Binary Numbers and Denary Conversion

~35 min · 7 sections · 0984

Why do computers use binary?

All data inside a computer is stored and processed as sequences of binary digits (bits). Each bit can only be in one of two states: 0 (off) or 1 (on). This maps directly onto the physical switches inside a processor — a transistor that is not conducting current represents 0, and one that is conducting represents 1.

Using a two-state system makes hardware reliable and simple. It is much easier to distinguish reliably between two voltage levels than between ten, which is why denary (base 10) arithmetic is not used internally.

Key vocabulary

Term	Definition
Bit	A single binary digit: 0 or 1
Nibble	A group of 4 bits
Byte	A group of 8 bits
Kilobyte (KB)	1,024 bytes (2¹⁰)
Megabyte (MB)	1,024 KB = 1,048,576 bytes
Gigabyte (GB)	1,024 MB
Terabyte (TB)	1,024 GB

**Exam tip:** Cambridge papers use the binary prefixes above (1 KB = 1,024 bytes), not the SI prefixes (1 kB = 1,000 bytes). Always use the powers-of-two definitions in answers.

Binary place values

Binary is a base-2 number system. Each column position represents a power of 2, just as each column in denary represents a power of 10.

2⁷	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰
128	64	32	16	8	4	2	1

An 8-bit byte can therefore hold values from 0 (all bits 0) to 255 (all bits 1):

1111 1111₂ = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255₁₀

The subscript ₂ denotes a binary number and ₁₀ denotes denary (base 10).

Converting binary to denary

Method: write the binary number under the column headings, then add up the values of columns that contain a 1.

Step-by-step approach:

1. Write the column headings from right to left: 1, 2, 4, 8, 16, 32, 64, 128.

2. Write each bit under its column.

3. For each column that has a 1, add its heading value to a running total.

4. The total is the denary value.

Worked Example

Worked example: binary to denary

Convert 1011 0110₂ to denary.

128	64	32	16	8	4	2	1
1	0	1	1	0	1	1	0

Columns with a 1: 128, 32, 16, 4, 2

128 + 32 + 16 + 4 + 2 = 182

Therefore 1011 0110₂ = 182₁₀

Convert 0100 1101₂ to denary.

128	64	32	16	8	4	2	1
0	1	0	0	1	1	0	1

Columns with a 1: 64, 8, 4, 1

64 + 8 + 4 + 1 = 77

Therefore 0100 1101₂ = 77₁₀

Converting denary to binary

Method — repeated subtraction:

1. Start with the largest power of 2 that is ≤ your number (for 8-bit: start with 128).

2. If the column value fits into the remaining number, write 1 and subtract the column value.

3. If it does not fit, write 0.

4. Move to the next column and repeat.

Method — repeated division by 2:

1. Divide the number by 2 and record the remainder (0 or 1).

2. Repeat with each quotient until you reach 0.

3. The binary number is the remainders read from bottom to top.

Both methods give the same result. The repeated-subtraction method maps more directly to column headings and is usually faster for exam questions.

Worked Example

Worked example: denary to binary

Convert 93₁₀ to an 8-bit binary number.

Using repeated subtraction:

Column	128	64	32	16	8	4	2	1
Fits?	No (93 < 128)	Yes (93 ≥ 64)	Yes (29 ≥ 32)? No — 93 − 64 = 29, 29 < 32	No (29 < 32)	Yes (29 ≥ 16, 29 − 16 = 13)	—	—	—

Let me redo this cleanly:

Start: 93

128: 93 < 128 → write 0, remaining = 93
64: 93 ≥ 64 → write 1, remaining = 93 − 64 = 29
32: 29 < 32 → write 0, remaining = 29
16: 29 ≥ 16 → write 1, remaining = 29 − 16 = 13
8: 13 ≥ 8 → write 1, remaining = 13 − 8 = 5
4: 5 ≥ 4 → write 1, remaining = 5 − 4 = 1
2: 1 < 2 → write 0, remaining = 1
1: 1 ≥ 1 → write 1, remaining = 0

Result: 0101 1101₂

Check: 64 + 16 + 8 + 4 + 1 = 93 ✓

Convert 200₁₀ to an 8-bit binary number.

128: 200 ≥ 128 → 1, remaining = 72
64: 72 ≥ 64 → 1, remaining = 8
32: 8 < 32 → 0
16: 8 < 16 → 0
8: 8 ≥ 8 → 1, remaining = 0
4: 0 < 4 → 0
2: 0 < 2 → 0
1: 0 < 1 → 0

Result: 1100 1000₂

Check: 128 + 64 + 8 = 200 ✓

Common Misconception

Common misconceptions

Misconception 1: "Leading zeros don't matter in binary"

In denary, 093 and 93 are the same. However, in binary contexts the number of bits matters. An 8-bit number uses all 8 positions. When asked to give an 8-bit answer, you must include leading zeros:

Correct: 0101 1101 (93 in 8-bit binary)
Wrong: 101 1101 (only 7 bits — marks may be lost for missing bits)

Misconception 2: "The largest 8-bit number is 256"

The largest value of an 8-bit *unsigned* binary number is 255 (1111 1111). The number 256 requires 9 bits (1 0000 0000). If someone says "8-bit numbers go up to 256", they are confusing the count of possible values (256 values: 0–255) with the maximum value.

Misconception 3: Misremembering place values

Students sometimes guess "powers of 2" as 2, 4, 8, 18, 32... The values are 1, 2, 4, 8, 16, 32, 64, 128. Notice 16 (not 18). Writing out the column headings first, before answering, avoids this mistake.

Exam-Style Practice Questions

Test your knowledge with original Cambridge-style questions including short-answer and pseudocode problems.

Practice questions Pseudocode exercises