Number Systems: Binary and Denary

Understanding number systems is the foundation of data representation in Computer Science. Computers operate using binary (base-2), while humans typically use denary (base-10, also called decimal). This lesson explains both systems and how to convert between them.

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The Denary (Decimal) System

The denary system is the number system you use every day. It is a base-10 system, meaning it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Each column in a denary number represents a power of 10:

Thousands (10³)	Hundreds (10²)	Tens (10¹)	Units (10⁰)
1000	100	10	1

For example, the number 3742 means:

• 3 × 1000 = 3000
• 7 × 100 = 700
• 4 × 10 = 40
• 2 × 1 = 2
• Total = 3742

This is called positional notation — the value of a digit depends on its position in the number.

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The Binary System

The binary system is a base-2 system. It uses only two digits: 0 and 1. Each digit in a binary number is called a bit (short for binary digit).

Each column in a binary number represents a power of 2:

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

Why Do Computers Use Binary?

Computers are built from billions of tiny electronic switches called transistors. Each transistor can be in one of two states:

• On (represented by 1)
• Off (represented by 0)

Because there are only two states, binary is the natural number system for computers. Everything a computer processes — numbers, text, images, sound — is ultimately stored as sequences of 0s and 1s.

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Converting Binary to Denary

To convert a binary number to denary, write out the place values and add up the columns that contain a 1.

Worked Example 1: Convert 10110101 to denary

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

Worked Example 2: Convert 01001100 to denary

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

• 64 + 8 + 4 = 76

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Converting Denary to Binary

To convert a denary number to binary, use the successive division by 2 method or the place value method.

Method 1: Place Value Method

1. Write out the column headings: 128, 64, 32, 16, 8, 4, 2, 1.
2. Starting from the largest value, ask: "Does this value fit into the remaining number?"
3. If yes, write a 1 and subtract that value. If no, write a 0.
4. Repeat for each column.

Worked Example: Convert 200 to binary

• 200 ≥ 128? Yes → 1, remainder = 72
• 72 ≥ 64? Yes → 1, remainder = 8
• 8 ≥ 32? No → 0
• 8 ≥ 16? No → 0
• 8 ≥ 8? Yes → 1, remainder = 0
• 0 ≥ 4? No → 0
• 0 ≥ 2? No → 0
• 0 ≥ 1? No → 0

200 in binary = 11001000

Method 2: Successive Division by 2

1. Divide the number by 2 and note the remainder.
2. Keep dividing the quotient by 2, noting each remainder.
3. Read the remainders from bottom to top to get the binary number.

Worked Example: Convert 53 to binary

Division	Quotient	Remainder
53 ÷ 2	26	1
26 ÷ 2	13	0
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading remainders bottom to top: 53 in binary = 110101

(As an 8-bit binary number with leading zeros: 00110101)

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Key Terminology

• Bit: A single binary digit (0 or 1). The smallest unit of data.
• Nibble: 4 bits (e.g. 1010).
• Byte: 8 bits (e.g. 10110101). A byte can represent values from 0 to 255 in unsigned binary.
• Most Significant Bit (MSB): The leftmost bit, with the highest place value.
• Least Significant Bit (LSB): The rightmost bit, with the lowest place value (1).

Exam Tip: Always double-check your binary-to-denary conversions by adding up the place values. A common mistake is misaligning the place values. An 8-bit number has a maximum unsigned value of 255 (11111111 in binary = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1).

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Summary

• Denary (base-10) uses digits 0–9 and powers of 10 for place values.
• Binary (base-2) uses digits 0 and 1 and powers of 2 for place values.
• Computers use binary because transistors have two states: on (1) and off (0).
• To convert binary to denary, add together the place values of all the 1-bits.
• To convert denary to binary, use the place value method or successive division by 2.
• An 8-bit binary number can represent unsigned values from 0 to 255.