9618 · 13.3
Floating-point numbers, representation and manipulation — FAQ
Frequently asked questions for 9618 Floating-point numbers, representation and manipulation. Direct answers first, then deeper explanation — then practise with marking.
Why is the normalisation rule for negative numbers '1.0...'?
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.
What is the difference between floating-point overflow and underflow?
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.
How do I convert from a normalised floating-point number back to denary?
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.
Can the exponent be negative?
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.