⌚ September 10, 2014
We can express fractional values in binary just as we can in decimal. A radix point separates the whole number portion from the fractional portion in binary just as a decimal point separates the whole number portion from the fractional portion in decimal.
The dot (.) serves the same purpose as a decimal point to separate the whole number part from the fractional part, but it is called a radix point in binary. The only difference is the name. All place values to the left of the radix point are 1 or greater, while all place values to the right of the radix point are less than one.
Fractional Place Value
The value of a bit is determined by the value of the place it occupies in a place value chart. Values to the left are greater while values to the right are smaller. We find the place values of places right of the radix point by raising 2 to negative powers starting with 2^-1. (The caret symbol ^ means to raise to a power.)
Recall the definition of negative exponents from algebra.
We use this definition to form a binary place value chart containing fractional places.
Immediately to the right of the radix point, start with 2^-1. There is only one 2^0, and it is located immediately to the left of the radix point as before. For each place to the right, raise 2 to a successive negative power: 2^-2, 2^-3, 2^-4, and so on. To find the value of each place, convert the negative exponent into a fraction. The rules are the same as in algebra.
Just as we raise 2 to a successive positive power to find the place values to the left of the radix point, we can raise 2 to successive negative powers to find the place values to the right of the radix point.
The first place to the right of radix point is 1/2, not 1, because 1 is a whole number and whole numbers are located to the left of the radix point. We obtain the first fractional place, 1/2, by dividing 1 by 2. This technique leads to another way to find the fractional place values. We can divide each successive place on the right of the radix point by 2.
For an easier shortcut in finding fractional place values, count 2, 4, 8, 16, 32, and so on and let the number be the denominator while keeping the numerator as 1. This produces 1/2, 1/4, 1/8, 1/16, and 1/32. There is no limit to the number of fractional bits possible, so use only those needed.
Zeros to the right of the least significant bit have no value just as zeros to the left of the most significant bit have no value. Both can be discarded unless padding is required for readability.
0001b = 1b
.1100b = .11b
01.10b = 1.1b
Converting Binary Fractions to Decimal
Convert the whole part to decimal the same as before, and convert the fractional part to decimal by adding the fractions. Line up the radix point of the binary number with the radix point of the place value chart.
Convert .001b to decimal.
First, create a place value table for each fractional place. Since no whole part exists, we can create a place value chart consisting only of fractional places. Remember to start with 2^-1 immediately to the right of the radix point, not 2^0. Sum all of the fractional values whose place contains a 1 bit.
.001b = 1/8 (base 10)
Convert 1.11b to decimal.
Create a binary place value chart. Line up the binary number by its radix point to ensure that each bit is assigned to the correct place value.
1.11b = 7/4d
Mixed number and improper fraction sums are both correct. In the mixed number sum, the 1 is obtained from the left side of the radix point, and the fractional part is obtained from the right side. The two sums are then added together to form a mixed number. This allows us to calculate the whole number part and the fractional part separately.
Convert 011.101000b to decimal.
The least significant bit is the 1/8 place, so we can drop the three trailing zeros and create a place value chart with three fractional places instead of six. The leading zero before the most significant bit can also be dropped, so the whole number place value chart will contain two relevant places.
Do not use a calculator. All of these problems can be solved with pencil, paper, and some thought.
1. Create a binary place value chart containing four fractional places. Label each place with a power (base 2) and its place value.
2. Create a binary place value chart with five whole number places and three fractional places.
3. True or false: A binary place value chart containing a fractional part may use a decimal point.
4. Create a binary place value chart that can accommodate 1.1b.
5. Create a binary place value chart that can accommodate 10110.001101b.
6. Convert 1.0001b to decimal.
7. Convert 11.11b to decimal.
8. Convert 10101 to decimal.
9. Convert .10001 to decimal.
10. Create a binary place value chart containing ten whole places and twelve fractional places. Label the powers and place values.
Did you remember to include the radix point? If not, then the answer is wrong because a place value chart without a radix point indicates whole numbers, not fractions. The radix point helps avoid ambiguity.
3. False. Though the meaning is the same, the names are different. The “dot” is called a decimal point when used with decimal numbers in a base 10 number system, and the “dot” is called a radix point when used in the base 2 number system.
1.0001b = 1 + 1/16 = 17/16d
Another way to think of this is 2^0 + 2^-4 = 1 + 1/16 = 17/16d.
11.11b = 3 + 1/2 + 1/4 = 12/4 + 2/4 + 1/4 = 15/4d
8. 10101b = 21d (No fractional part exists. This is a whole binary value.)
.10001b = 1/2 + 1/32 = 16/32 + 1/32 = 17/32d