How to Convert from Binary to Decimal?
- Identify the Base of the Number to be Converted to Decimal.
- Split the Number into Whole and Fractional Part.
- The part before the point is the whole number part.
- The part after the point is the fractional part.
- If there’s no point, just treat it as a whole number.
- Convert the Whole Number Part.
- Assign index 0 to the rightmost digit, then increase the index as you move left.
- For each digit:
- Multiply the digit by the base raised to the power of its index.
- Add the result to a running total.
- Convert the Fractional Part.
- Assign index -1 to the leftmost digit, then decrease the index as you move right.
- For each digit:
- Multiply the digit by the base raised to the power of its index.
- Add the result to a running total.
- Combine the Results.
- The total from the whole part and the total from the fractional part give you the final decimal value.
To confirm that our decimal-to-binary conversion was correct, we’ll now take the binary number we obtained earlier and convert it back to decimal using the steps we just outlined.
Example: Let’s convert \(\mathsf{1100101.0001100_{2}}\) into base-10.
To begin the conversion, we separate the binary number into its whole and fractional parts. The digits before the binary point form the whole number part, while those after it make up the fractional part. Hence \(\mathsf{1100101_2}\) represents the whole number part, and \(\mathsf{0001100_2}\) is the fractional part.
The whole number conversion simply involves assigning index values to each digit, multiplying each digit by \(\mathsf{2}\) raised to its index, and then summing all the resulting products.
This gives us \(\mathsf{101_{10}}\), the decimal equivalent of \(\mathsf{1100101_{2}}\).
For the fractional part, the method is the same, except the indexing starts from the left with the first digit assigned an index of \(\mathsf{-1}\), decreasing further to the right.
The decimal equivalent of the fractional part, computed through this method, is \(\mathsf{0.09375_{10}}\).
After separately converting both the whole and fractional components using their respective methods, we combine the results to form the complete decimal representation. Thus, the decimal number \(\mathsf{1100101.0001100_{2}}\) is equivalently expressed in decimal as \(\mathsf{101.09375_{10}}\).
Did we Get it Right?
Now, if you remember, we originally got this binary number by converting \(\mathsf{101.101_{10}}\) from decimal. When we reversed the process, the result was close—but not exactly the same. That’s because the decimal version had just \(\mathsf{3}\) digits of precision in the fractional part, while the binary equivalent had \(\mathsf{7}\). Since binary only uses two symbols (\(\mathsf{0}\) and \(\mathsf{1}\)), it needs more digits to represent the same level of detail. So even with \(\mathsf{7}\) binary digits in the fractional part, we couldn’t fully capture the precision of just \(\mathsf{3}\) decimal digits.
A Shortcut!
There’s a shortcut when converting binary to decimal, thanks to the simplicity of binary digits. Since binary only uses \(\mathsf{0}\) and \(\mathsf{1}\), any digit multiplied by \(\mathsf{0}\) becomes \(\mathsf{0}\), and any digit multiplied by \(\mathsf{1}\) just gives the original value. So, instead of calculating every digit, you can simply line up all the binary digits, assign index values based on their position relative to the binary point, and focus only on the digits that are \(\mathsf{1}\). For each of those, calculate the base (which is \(\mathsf{2}\)) raised to their index, then add up all those values to get the final decimal result.
For octal to decimal and hexadecimal to decimal conversions, you may go to the conversion from other number system to decimal page..
Number Base Conversion Calculator (Including Fractional Part) Between Binary, Octal, Decimal and Hexadecimal
Below you will find a calculator that can be used to convert numbers between systems.