How to Convert Decimal to Binary and Binary to Decimal
Decimal, the Base 10 Numbering System
The decimal, denary or base 10 numbering system is what we use in everyday life for counting. The fact that there are ten symbols is more than likely because we have 10 fingers.
We use ten different symbols or numerals to represent the numbers from zero to nine.
Those numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
When we get to the number ten, we have no numeral to represent this value, so it is written as:
10
The idea is to use a new place holder for each power of 10 to make up any number we want.
So 134 means one hundred, 3 tens and a 4 although we just interpret and read it as one hundred and thirty four.
Placeholder Value in the Decimal Numbering System
Binary, the Base 2 Numbering System
In the decimal number system, we saw that ten numerals were used to represent numbers from zero to nine.
Binary only uses two numerals 0 and 1. Place holders in binary each have a value of powers of 2. So the first place has a value 2^{0} = 1, the second place 2^{1} = 2, the third place 2^{2} = 4, the fourth place 2^{3} = 8 and so on.
In binary we count 0, 1 and then since there's no numeral for two we move onto the next place holder so three is written as 10 binary. This is exactly the same as when we get to ten decimal and have to write it as 10 because there's no numeral for ten.
Placeholder Value in the Binary Numbering System
Most Significant Bit (MSB) and Least Significant Bit (LSB)
For a binary number, the most significant bit (MSB) is the digit furthermost to the left of the number and the least significant bit (LSB) is the rightmost digit.
Decimal and Binary Equivalents
Decimal
 Binary


0
 0

1
 1

2
 10

3
 11

4
 100

5
 101

6
 110

7
 111

8
 1000

Steps to Convert Decimal to Binary
If you don't have a calculator to hand, you can easily convert a decimal number to binary using the remainder method. This involves dividing the number by 2 recursively until you're left with 0, while taking note of each remainder.
 Write down the decimal number.
 Divide the number by 2.
 Write the result underneath.
 Write the remainder on the right hand side. This will be 0 or 1.
 Divide the result of the division by 2 and again write down the remainder.
 Continue dividing and writing down remainders until the result of the division is 0.
 The most significant bit (MSB) is at the bottom of the column of remainders and the least significant bit (LSB) is at the top.
 Read the series of 1s and 0s on the right from the bottom up. This is the binary equivalent of the decimal number.
Steps to Convert Binary to Decimal
Converting from binary to decimal involves multiplying the value of each digit (i.e. 1 or 0) by the value of the placeholder in the number
 Write down the number.
 Starting with the LSB, multiply the digit by the value of the place holder.
 Continue doing this until you reach the MSB.
 Add the results together.
Indicating the Base of a Number
The binary number 1011011 can be written as 1011011_{2} to explicitly indicate the base. Similarly 54 base 10 can be written 54_{10} Often however, the subscript is omitted to avoid excessive detail when the context is known. Usually subscripts are only included in explanatory text or notes in code to avoid confusion if several numbers with different bases are used together.
What is Binary Used For?
For more details on how binary is used in computer systems and digital electronics, see my other article:
What Other Bases Are There Apart From 2 and 10?
Base 16 or hexadecimal (hex for short) is a shorthand used when programming computer systems. It uses sixteen symbols, representing 10, 11, 12, 13, 14 and 15 decimal with the letters A, B, C, D, E, and F respectively. You can read more about converting hex to binary and binary to hex here:
Questions & Answers
How would you convert a decimal like this 25.32 to binary?
Have a look at this article which explains the basics
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© 2018 Eugene Brennan
Comments
Very good explanation. Thank you very much sir.Can understand easily.
This information was very useful to me.
i was looking how to convert (binary to decimal) and (decimal to binary) and this was in very good pattern to understand it.
Good explanation, thanks
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Excellent article. I really liked the technique you presented here. As I read this article, I had fond memories of writing programs in DOS and my teacher going through the binary system with us. Negative or positive? Off or on?
Thanks a lot for a well written and informative article.
Sincerely,
Tim
It's interesting to me that as a (now retired) engineer who spent years designing logic for chips, your decimal to binary method didn't bring back any fond memories. As I think of it, although I had to convert from binary to decimal all the time, I don't recall having to go the other way very often. That's probably not the case for every logic designer. In my case I think it's because I wasn't really dealing with numbers per se, but usually with enacting state machines. In any case, good article.
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