Binary fraction to decimal
A binary fraction uses powers of 2 instead of powers of 10. To convert it to decimal, read each bit by place value, just as you would with a decimal number.
Method
- Identify the place values.
- Digits to the left of the binary point represent $2^0, 2^1, 2^2, \dots$
- Digits to the right represent $2^{-1}, 2^{-2}, 2^{-3}, \dots$
- Multiply each bit by its place value.
- A digit 1 keeps the place value.
- A digit 0 contributes nothing.
- Add the results.
- This gives the decimal value.
Example idea
For a number like $101.01_2$:
- $1\cdot 2^2 = 4$
- $0\cdot 2^1 = 0$
- $1\cdot 2^0 = 1$
- $0\cdot 2^{-1} = 0$
- $1\cdot 2^{-2} = 0.25$
So the decimal value is $4 + 1 + 0.25 = 5.25$.
Check your answer
- Recompute the sum carefully.
- If there is a fractional part, make sure the powers of 2 on the right side become smaller: $\tfrac12, \tfrac14, \tfrac18, \dots$
- For exact answers, write the decimal clearly and simplify if the result has a terminating decimal.
Helpful habit
Work from left to right, and keep track of the power of 2 under each digit. This makes it much easier to avoid place-value mistakes.