Floating-Point Basics (Cambridge International A-Level Computer Science)

Floating-point binary

What is floating-point binary?

• Floating-point binary represents real numbers in binary: numbers with a fractional part, and numbers that are very large or very small

• It works like scientific notation, but in base 2

• The number is written as a mantissa multiplied by 2 to the power of an exponent

• In denary, 3140 = 3.14 × 10³

  • Here 3.14 is the mantissa (the significant digits) and 3 is the exponent (how far to move the decimal point)

• Floating-point binary does exactly this using powers of 2

Why use floating-point?

• It represents a much wider range of numbers than fixed-point using the same number of bits
• The binary point can "float", giving good precision for fractional values

The two parts

• Both the mantissa and the exponent are stored in two's complement

Positive numbers: example +6.25

• 6.25 = 110.01₂
• Normalise: 110.01 = 0.11001 × 2³
• Mantissa (12-bit) = 0 then fraction 11001, padded = 011001000000
• Exponent (4-bit) = +3 = 0011

Negative numbers: example −6.25

• Work out the positive mantissa first, then take its two's complement (flip the bits and add 1)
• Do not just change the sign bit

• So −6.25: Mantissa = 100111000000, Exponent = 0011
• The mantissa now begins 10, which is the correct start for a normalised negative number

Normalising floating-point numbers

What is normalisation?

• A floating-point number is normalised when the first two bits of the mantissa are different:

  • 01… for a positive number (the form 0.1xxxx)
  • 10… for a negative number (the form 1.0xxxx)

Why normalise?

• Precision: removing leading zeros means the mantissa uses all of its bits for significant figures, giving the most accurate value possible for that number of bits
• Uniqueness: each number has exactly one normalised form, which makes comparison and arithmetic simpler

How to normalise

1. Shift the bits until the first two bits differ (01 for positive, 10 for negative)

2. Adjust the exponent by the number of places the binary point moved:

3. point moved left → increase the exponent

4. point moved right → decrease the exponent

Example (positive)

• Before: mantissa 0.0011, exponent +2
• Move the binary point 2 places right so the mantissa becomes 0.1100 (first two bits are now 01)
• Decrease the exponent by 2: +2 → 0
• Normalised: mantissa 01100, exponent 0000 (the value is unchanged: 0.0011 × 2² = 0.11 × 2⁰)

Exam Hint

• The whole mantissa is two's complement. For a negative number, find the positive mantissa, then flip the bits and add 1. Changing only the sign bit is a common mistake that loses every mark on the conversion.
• Both the mantissa and the exponent are in two's complement.
• Check your answer with the first two bits: a normalised positive number starts 01, a normalised negative number starts 10. If a negative number starts 11, it is not normalised (or you used sign-and-magnitude by mistake).
• Write the normalised form as 0.xxxxx × 2ⁿ (CIE), not 1.xxxxx × 2ⁿ (that is IEEE).