Binary Fractions

So far we have worked with representing whole numbers or integers but what about fractions?

Unsigned Binary Fractions (radix point):

  • Conceptually, the radix point, . delimits or separates integers from fractions
  • Note: the decimal point is a radix point.
  • Fraction digit/column values inversely mirror the values of the integers:

. . . 8 4 2 1 .1/2 1/4 1/8 1/6 1/32 . . .

So, the unsigned binary number 1011.1100 converts to 8 + 0 + 2 + 1 + 1/2 + 1/4 + 0 + 0 = 11 and 3/4

Summing up the integer part as per standard binary number.

  • To sum up the fraction part, simply find the least significant bit column with a "1" which will determine the lowest common denominator.

  • Each column to the left will be 2x the value and each column to the right will be 1/2 the value.

  • In the example above, 1/4 column is the least significant column with a 1, so you know that there is at least one 1/4.

  • How many more 1/4s are there?

  • Since there is a 1 in the 1/2 column which is 2x 1/4, there are 2 more 1/4s. (Following this scheme, the next column to the left is the 1s column which of course is 4x 1/4 and so on.)

  • Therefore there are 3 in total 1/4 or 3/4 being represented in the fractions part.

  • The number, 1011.1100 in decimal then is 11 3/4.

Now we are ready to combine signed integers and fractions to represent rational numbers.