4. Fractional components

Numbers in any base can have a fractional component.

• $51.5_{10}$ (decimal point)
• $1011.1101_2$ (binary point)

We can use negative powers to easily convert these components:

• Take for example, $0.101_2$.
• We just continue in the downward trend: $$\begin{array}{rl}0.101_2& =2^{-1}+2^{-3}\\ & =0.625_{10}\end{array}$$

Some fractional components may have an infinite series of digits in other bases.

• For example, $0.3_{10}$ will never terminate in binary.
• Or, $\frac{2}{3}=0.{\bar{6}}_{10}=0.2_3$.

As such, we may not always be able to accurately convert fractional components to other bases. A simple way we use to deal with loss of accuracy is to specify how many digits of accuracy you are using.

• For example $0.3_{10}$ in binary is $0.01100_2$ to 5 bits of accuracy.