Floating Point

Representing Fractions

We have a binary point that signifies the boundary between the integer and fractional portion. That is, the number to the right of the decimal point represents $2^{-1}$, and the number to the right of the decimal number represents $2^1$. $10.1010_2 = 2^1 + 2^{-1} + 2^{-3}$ .

Addition with a fixed point:

Same as traditional addition.

Multiplication with a fixed point:

Same as traditional multiplication.

Floating Binary Points

If we could just move the binary point wherever we wanted, we could store much more varied information. To do this we use a variant on scientific notation. It has a few components: given a scientific number $a \times 2^{23}$, $a$is the mantissa, $23$is the exponent, $2$is the radix, or base. In IEEE 754 Floating Point Standard, we dedicate one bit to the sign, 8 bits for our exponent, 23 bits for our significand.

We can use bias notation to represent more numbers, where the bias is a number subtracted to get the real number.