Number Representation

Binary Floating Point

We can develop a similar notation in binary.

Example:

    001011001 = 1.011001 * 26

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)

How many bits do we use for each component?

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.

IEEE 754 Standard

32-bit word

31	30      23	22                   0
S	Exp	Significand
	↑ 8 bits	↑ 23 bits

Range is approx. 2.0 * 10-38 → 2.0 * 1038

64-bit word

63	62        52	51                                       0
S	Exp	Significand
	↑ 11 bits	↑ 52 bits

Range is roughly 2.0 * 10-308 → 2.0 * 10308 (bigger range and more accurate)

Note: A number can be too big, but also too small.
Overflow: exponent too large
Underflow: negative exponent too large