This free online binary to decimal calculator will convert binary numbers into decimal numbers and display a conversion chart to show how it arrived at the answer.
A binary number is a number that consists of only 1s and 0s. Binary numbers use the base 2 system (hence the "bi" in binary), as opposed to decimal numbers that use the base 10 system.
In other words, the decimal system (base 10) uses only the digits 0,1,2,3,4,5,6,7,8 and 9, whereas the binary system (base 2) uses only the digits 0 and 1.
To differentiate between a base 2 and a base 10 number, base 2 numbers are usually written with a 2 as the subscript. For example, 1012 would tell you that the number is the binary number one-zero-one, and not the decimal number one-hundred-one.
In order to help you to understand how to convert binary to decimal, it may help to look at how we translate the value of a decimal number. Lets use the decimal number 1234 (123410, or one-thousand, two-hundred and thirty-four) as an example:
|A||Power of 10:||103||102||101||100|
|B||Place value (A result):||1000||100||10||1|
|C||Entered decimal digit:||1||2||3||4|
|D||Product of B * C:||1000||200||30||4|
|E||Cumulative total of D:||1000||1200||1230||1234|
With the above base 10 translation in mind, here is how you would convert the base 2 number 1111 (11112 or one-one-one-one) into a base 10 number:
|A||Power of 2:||23||22||21||20|
|B||Place value (A result):||8||4||2||1|
|C||Entered binary digit:||1||1||1||1|
|D||Product of B * C:||8||4||2||1|
|E||Cumulative total of D:||8||12||14||15|
As you can see, converting a binary number to a decimal number is a simple process of identifying the place value of each digit, multiplying each digit by its place value, and then adding up all of the products.
If the base 2 you want to convert has a decimal point in it, you simply continue subtracting 1 from each exponent (line A below) as you move from left to right. For example, here is how you would convert 111.1012 into its base 10 equivalent:
|A||Power of 2:||22||21||20||2-1||2-2||2-3|
|B||Place value (A result):||4||2||1||0.5||0.25||0.125|
|C||Entered binary digit:||1||1||1||1||0||1|
|D||Product of B * C:||4||2||1||0.5||0||0.125|
|E||Cumulative total of D:||4||6||7||7.5||7.5||7.625|
So you see, in a binary number each place value to the right of the decimal point decreases by 1/2 (--> 1/2, 1/4, 1/8, etc.) as opposed to the place values to the left of the decimal point whose values double with each place you move to the left (8, 4, 2, 1 <--).
With that, let's use the Binary to Decimal Converter to convert base 2 to base 10.
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Binary system: Number system that uses the base 2 system of expressing values, which consist only of the digits 0 and 1. Base 2 numbers are usually written with a subscripted 2 behind them (10102). The binary system is what electronic devices use for counting and displaying information, as their internal switches recognize a 0 as "off" and a 1 as "on." Each digit within a base 2 number is called a "bit" (comes from binary digit).
Binary number entry field: Enter the binary number you would like to convert into a decimal number. Note that the entered number may only consist of ones, zeros, and a single decimal point, and the leading number must not be a zero (leading zeros in binary conversions are dropped since they have no value, just like in the decimal system).
Equivalent decimal number result field: This is the base 10 equivalent to the entered base 2 number. Note that after clicking the Convert Binary to Decimal button the binary to decimal converter will display a conversion chart showing how it arrived at the result.