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Non-integral mathematics 253

2. 25 00010. 010 11. 125 01011. 001
+ 1. 50 = + 00001. 100 − 5. 625 = + 11010. 011
3. 75 00011. 110 5. 500 00101. 100

−12. 375 10011. 101
+ 5. 250 = + 00101. 010

− 7. 125 11000. 111
Figure 8.1: Examples of fixed point signed arithmetic.

changing its format, until the radix points are aligned. The choice of which one to shift de-
pends on what format we desire for the result. If we desire eight bits of fraction in our result,
then we would shift the S(7,4) left by four bits, converting it into an S(7,8). With the radix
points aligned, we simply use an integer addition operation to add the two numbers. The result
will have it’s radix point in the same location as the two numbers being added.

8.4.2 Fixed point multiplication

Recall that the result of multiplying an n bit number by an m bit number is an n + m bit num-
ber. In the case of fixed point numbers, the size of the fractional part of the result is the sum
of the number of fractional bits of each number, and the total size of the result is the sum of
the total number of bits in each number. Consider the following example where two U(5,3)
numbers are multiplied together:

00011. 110
× 00010. 100
0001. 1110
000111. 10
0000001001. 011000

The result is a U(10,6) number. The number of bits in the result is the sum of all of the bits
of the multiplicand and the multiplier. The number of fractional bits in the result is the sum
of the number of fractional bits in the multiplicand and the multiplier. There are three simple
rules to predict the resulting format when multiplying any two fixed point numbers.