(A Level) Floating-point representation and normalisation

Learning objectives

Represent a real number in binary floating-point form as a two's complement mantissa with a two's complement exponent, and convert such a representation back into denary

Normalise a floating-point number by shifting the mantissa until its first two bits differ and adjusting the exponent to match, and explain why normalisation gives the greatest precision available from a fixed number of bits

Explain the trade-off made when bits are allocated between the mantissa and the exponent, relating mantissa bits to precision and exponent bits to the range of values that can be held

Describe the causes of rounding and truncation errors when real numbers are stored in a fixed number of bits, and explain why testing two floating-point values for exact equality is unsafe

Compare the storage of integers with the storage of real numbers, and justify choosing a floating-point type rather than an integer type for a stated calculation

Explain how overflow and underflow occur when a result lies outside the range of a floating-point format, and work out the largest and the smallest positive normalised values a given format can hold