478) · lesson 1 of 35 · 30 min
What Is the Binary Number System?
Computers store and process all data — numbers, text, images, sound — using only two states: on (1) and off (0). This two-symbol system is called binary (base-2). Every piece of information inside a computer is ultimately represented as a sequence of 1s and 0s called bits (binary digits).
The binary system works exactly like the decimal (base-10) system you use every day, except each column represents a power of 2 instead of a power of 10.
Place Values in Binary
In decimal, the columns from right to left are: 1, 10, 100, 1000 (powers of 10). In binary, the columns from right to left are powers of 2:
- Bit position
- 7
- 6
- 5
- 4
- 3
- 2
- 1
- 0
- Place value
- 128
- 64
- 32
- 16
- 8
- 4
- 2
- 1
To find the denary (decimal) value of a binary number, multiply each bit by its place value and add the results.
Worked Examples
Convert 10110101 to denary
Write out the bits under their place values:
- 128
- 64
- 32
- 16
- 8
- 4
- 2
- 1
- 1
- 0
- 1
- 1
- 0
- 1
- 0
- 1
Add the place values where the bit is 1:
128 + 32 + 16 + 4 + 1 = 181
Convert denary 75 to binary
Subtract the largest power of 2 that fits, place a 1, then continue:
- 75 ≥ 64 → bit 6 = 1, remainder = 11
- 11 < 32 → bit 5 = 0
- 11 < 16 → bit 4 = 0
- 11 ≥ 8 → bit 3 = 1, remainder = 3
- 3 < 4 → bit 2 = 0
- 3 ≥ 2 → bit 1 = 1, remainder = 1
- 1 ≥ 1 → bit 0 = 1, remainder = 0
75 = 01001011
Key Points
Binary is base-2; only digits 0 and 1 are used. Each bit position represents a power of 2, doubling from right to left. An 8-bit byte can store values from 0 (00000000) to 255 (11111111). To convert binary → denary: sum the place values where bits are 1. To convert denary → binary: repeatedly subtract the largest fitting power of 2.
What is the denary value of 00110010?
- 48
- 50
- 52
- 46
Answer: 50
32 + 16 + 2 = 50.
How many different values can a 4-bit number represent?
- 8
- 15
- 16
- 32
Answer: 16
2⁴ = 16 values (0 to 15).
What is 45 in 8-bit binary?
- 00101100
- 00101101
- 00101110
- 00110101
Answer: 00101101
32+8+4+1 = 45 → 00101101.
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