Number Systems: Binary and Denary (Decimal)

Understanding number systems is essential for OCR J277 Section 2.6 (Data Representation). Computers store and process all data using binary (base-2), but humans normally use denary (base-10, also called decimal). You must be able to convert fluently between these two systems.

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Denary (Base-10)

The denary number system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each position in a denary number represents a power of 10.

Position	Thousands (10^3)	Hundreds (10^2)	Tens (10^1)	Units (10^0)
Example: 4527	4	5	2	7
Value	4000	500	20	7

4527 = (4 x 1000) + (5 x 100) + (2 x 10) + (7 x 1) = 4527

This is the system you use every day, so it feels natural. Computers, however, use binary because their electronic circuits can only distinguish between two states: on (1) and off (0).

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Binary (Base-2)

The binary number system uses only two digits: 0 and 1. Each position represents a power of 2. A single binary digit is called a bit.

Position	128 (2^7)	64 (2^6)	32 (2^5)	16 (2^4)	8 (2^3)	4 (2^2)	2 (2^1)	1 (2^0)
Example: 10110011	1	0	1	1	0	0	1	1

To convert to denary: 128 + 32 + 16 + 2 + 1 = 179

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Converting Denary to Binary

Use the successive division method (divide by 2 and record remainders):

Example: Convert 156 to binary

Division	Quotient	Remainder
156 / 2	78	0
78 / 2	39	0
39 / 2	19	1
19 / 2	9	1
9 / 2	4	1
4 / 2	2	0
2 / 2	1	0
1 / 2	0	1

Read remainders from bottom to top: 156 = 10011100

Alternatively, use the place value method: find the largest power of 2 that fits and subtract:

• 156 - 128 = 28 (1 in 128 column)
• 28 - 16 = 12 (1 in 16 column, 0 in 64 and 32)
• 12 - 8 = 4 (1 in 8 column)
• 4 - 4 = 0 (1 in 4 column)
• Result: 10011100

OCR Exam Tip: The place value method is usually quicker in the exam and less prone to error. Always double-check by converting your binary answer back to denary.

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Converting Binary to Denary

Write out the place values and add up the columns where there is a 1.

Example: Convert 11010110 to denary

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

128 + 64 + 16 + 4 + 2 = 214

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Why Computers Use Binary

Reason	Explanation
Electronic circuits	Transistors have two states: on (1) and off (0)
Reliability	It is easy to distinguish between two voltage levels, reducing errors
Simplicity	Binary arithmetic is simpler to implement in hardware
Boolean logic	Binary maps directly to true/false logic operations

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Key Terms

Term	Definition
Bit	A single binary digit (0 or 1)
Nibble	4 bits
Byte	8 bits
Denary	Base-10 number system (0-9)
Binary	Base-2 number system (0, 1)
Place value	The value of a digit based on its position

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Practice

1. Convert 200 to 8-bit binary: 11001000
2. Convert 01101001 to denary: 64 + 32 + 8 + 1 = 105
3. What is the maximum value that can be stored in 8 bits? 255 (11111111 = 128+64+32+16+8+4+2+1)

OCR Exam Tip: The maximum value for an n-bit binary number is 2^n - 1. For 8 bits, this is 2^8 - 1 = 255. You should memorise this.

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Summary

• Denary uses digits 0-9 with place values as powers of 10
• Binary uses digits 0 and 1 with place values as powers of 2
• Convert denary to binary using division by 2 or the place value method
• Convert binary to denary by adding the place values where there is a 1
• Computers use binary because transistors have two states (on/off)
• The maximum value of an n-bit number is 2^n - 1