How to Convert Decimal to Binary and Binary to Decimal

Updated on May 24, 2019
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Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

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.

Binary number and its decimal equivalent.
Binary number and its decimal equivalent. | Source

Placeholder Value in the Decimal Numbering System

Placeholder value in the base 10 number system
Placeholder value in the base 10 number system | Source

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 20 = 1, the second place 21 = 2, the third place 22 = 4, the fourth place 23 = 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 two 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

Placeholder value in the binary number system
Placeholder value in the binary number system | Source

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.

Most significant bit (MSB) and least significant bit (LSB).
Most significant bit (MSB) and least significant bit (LSB). | Source

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.

  1. Write down the decimal number.
  2. Divide the number by 2.
  3. Write the result underneath.
  4. Write the remainder on the right hand side. This will be 0 or 1.
  5. Divide the result of the division by 2 and again write down the remainder.
  6. Continue dividing and writing down remainders until the result of the division is 0.
  7. The most significant bit (MSB) is at the bottom of the column of remainders and the least significant bit (LSB) is at the top.
  8. Read the series of 1s and 0s on the right from the bottom up. This is the binary equivalent of the decimal number.


Converting decimal to binary
Converting decimal to binary | Source

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

  1. Write down the number.
  2. Starting with the LSB, multiply the digit by the value of the place holder.
  3. Continue doing this until you reach the MSB.
  4. Add the results together.

Converting binary to decimal
Converting binary to decimal | Source

Indicating the Base of a Number

The binary number 1011011 can be written as 10110112 to explicitly indicate the base. Similarly 54 base 10 can be written 5410 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:

Why is Binary Used In Computers and Electronics?

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:

How to Convert Hex to Binary and Binary to Hexadecimal

Questions & Answers

© 2018 Eugene Brennan

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    • profile image

      francis khayeka 

      2 months ago

      thanks alot be blessed excellent explanations

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 months ago from Ireland

      Yes, you're right!

      Thanks for spotting the silly error.

    • profile image

      garyZ 

      3 months ago

      I believe this is not correct:

      "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."

      Decimal 3 is binary 11.

      Binary 10 is decimal 2.

    • profile image

      kwizera mugara gentil 

      3 months ago

      thank you for good lectures may good Lord God bless you

    • profile image

      Janaki 

      5 months ago

      Very good explanation. Thank you very much sir.Can understand easily.

    • profile image

      shahzar khan 

      5 months ago

      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.

    • profile image

      RajaSri S 

      5 months ago

      Good explanation, thanks

    • profile image

      zamani ezekiel 

      6 months ago

      Am a 200level student in the university of jos,plateau state

      here in Nigeria. I found your site very help ful to me.

      Am a computer science student.I just want to say thank you you are really doing a great job.can u help me in any way u can so as i put my best in my study. thank you sir.

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      11 months ago from Ireland

      Thanks Tim. I'm a veteran from the DOS age too!

    • Tim Truzy info4u profile image

      Tim Truzy 

      11 months ago from U.S.A.

      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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      11 months ago from Ireland

      Thanks Ron, always good to get feedback from you! Now hopefully this short guide will become as popular as the hex to binary one! (BTW, I just had to Google "Mechanicsburg". I'm always curious about the history of how places got their names!)

    • RonElFran profile image

      Ronald E Franklin 

      11 months ago from Mechanicsburg, PA

      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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