In daily life, we mostly use the **decimal number system**. The decimal number system has a base of ten (10). It means that there are ten digits with the help of which we can represent any number of the decimal family. The digits are from 0 to 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

**Representation of decimal number**

It is represented as (X)_{10}

where, X is the decimal number and the base 10 is written as suffix or radix.

example: (4)_{10}, (56)_{10}, (78)_{10}, (24)_{10}, etc. are the examples of a decimal number.

**Note:** If radix 10 is not written then we must understand that it is a decimal number by default. example, 4, 10, 98, 100, etc. are treated as a decimal number by default, if radix is not given.

Any decimal number can be written in the form of

(1342365)_{10} = 10^{6} × 1 + 10^{5} × 3 + 10^{4} × 4 + 10^{3} × 2 + 10^{2} × 3 + 10^{1} × 6 + 10^{0} × 5

(4)_{10} = 10^{0} × 4

(24)_{10} = 10^{1} × 2 + 10^{0} × 4

(56)_{10} = 10^{1} × 5 + 10^{0} × 6

(78)_{10} = 10^{1} × 7 + 10^{0} × 8

(100)_{10} = 10^{2} × 1 + 10^{1} × 0 + 10^{0} × 0

In this article, we will learn another numbering system, known as **the binary number system**. It has a base of two (2). It means that there are two digits with the help of which we can represent any number of the binary family. The digits are from 0 to 1 i.e. (0 and 1).

**Representation of binary number**

It is represented as (X)_{2}

where, X is the binary number and the base 2 is written as suffix or radix.

example: (101)_{2}, (11011)_{2}, (1011)_{2}, (1111101)_{2}, etc. are the examples of a binary number.

Consider a binary number (10101110)_{2} as shown below. Here leftmost digit is known as a **most significant bit (MSB)** and rightmost digit is known as a **least significant bit (LSB).**

## Binary to Decimal conversion

Expand the number given in binary form in the power of 2 and sum the values, the result which we will get will be in the decimal form. For example-

(101)_{2} = 2^{2} × 1 + 2^{1} × 0 + 2^{0} × 1

= 4 × 1 + 2 × 0 + 1 × 1

= 4 + 0 + 1

= 5

= (5)_{10}

## Decimal to Binary conversion

Divide the number by 2 and take only the remainder, if the division is completed than take only the remainder which gives the binary number. First of all, MSB is written on the left and move until LSB is reached.

Consider an example.

Find the binary equivalent for decimal 35.

MSB = most significant bit = 1

LSB = least significant bit = 1

We have to write remainder values moving from bottom to top (left to write on paper) as shown above.

Therefore, the binary equivalent for decimal 35 is (100101)_{2}.

Here is the table for a binary equivalent of first 16 decimal numbers.

Decimal number (base = 10) |
Binary number (base = 2) |

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

10 | 1010 |

11 | 1011 |

12 | 1100 |

13 | 1101 |

14 | 1110 |

15 | 1111 |