🔢 Component 1 · 1.2 Data Representation
1.2.4a Binary & Denary Conversion
OCR J277 · GCSE Computer Science · ~11 min read
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Number Systems Used in Computing

OCR J277 requires knowledge of denary (base 10), binary (base 2), and hexadecimal (base 16). This lesson focuses on converting between denary and binary.

  • Denary (decimal) — the everyday number system. Digits 0–9. Base 10.
  • Binary — the number system computers use internally. Digits 0 and 1 only. Base 2.

Why Computers Use Binary

All data in a computer is represented as electrical signals — either ON (1) or OFF (0). Binary maps directly to these two states. Every piece of data — text, images, sound, programs — is ultimately stored and processed as sequences of 0s and 1s.

Binary Column Values (8-bit)

For an 8-bit binary number, each position (bit) has a place value that is a power of 2:

Bit position76543210
Column value1286432168421
Power of 22⁷2⁶2⁵2⁴2³2²2¹2⁰

The most significant bit (MSB) is on the left (value 128); the least significant bit (LSB) is on the right (value 1).

Binary to Denary Conversion

Method: Identify which bits are 1, then add up their column values.

Example: Convert 10110101 to denary

1286432168421
10110101

1s are in columns: 128, 32, 16, 4, 1

128 + 32 + 16 + 4 + 1 = 181

Denary to Binary Conversion

Method: Work from the largest column value downward. Subtract if it fits (put 1), skip if it doesn't (put 0).

Example: Convert 75 to binary

  • 128 > 75 → 0
  • 64 ≤ 75 → 1 (75 − 64 = 11 remaining)
  • 32 > 11 → 0
  • 16 > 11 → 0
  • 8 ≤ 11 → 1 (11 − 8 = 3 remaining)
  • 4 > 3 → 0
  • 2 ≤ 3 → 1 (3 − 2 = 1 remaining)
  • 1 ≤ 1 → 1 (1 − 1 = 0)

Result: 01001011

Check: 64 + 8 + 2 + 1 = 75 ✓

Key Worked Examples

DenaryBinary (8-bit)Check
000000000—
1000000011
10000010108+2=10
250001100116+8+1=25
1000110010064+32+4=100
20011001000128+64+8=200
25511111111128+64+32+16+8+4+2+1=255

Note: 255 is the maximum value for an 8-bit binary number (all bits = 1). An 8-bit number can represent 256 values: 0–255.

Exam tip: Always write your column headers (128, 64, 32, 16, 8, 4, 2, 1) before starting a conversion. This avoids place value errors. Always show your working in conversion questions — you can gain method marks even if your final answer is wrong.
⚠️ Common Mistakes
  • Starting from the wrong end — always start from the largest column value (128) and work right
  • Writing 9 bits instead of 8 — always use exactly 8 bits and include leading zeros (e.g. 00000101 not 101)
  • Adding column values incorrectly — write the check sum to verify
  • Confusing binary with denary — binary uses only 0 and 1; any digit 2–9 is NOT binary
  • Forgetting that 255 is the max 8-bit value and 256 values in total (0–255)
✅ Notes completed!
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What's in this video

  • • Why computers use binary (two-state logic)
  • • The 8-bit column header method: 128 64 32 16 8 4 2 1
  • • Binary → Denary: sum of active column values
  • • Denary → Binary: repeated subtraction method
Click slide or press arrow keys to navigate

Worksheet — 1.2.4a Binary & Denary

8 questions · 18 marks · Show all working

Q1State the column values for an 8-bit binary number, from MSB to LSB.[2]
✅ Mark scheme
128, 64, 32, 16, 8, 4, 2, 1 [2 — all 8 correct for 2 marks; accept 6+ for 1 mark]
Q2Convert the binary number 01001010 to denary. Show your working.[2]
✅ Mark scheme
Column values with 1s: 64, 8, 2 [1 for correct method]; 64 + 8 + 2 = 74 [1].
Q3Convert 10111001 to denary.[2]
✅ Mark scheme
128 + 32 + 16 + 8 + 1 [1] = 185 [1].
Q4Convert the denary number 47 to 8-bit binary. Show your working.[3]
✅ Mark scheme
128 → 0; 64 → 0; 32 ≤ 47 → 1 (15 left); 16 → 0 (15<16); 8 ≤ 15 → 1 (7 left); 4 ≤ 7 → 1 (3 left); 2 ≤ 3 → 1 (1 left); 1 → 1 [1 for method]; result 00101111 [1]; check 32+8+4+2+1=47 [1].
Q5Convert 130 to 8-bit binary.[2]
✅ Mark scheme
128 → 1 (2 left); 64 → 0; 32 → 0; 16 → 0; 8 → 0; 4 → 0; 2 → 1 (0 left); 1 → 0 [1 for method]; 10000010 [1].
Q6What is the maximum value that can be stored in an 8-bit binary number? State it in both binary and denary.[2]
✅ Mark scheme
11111111 in binary [1]; 255 in denary [1].
Q7How many different values can be represented by an 8-bit binary number? Explain how you calculated this.[2]
✅ Mark scheme
256 different values [1]; because 2⁸ = 256 (each bit has 2 possible states, and with 8 bits: 2×2×2×2×2×2×2×2 = 256) [1].
Q8Explain why computers use binary rather than denary to represent data.[3]
✅ Mark scheme
• Electronic components (transistors) have two stable states: on or off [1]
• Binary (0 and 1) maps directly to these two states [1]
• It is easier to build reliable circuits that distinguish two states than ten (0–9 in denary) [1]
• All data types can be encoded as binary patterns [1] (any 3)
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Mini Test — 1.2.4a Binary & Denary

10 questions · 10 marks · 10 minutes

  • • 5 multiple choice + 5 conversion questions
  • • Show all working for conversions
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