Computer Number Representation

Say we have six bit representation (not single, double) (see the Fig.)

• $1~bit\rightarrow sign$
• $3(+1)~bits\rightarrow mantissa$
• $2~bits\rightarrow exponent$

Figure 2.5: Computer numbers with six bit representation.

• For positive range $\frac{9}{32}\longleftrightarrow \frac{15}{4}$
• For negative range $\frac{-15}{4}\longleftrightarrow \frac{-9}{32}$; even discontinuity at point zero since it is not in the ranges.

Figure 2.6: Upper: number line in the hypothetical system, Lower: IEEE standard.

• Very simple computer arithmetic system $\Rightarrow$ the gaps between stored values are very apparent.
• Many values can not be stored exactly. i.e., 0.601, it will be stored as if it were 0.6250 because it is closer to $\frac{10}{16}$, an error of $4\%$
• In IEEE system, gaps are much smaller but they are still present. (see the lower Fig. 2.6)