Binary to Decimal Converter

What the binary to decimal converter does

You have a string of 0s and 1s and you want to know what ordinary number it stands for. That is what this does. You type in the binary, press the button, and it gives you the plain decimal value, ready to copy.

This is the other direction from writing a number in binary. Here you are reading binary that already exists, the kind you might pull off a register, a permissions bit, or a line of someone else's code, and turning it back into a number you can actually picture.

How to use it

1. Enter your binary number. Type the 0s and 1s, for example 101010.
2. Press Convert. The decimal value appears in the output box.
3. Click the result to copy it. Tap the output and it lands on your clipboard.

Press Reset to clear both boxes.

Place value is the whole trick

Reading binary is the same skill as reading ordinary numbers, only the columns are worth different amounts. In a decimal number the columns go ones, tens, hundreds, each ten times the last. In binary the columns go ones, twos, fours, eights, sixteens, each one twice the last. Those are the powers of two.

So to read a binary number, you look at which columns hold a 1, and you add up what those columns are worth. The 0s contribute nothing. That is the entire method, and it is worth seeing on a real number.

A worked example with real numbers

Take 101010. Write each digit over its place value, starting from the right.

• Rightmost digit 0, worth 1, contributes 0
• Next digit 1, worth 2, contributes 2
• Next digit 0, worth 4, contributes 0
• Next digit 1, worth 8, contributes 8
• Next digit 0, worth 16, contributes 0
• Leftmost digit 1, worth 32, contributes 32

Add up the columns that held a 1: 32 + 8 + 2 = 42. So 101010 in binary is 42 in decimal. Every binary-to-decimal conversion is just this, find the 1s, add their place values.

A faster way: double and add

Once the place value idea makes sense, here is a trick that is quicker for long numbers, and it works left to right so you can do it as you read. Start with 0. For each binary digit in turn, double what you have so far, then add the digit.

Run it on 101010, reading left to right:

• Start at 0. First digit 1: (0 × 2) + 1 = 1
• Next digit 0: (1 × 2) + 0 = 2
• Next digit 1: (2 × 2) + 1 = 5
• Next digit 0: (5 × 2) + 0 = 10
• Next digit 1: (10 × 2) + 1 = 21
• Last digit 0: (21 × 2) + 0 = 42

Same answer, 42, with no powers of two to remember. Doubling as you go quietly builds the place values for you. It is the method most people end up using once they are comfortable.

Reading a byte at a glance

Binary often comes in groups of eight, called a byte, because a byte is the usual unit a computer stores. A full byte of all 1s, 11111111, comes to 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255, and a byte of all 0s is just 0. So any single byte holds a number from 0 to 255, which is a handy range to keep in your head. If a binary number you are reading is longer than eight digits, it simply needs more than one byte to store.

Questions people ask

How do I read a binary number?

Give each digit a place value, doubling from right to left as 1, 2, 4, 8, 16 and so on, then add up the place values wherever there is a 1. The 0s add nothing.

What is the quickest way to do it by hand?

Double and add. Start at 0, then for each digit left to right, double your running total and add the digit. You finish with the decimal value and never have to recall the powers of two.

What is the largest value in eight bits?

255. Eight 1s in a row, 11111111, add up to 255, so a single byte covers 0 to 255.

Do leading zeros change the value?

No. 0101 and 101 are the same number, exactly as 07 and 7 are in decimal. Leading zeros only pad the number out to a set width.

References

1. Weisstein, Eric W. "Binary." From MathWorld, A Wolfram Resource. https://mathworld.wolfram.com/Binary.html