floating-point-basics

Floating-point binary

What is floating-point binary?

• Floating-point binary is a method of representing real numbers in binary, including fractions and very large or very small values

• Uses scientific notation in base-2

• Unlike fixed-point (where the binary point is fixed), the binary point can “float” using a mantissa and an exponent

Why use floating-point?

• Can represent a much wider range of numbers

• Provides more precision for fractional values

• Makes efficient use of limited storage (e.g. 8-bit, 16-bit representations)

Components of floating-point

Example in decimal

3.14 × 10³

• Mantissa = 3.14

• Exponent = 3

Example in binary

110.01₂ = 0.11001 × 2³

• Mantissa = 011001… (sign 0, fraction 11001 padded to available bits)

• Exponent = 0011 (+3 in two’s complement)

Positive floating-point representation

• Sign bit = 0

• Fraction part stored in the remaining mantissa bits

• Exponent shifts the binary point

Negative floating-point representation

• Sign bit = 1

• Fraction part stored in the remaining mantissa bits

• Exponent is in two’s complement

• Example:

  • −6.25 = 110.01₂ = 0.11001 × 2³

  • Mantissa = 111001000000 (sign = 1, fraction = 11001 padded)

  • Exponent = 0011

Normalising floating-point numbers

• A floating-point number is normalised when the mantissa begins with:

  • 01 for positive numbers

  • 10 for negative numbers

Why normalise?

Steps to normalise a floating-point number

1. Shift the binary point until the mantissa begins with 01 (positive) or 10 (negative)

2. Adjust the exponent by the number of shifts:

   • Moving left → increase exponent

   • Moving right → decrease exponent

Example

• Before normalisation:

  • Mantissa = 00011

  • Exponent = 0010 (+2)

• Process:

  • Shift binary point 2 places right → mantissa = 01100

  • Decrease exponent by 2 → exponent = 0000

• After normalisation:

  • Mantissa = 01100

  • Exponent = 0000