Representation of Floating-Point Numbers

We saw that the exponents of fractions are negative (e.g., $0.0012 = -1.2 \times 10^{-4}$). Nonetheless, the sign bit we have is for the significand, not the exponent. So, how do we represent negative exponents?

To represent negative exponents, we use bias (also called excess,) which is a number that is added to each exponent to shift all the numbers forward. The bias will always be equal to $2^{\text{exponent bits} - 1} - 1$, and will allow us to store negative numbers within the exponent bits, without the need to add an additional sign bit.

For example, in our 5-bit exponent format, the bias will be equal to: $2^{5 - 1} - 1 = 2^4 - 1 = 16 - 1 = 15$. This creates an excess-15 exponent. Therefore, the exponent $1$ will be represented as $1 + 15 = 16_{10} = 10000_2$, $2$ as $17_{10} = 10001_2$, etc., while the exponent $-5$, for example, is represented as $-5 + 15 = 10_{10} = 001010_2$.

Note that we don't use 1's or 2's complement neither for the exponent nor for the significand. In fact, in our floating-point representation, both the exponent and significand are represented in a manner similar to the sign & magnitude format.