We can develop a similar notation in binary.

Example:

`001011001 = 1.011001 * 2`

^{6}Since we know the base (

`2`

), we need to keep track of

- the sign (positive or negative)
- the significand (the significant digits,
e.g.,`1011001`

)- the exponent (
e.g.,`6`

)

Sign: 1 bit

Accuracy vs. Range

- If we give more bits to the significand, we have greater accuracy.
- If we give more bits to the exponent, we have greater range.

## 32-bit word

31 30 23 22 0 S Exp Significand ↑ 8 bits ↑ 23 bits Range is approx.

`2.0 * 10`

^{-38}→ 2.0 * 10^{38}## 64-bit word

63 62 52 51 0 S Exp Significand ↑ 11 bits ↑ 52 bits Range is roughly

`2.0 * 10`

(bigger range^{-308}→ 2.0 * 10^{308}andmore accurate)Note: A number can be

too big, but alsotoo small.

Overflow: exponent too large

Underflow: negative exponent too large

Alyce Brady, Kalamazoo College