Fractional Binary

Using a similar mechanisms to unsigned binary, we can represent fractional values in binary using what we refer to as fractional binary. Note that we cover this because this is a representation that will appear as an intermediary value when performing conversion from the standard IEEE 754 conversion of a floating point value to and from their base 10 representations.

Example 1

Convert the fractional binary value 101.11 to decimal.

Solution

We currently have an understanding that each place value on the left hand side of the radix point is represented by a positive power of two. We can extend this understanding to the right hand side of the radix point by using negative powers of 2.

Binary value	1	0	1	.	1	1
Power of 2	2²	2¹	2⁰	.	2⁻¹	2⁻²
Decimal value	4	2	1	.	0.5	0.25

Thus, we just need to find the sum of \(4 + 1 + 0.5 + 0.25\). This give the decimal value of \(5.75\).

Example 2

Convert the fraction \(\dfrac{26}{128}\) to fractional binary.

Solution

We can rewrite this value as \(26 \times \dfrac{1}{128}\) or \(26 \times 2^{-7}\).

Convering 26 to binary gives \(11010\).

Multiplying by \(2^{-7}\) moves the radix point 7 places to the left. Leading 0s are added as needed.

\[ \begin{aligned} & 00011010. \quad (0)\\ & 0001101.0 \quad (1)\\ & 000110.10 \quad (2)\\ & 00011.010 \quad (3)\\ & 0001.1010 \quad (4)\\ & 000.11010 \quad (5)\\ & 00.011010 \quad (6)\\ & 0.0011010 \quad (7) \end{aligned} \]

The final result is \(0.0011010\)