Table of Contents

Last modified on February 7th, 2024

The word binary comes from ‘Bi’ means 2. Thus, a binary number system consists of 2 numbers, 0 and 1. It starts with 0 and ends in 1 and, therefore, has a base 2. The base-2 system in the positional notation is represented as (11101)_{2}.

It is widely used in making the latest computers and computer-based devices in their electronic circuits using logic gates. Each digit in the machine is referred to as a bit.

The numbers 0 to 20 in decimal are written in binary as follows:

Decimal | Binary |
---|---|

0 | 0 |

1 | 1 |

2 | 10 |

3 | 11 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

8 | 1000 |

9 | 1001 |

10 | 1010 |

11 | 1011 |

12 | 1100 |

13 | 1101 |

14 | 1110 |

15 | 1111 |

16 | 10000 |

17 | 10001 |

18 | 10010 |

19 | 10011 |

20 | 10100 |

The binary number is converted to the decimal number by expressing each digit as the product of each number (1 or 2) to the power of 2 based on its place value.

If a binary number has n digits a_{n-1}….a_{3}a_{2}a_{1}a_{0}, its corresponding decimal number is obtained as:

(a_{0}×2^{0}) + (a_{1}×2^{1}) + (a_{2}×2^{2}) +….

Let us convert the binary number 11101 to its corresponding decimal number.

The binary number 11101 is expressed as:

(11101)_{2} = (1 × 2^{4}) + (1 × 2^{3}) + (1 × 2^{2}) + (0 × 2^{1}) + (1 × 2^{0})

= (29)_{10}

A decimal number is converted to its corresponding binary number by dividing the number by 2 until we get 1 as the quotient. The quotients are written from down to up.

Let us convert the decimal number 20 to its corresponding binary number.

Dividing the number by 2 in each step, we get:

Dividend | Quotient | Remainder |
---|---|---|

20 ÷ 2 | 10 | 0 |

10 ÷ 2 | 5 | 0 |

5 ÷ 2 | 2 | 1 |

2 ÷ 2 | 1 | 0 |

1 ÷ 2 | 0 | 1 |

Thus, the decimal number (20)_{10} is expressed as (10100)_{2}

Like decimal numbers, binary numbers are used to perform mathematical operations: addition, subtraction, multiplication, and division.

Binary numbers are added digit by digit to obtain the result of addition.

We will use the following 4 rules for addition:

- 0 + 0 = 0
- 1 + 0 = 1
- 0 + 1 = 1
- 1 + 1 = 0

Like addition, binary subtraction is done digit by digit to obtain the result.

- 0 – 0 = 0
- 1 – 0 = 1
- 0 – 1 = 0
- 1 – 1 = 0

The rules for multiplying 2 binary numbers are given below:

- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 ×1 = 1

The rules for dividing 2 binary numbers are given below:

- 0 ÷ 1 = 0
- 1 ÷ 1 = 1
- 1 ÷ 0 = Meaningless
- 0 ÷ 0 = Meaningless

Click here to learn more about addition, subtraction, multiplication, and division in the binary number system.

- 1’s complement is obtained by inverting the digits of the binary number. For example, the complement of (110)
_{2}is (001)_{2}. - 2’s complement is given by inverting the digits of the number and adding 1 to the least significant bit. For example, 2’s complement of (111)
_{2}is (001)_{2}.

**Convert the binary number (111001) _{2} to its decimal number.**

Solution:

The given binary number is (111001)_{2}

To find the corresponding decimal number:

= (1 × 2^{0}) + (1 × 2^{1}) + (1 × 2^{2}) + (0 × 2^{3}) + (0 × 2^{4}) + (1 × 2^{5})

= 1 + 2 + 4 + 8 + 16 + 32

= 57

Thus, the corresponding decimal number is (57)_{10.}

**Convert the decimal number (134) _{10} to its binary number.**

Solution:

The given decimal number is (134)_{10}

To find the corresponding binary number:

134 ÷ 2 = 81, R = 0

81 ÷ 2 = 40, R = 1

40 ÷ 2 = 20, R = 0

20 ÷ 2 = 10, R = 0

10 ÷ 2 = 5, R = 1

5 ÷ 2 = 2, R = 1

2 ÷ 2 = 1, R = 0

1 ÷ 2 = 0, R = 1

Thus, the corresponding binary number is (10110010)_{2}

Last modified on February 7th, 2024