SLIDE 1 / 10
CSZone.co.uk
Click anywhere to advance · Arrow keys also work
AQA 7517 · Paper 2 · 4.5.2

Fixed-Point
Binary

Representing real numbers · Range vs precision · Section 4.5

WHAT YOU'LL LEARN
Fixed binary point · Fractional values · Negative place values · Range vs precision trade-off
AQA SPEC LINK
4.5.2 — Representation of numbers in fixed-point binary
The Concept

What is Fixed-Point Binary?

In fixed-point binary, a binary point is assumed to be at a fixed position in the bit pattern. Bits to the right of the point have fractional values (negative powers of 2).
2³2²2¹2⁰·2⁻¹2⁻²2⁻³
8421.½¼⅛
Conversion

Fixed-Point Conversion Examples

Assume binary point is after bit 4 (8 bits: 4 integer, 4 fractional).
Convert 0101.1010 to denary:
8421.½¼⅛1/16
0101.1010
4 + 1 + 0.5 + 0.125 = 5.625
Denary to Fixed-Point

Converting Fraction to Binary

Multiply fraction by 2; take integer part as next bit. Repeat.
Convert 0.625 to binary:
0.625 × 2 = 1.25 → bit = 1
0.25 × 2 = 0.5 → bit = 0
0.5 × 2 = 1.0 → bit = 1
Result: 0.101
Check: ½ + ⅛ = 0.5 + 0.125 = 0.625 ✓
Range vs Precision

Range vs Precision Trade-off

Given a fixed number of bits, moving the binary point left increases range but decreases precision, and vice versa.
MORE FRACTIONAL BITS
Higher precision (finer detail)
Smaller range (can't store large numbers)
MORE INTEGER BITS
Larger range
Less precision (cruder fractions)
Limitations

Limitations of Fixed-Point Binary

Cannot represent all real numbers exactly — e.g. 0.1 in binary is recurring (0.000110011…)
Fixed range — once the point position is set, you cannot represent very large AND very precise numbers simultaneously
Rounding errors — truncating the binary representation introduces small errors
Floating-point was developed to overcome these limitations by allowing the point to move
Negative Fixed-Point

Negative Fixed-Point Numbers

Use two's complement with fixed binary point. MSB has negative weight of -2^(n-1). Same inversion + add 1 process applies.
8-bit (4 int, 4 frac): -3.5
+3.5 = 0011.1000
Invert = 1100.0111
Add 1 = 1100.1000
Check: -8 + 4 + 0.5 = -3.5 ✓
Applications

When is Fixed-Point Used?

Embedded systems — microcontrollers without floating-point hardware (sensors, actuators)
Financial — money values where precision is more important than large range (£999.99 max)
Digital signal processing — audio/video processing chips use fixed-point for speed
Game physics — older games used fixed-point as floating-point was slow on CPUs
AQA Exam Style

Practice Question

AQA 7517 — Paper 2 Style
An 8-bit fixed-point number uses 5 bits for the integer part and 3 bits for the fractional part.
(a) Convert 01011.011 to denary. [2]
(b) Convert the denary value 6.75 to 8-bit fixed-point binary. [2]
(c) Explain the trade-off between range and precision in fixed-point representation. [2]
[6 marks]
2 marks
(a) 8+2+1 + 0.25+0.125 = 11 + 0.375 = 11.375
2 marks
(b) 6 = 00110; 0.75 = 0.11 → 00110.110
2 marks
(c) More fractional bits = greater precision but smaller maximum value (reduced range); more integer bits = larger range but less precise fractional representation
Summary

Key Points to Remember

Fixed-point — binary point at a fixed position; bits right of point are fractions (½, ¼, ⅛…)
Convert fraction: multiply by 2, take integer parts as bits
Range vs precision — more fractional bits = more precise but smaller range
Limitation: cannot represent all real numbers exactly; rounding errors occur
Floating-point overcomes fixed-point limitations at the cost of complexity
🎉 Lesson complete — move to the quiz!