Floating-point numbers, representation and manipulation — common mistakes

Pay close attention to the normalisation rules: the mantissa for a positive number must start 0.1, and for a negative number, it must start 1.0. Always double-check the number of bits specified for both the mantissa and the exponent in the exam question, as this is a common source of error.

Exam questions frequently ask about the consequences of changing the bit allocation. For example, 'What would be the effect of increasing the number of bits used for the exponent?'. The answer should always mention the impact on both range (increases) and precision (decreases).

This is a convention to ensure a unique representation. A normalised negative number in two's complement will always have its most significant bits as '10...'. The '1' indicates it's negative, and the '0' that follows is the most significant bit of the magnitude, ensuring no leading zeros are wasted after the sign bit, thus maximising precision.

Overflow occurs when a calculation results in a number whose magnitude is too large to be represented by the exponent (e.g., trying to store a number larger than the maximum possible value). Underflow occurs when a number is too small (i.e., too close to zero) to be represented, and the computer must round it down to zero.

First, convert the mantissa and exponent from two's complement binary to denary fractions/integers. Then, apply the formula: Denary Value = Mantissa × 2^Exponent. For example, if the mantissa is 0.11 (0.75) and the exponent is 3, the value is 0.75 × 2^3 = 0.75 × 8 = 6.

Yes. A negative exponent means the binary point should be shifted to the right in the mantissa, which results in a number smaller than the mantissa's value (i.e., a number between -1 and 1, excluding 0). The exponent is stored in two's complement, so it can represent both positive and negative values.