Binary Fractions
8 - 8 C2000 Microcontroller Workshop - Numerical Concepts
Binary Fractions
Given the problems associated with integers and multiplication, consider the possibilities of using
fractional values. Fractions do not grow when multiplied, therefore, they remain representable
within a given word size and solve the problem. Given the benefit of fractional multiplication,
consider the issues involved with using fractions:
• How are fractions represented in two's complement?
• What issues are involved when multiplying two fractions?
Representing Fractions in Binary
In order to represent both positive and negative values, the two's complement process will again
be used. However, in the case of fractions, we will not set the LSB to 1 (as was the case for
integers). When one considers that the range of fractions is from -1 to ~+1, and that the only bit
which conveys negative information is the MSB, it seems that the MSB must be the “negative
ones position.” Since binary representation is based on powers of two, it follows that the next bit
would be the “one-halves” position, and that each following bit would have half the magnitude
again. Considering, as before, a 4-bit model, we have the representation shown in the following
example.
1
.
0 1 1 = -1 + 1/4 + 1/8 = -5/8
-1 1/2 1/4 1/8
Fraction Basics
Fraction Basics
-2
0
2
-1
2
-2
2
-3
•
1101b = (1*-2
0
)+(1*2
-1
)+(0*2
-2
)+(1*2
-3
)
= -1 + 1/2 + 1/8
= -3/8
Fractions have the nice property that
fraction x fraction = fraction
2
-(n-1)
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