Convert Binary to Decimal number step-by-step explanation with illustrated examples

In earlier posts, we have covered the various number systems and their representation. Here we will see how we can convert a given binary number to its equivalent in another number system(Decimal number)

## Binary to Decimal Converter

## How to Convert Binary to Decimal

**Example 1: Convert ( 10110 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

```
```**= 1 0 1 1 0
↑ ↑
MSB LSB**

**= 1 x 2**^{4} + 0 x 2^{3} + 1 x 2^{2} + 1 x 2^{1} + 0 x 2^{0}

** = 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 0 x 1**

** = 16 + 0 + 4 + 2 + 0**

**= 22
**

** Therefore ( 1 0 1 1 0 )**_{2} = ( 22 )_{10}

**Example 2: Convert ( 111000 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

**= 1 1 1 0 0 0
↑ ↑
MSB LSB**

**= 1 x 2**^{5} + 1 x 2^{4} + 1 x 2^{3} + 0 x 2^{2} + 0 x 2^{1}+ 0 x 2^{0}

** = 1 x 32 + 1 x 16 + 1 x 8 + 0 x 4 + 0 x 2 + 0 x 1**

** = 32 + 16 + 8 + 0 + 0 + 0**

**= 56**

** Therefore ( 1 1 1 0 0 0 )**_{2} = ( 56 )_{10}

**Example 3: Convert ( 010101 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

**= 0 1 0 1 0 1
↑ ↑
MSB LSB**

**= 0 x 2**^{5} + 1 x 2^{4} + 0 x 2^{3} + 1 x 2^{2} + 0 x 2^{1}+ 1 x 2^{0}

** = 0 x 32 + 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 1 x 1**

** = 0 + 16 + 0 + 4 + 0 + 1**

**= 21**

** Therefore ( 0 1 0 1 0 1 )**_{2} = ( 21 )_{10}

**Example 4: Convert ( 101 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

**= 1 0 1
↑ ↑
MSB LSB**

```
```**= 1 x 2**^{2} + 0 x 2^{1}+ 1 x 2^{0}

** = 1 x 4 + 0 x 2 + 1 x 1**

** = 4 + 0 + 1**

**= 5**

**Therefore ( 1 0 1 )**_{2} = ( 5 )_{10}

**Example 5: Convert ( 01111 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

**= 0 1 1 1 1
↑ ↑
MSB LSB**

Here, zero at the beginning does not make any sense and we can safely eliminate the zero at the MSB position.

**= 1 1 1 1
↑ ↑
MSB LSB**

** = 1 x 2**^{3} + 1 x 2^{2} + 1 x 2^{1}+ 1 x 2^{0}

** = 1 x 16 + 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1**

** = 8 + 4 + 2 + 1**

`= 15`

```
```**Therefore ( 1 1 1 1 )**_{2} = ( 15 )_{10}

The final answer of binary 01111 to decimal is 15.

## Convert Binary fraction to Decimal

Converting binary to decimal number, which includes a fractional part, can be done by following the step-by-step process below, explaining each step on how this method can convert binary fractions to decimal fractions with illustrated examples for better understanding.

## How to Convert Binary Fraction to Decimal

**Example 1: Convert ( 101.101 )**_{2} Binary fraction to decimal fraction ( ? )_{10}

_{2}Binary fraction to decimal fraction ( ? )

_{10}

**= 1 0 1 . 1 0 1
↑ ↑
MSB LSB**

**= 1 x 2**^{2} + 0 x 2^{1} + 1 x 2^{0} . 1 x 2^{-1} + 0 x 2^{-2} + 1 x 2^{-3}

** = 1 x 4 + 0 x 2 + 1 x 1 . 1 x ( 1 / 2 ) + 0 x ( 1 / 4 ) + 1 x ( 1 / 8 )
**

** = 4 + 0 + 1 . ( 1 / 2 ) + 0 + ( 1 / 8 )
**

** = 5 . 0.5 + 0.125**

** = 5 . 625**

**Therefore ( 1 0 1 . 1 0 1 )**_{2} = ( 5.625 )_{10}

**Example 2: Convert Binary Fraction ( 0.0001 )**_{2} to Decimal ( ? )_{10}

_{2}to Decimal ( ? )

_{10}

**= 0 . 0 0 0 1
↑ ↑
MSB LSB**

** = 0 x 2**^{0 }. 0 x 2^{-1} + 0 x 2^{-2} + 0 x 2^{-3}+ 1 x 2^{-4}

** = 0 x 1 . 0 x ( 1 / 2 ) + 0 x ( 1 / 4 ) + 0 x ( 1 / 8 ) + 1 x ( 1 / 16 ) **

** = 0 . 0 + 0 + 0 + ( 1 / 16 )**

**= 0 . 0.0625**

**
= 0 . 0625**

**Therefore ( 0 . 0 0 0 1 )**_{2} = ( 0.0625 )_{10}

**Example 3 : Convert ( 101010.1111 )**_{2}= ( ? )_{10}

_{2}= ( ? )

_{10}

**= 1 0 1 0 1 0 . 1 1 1 1
↑ ↑
MSB LSB **

**= 1x2**^{5}+ 0x2^{4}+ 1x2^{3}+ 0x2^{2}+ 1x2^{1}+ 0x2^{0} . 1x2^{-1}+ 1x2^{-2}+ 1x2^{-3}+ 1x2^{-4}

** = 1 x 32 + 0 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 0 x 1 . 1 x (1/2) + 1 x (1/4) + 1 x (1/8) + 1 x (1/16) **

** = 32 + 0 + 8 + 0 + 2 + 0 . (1/2) + (1/4) + (1/8) + (1/16)
**

** = 32 + 8 + 2 . ( 0.5 ) + ( 0.25 ) + ( 0.125 ) + ( 0.0625 )**

** = 42 . 9375**

** Therefore ( 1 0 1 0 1 0 . 1 1 1 1 )**_{2} = ( 42.9375 )_{10}

## Binary to Decimal Shortcut

We have till now discussed the conventional method of converting binary to decimal numbers. Instead of following the long procedure here, you will learn a simple shortcut or a memory-based technique to perform the binary to decimal conversions in seconds. As you practice, you will mentally convert any binary to decimal without the pen and the paper.

However, the earlier procedure is a must in examination point of view. This shortcut is only for knowledge and for daily usage.

**Step 1:** First, we will have to make a tabular column with three rows R1,R2,R3 and 8 columns C0,C1,C2,C3,C4,C5,C6,C7 as shown below:

Here, each element of the table will hold one bit, and there are 8 columns which mean 8 bits in one row (i.e., 1 byte). In this shortcut, the maximum value that we cover is (255) in the decimal number system, and you can extend it if you wish.

**Step 2:** Now insert the values into the first row of the table (R1) as shown below

The empty columns can be ignored

**Step 3: **Multiply R1 and R2 with the corresponding elements and put the product in **R3**

Step 4: Now add the individual elements of third row **R3, **this sum gives the decimal equivalent of the given binary number as illustrated

**8 + 0 + 0 + 1 = ( 9 ) _{10}**

## Example 2 : ( 1 0 1)_{2} = ( ? )_{10}

** 4 + 0 + 1 = ( 5 )**_{10}

Try the following yourself

**Example 3: (101010)**_{2}= ( ? )_{10}**Example 4: (111111)**_{2}= ( ? )_{10}**Example 5: (00011)**_{2}= ( ? )_{10}

You can also use the Binary to Decimal converter to test your answers.

## Binary to Decimal Table

Binary | Decimal |

0000 | 0 |

0001 | 1 |

0010 | 2 |

0011 | 3 |

0100 | 4 |

0101 | 5 |

0110 | 6 |

0111 | 7 |

1000 | 8 |

1001 | 9 |

1010 | 10 |

1011 | 11 |

1100 | 12 |

1101 | 13 |

1110 | 14 |

1111 | 15 |