To floating-point binary

Assume 10 bits for the mantissa and 6 bits for the exponent. Give your answer as a normalised number.

Convert the denary number to floating-point binary: 0

Floating-point uses a mantissa and exponent to represent numbers in the form: mantissa × 2exponent

Normalised form: The mantissa should start with 0.1 for positive or 1.0 for negative numbers.

Steps:

  1. Convert the number to binary (including fractional part)
  2. Normalise by shifting the binary point
  3. Store the mantissa (10 bits) and exponent (6 bits in two's complement)

Floating-point binary represents numbers using scientific notation in binary. A number is stored as mantissa × 2exponent.

Normalisation

A normalised floating-point number has the binary point positioned so that:

  • Positive numbers: Start with 0.1 (the first significant bit after the point is 1)
  • Negative numbers: Start with 1.0 (in two's complement form)
Method
  1. Convert the denary number to binary (whole and fractional parts)
  2. Normalise by moving the binary point to the correct position
  3. Count how many places you moved the point - this is your exponent
  4. Store the mantissa (10 bits) and exponent (6 bits in two's complement)
Example: Convert 6.5 to floating-point
6.5 in binary: 110.1
Normalise: 0.1101 × 2³
Mantissa: 0110100000 (pad to 10 bits)
Exponent: 3 = 000011 (in 6-bit two's complement)

References:

  • Wikibooks Computer Science A level (floating-point): AQA OCR
  • Wikibooks Computer Science A level (normalisation): AQA

Further reading: