** Binary Number System** uses two digits,

**, and is the foundation for all modern computing. The word binary is derived from the word “bi” which means two. But what makes it so essential, and how does it work? In this article, we will dive deep into**

**0 and 1****, binary decimal number conversion and vice versa, 1’s and 2’s complement, and how they are used in**

**binary numbers****.**

**computer systems**There are generally various types of number systems and among them the four major ones are,

- Binary Number System (Number system with Base 2)
- Octal Number System (Number system with Base 8)
- Decimal Number System (Number system with Base 10)
- Hexadecimal Number System (Number system with Base 16)

Table of Content

- Binary Number System
- Binary Number Table
- Binary to Decimal Conversion
- Decimal to Binary Conversion
- Arithmetic Operations on Binary Numbers
- 1’s and 2’s Complement of a Binary Number
- Uses of Binary Number System
- Binary Number System Example

## Binary Number System

Binary Number System is the number system in which we use two digits “0” and “1” to perform all the necessary operations. In the Binary Number System, we have a base of 2. The base of the Binary Number System is also called the radix of the number system.

In a binary number system, we represent the number as,

- (11001)
_{2}

In the above example, a binary number is given in which the base is 2. In a binary number system, each digit is called the “bit”. In the above example, there are 5 digits.

## Binary Number Table

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

1 | 001 | 11 | 1011 |

2 | 010 | 12 | 1100 |

3 | 011 | 13 | 1101 |

4 | 100 | 14 | 1110 |

5 | 101 | 15 | 1111 |

6 | 110 | 16 | 10000 |

7 | 111 | 17 | 10001 |

8 | 1000 | 18 | 10010 |

9 | 1001 | 19 | 10011 |

10 | 1010 | 20 | 10100 |

## Binary to Decimal Conversion

A binary number is converted into a decimal number by multiplying each digit of the binary number by the power of either 1 or 0 to the corresponding power of 2. Let us consider that a binary number has n digits, B = a_{n-1}…a_{3}a_{2}a_{1}a_{0}. Now, the corresponding decimal number is given as

D = (a_{n-1}× 2^{n-1}) +…+(a_{3}× 2^{3}) + (a_{2}× 2^{2}) + (a_{1}× 2^{1}) + (a_{0}× 2^{0})

Let us go through an example to understand the concept better.

**Example: Convert (10011)**_{2}** to a decimal number.**

**Solution:**

The given binary number is (10011)

_{2}.(10011)

_{2}= (1 × 2^{4}) + (0 × 2^{3}) + (0 × 2^{2}) + (1 × 2^{1}) + (1 × 2^{0})= 16 + 0 + 0 + 2 + 1 = (19)

_{10}Hence, the binary number (10011)

_{2}is expressed as (19)_{10}.

## Decimal to Binary Conversion

A decimal number is converted into a binary number by dividing the given decimal number by 2 continuously until we get the quotient as 1, and we write the numbers from downwards to upwards.

Let us go through an example to understand the concept better.

**Example: Convert (28)**_{10}** into a binary number.**

**Solution:**

Hence, (28)

_{10}is expressed as (11100)_{2}.

## Arithmetic Operations on Binary Numbers

We can easily perform various operations on Binary Numbers. Various arithmetic operations on the Binary number include,

- Binary Addition
- Binary Subtraction
- Binary Multiplication
- Binary Division

Now let’s learn about the same in detail.

**Binary Addition**

**Binary Addition**

The result of the addition of two binary numbers is also a binary number. To obtain the result of the addition of two binary numbers, we have to add the digit of the binary numbers by digit. The table added below shows the rule of binary addition.

Binary Number (1) | Binary Number (2) | Addition | Carry |
---|---|---|---|

1 | 1 | ||

1 | 1 | ||

1 | 1 | 1 |

**Binary Subtraction**

**Binary Subtraction**

The result of the subtraction of two binary numbers is also a binary number. To obtain the result of the subtraction of two binary numbers, we have to subtract the digit of the binary numbers by digit. The table added below shows the rule of binary subtraction.

Binary Number (1) | Binary Number (2) | Subtraction | Borrow |
---|---|---|---|

1 | 1 | 1 | |

1 | 1 | ||

1 | 1 |

**Binary Multiplication**

**Binary Multiplication**

The multiplication process of binary numbers is similar to the multiplication of decimal numbers. The rules for multiplying any two binary numbers are given in the table,

Binary Number (1) | Binary Number (2) | Multiplication |
---|---|---|

1 | ||

1 | ||

1 | 1 | 1 |

**Binary Division**

**Binary Division**

The division method for binary numbers is similar to that of the decimal number division method. Let us go through an example to understand the concept better.

**Example: Divide (11011)**_{2}** by (11)**_{2}

**Solution:**

## 1’s and 2’s Complement of a Binary Number

**1’s Complement of a Binary Number is obtained by inverting the digits of the binary number.**

**Example: Find the 1’s complement of (10011)**_{2}**.**

**Solution:**

Given Binary Number is (10011)

_{2}Now, to find its 1’s complement, we have to invert the digits of the given number.

Thus, 1’s complement of (10011)

_{2}is (01100)_{2}

**2’s Complement of a Binary Number is obtained by inverting the digits of the binary number and then by adding 1 to the least significant bit.**

**Example: Find the 2’s complement of (1011)**_{2}**.**

**Solution:**

Given Binary Number is (1011)

_{2}To find the 2’s complement, first find its 1’s complement, i.e., (0100)

_{2}Now, by adding 1 to the least significant bit, we get (0101)

_{2}Hence, the 2’s complement of (1011)

_{2}is (0101)_{2}

## Uses of Binary Number System

Binary Number Systems are used for various purposes and the most important use of the binary number system is,

- Binary Number System is used in all Digital Electronics for performing various operations.
- Programming Languages uses Binary Number System for encoding and decoding data.
- Binary Number System is used in Data Sciences for various purposes, etc.

**Read More,**

- Binary Formula
- Difference Between Decimal and Binary Number Systems

## Binary Number System Example

**Example 1: Convert Decimal Number (98)**_{10}** into Binary.**

**Solution:**

Thus, Binary Number for (98)

_{10}is equal to (1100010)_{2}See AlsoBinary to Decimal Converter

**Example 2: Convert Binary Number (1010101)**_{2}** to Decimal Number.**

**Solution:**

Given Binary Number, (1010101)

_{2}= (1 × 2

^{0}) + (0 × 2^{1}) + (1 × 2^{2}) + (0 × 2^{3}) + (1 × 2^{4}) + (0 × 2^{5}) + (1 ×2^{6})= 1 + 0 + 4 + 0 + 16 + 0 + 64

= (85)

_{10}Thus, Binary Number (1010101)

_{2}is equal to (85)_{10}in decimal system.

**Example 3: Divide (11110)**_{2}** by (101)**_{2}

**Solution:**

**Example 4: Add (11011)**_{2}** and (10100)**_{2}

**Solution:**

Hence, (11011)

_{2}+ (10100)_{2}= (101111)_{2}

**Example 5: Subtract (11010)**_{2}** and (10110)**_{2}

**Solution:**

Hence, (11010)

_{2 }– (10110)_{2}= (00100)_{2}

**Example 6: Multiply (1110)**_{2}** and (1001)**_{2}**.**

**Solution:**

Thus, (1110)

_{2}× (1001)_{2}= (1111110)_{2}

## Binary Number System- FAQs

### What is a Binary Number System?

Binary Number System is one of the four number system that is used to represent the numbers using only two digits, “0” and “1”. In binary number system the digits are called ‘bits’. Binary Number System is used by computers to perform various calculations.

### What is a **B**it?

**B**

A bit in Binary Number System is defined as a individual digits that holds the value ‘0’ or ‘1’.

**What is a Nibble?**

**What is a Nibble?**

A group of four digits is called the Niblle.

### What is Binary Value of 10?

Binary value of 10 is (1010)

_{2}

### What are Types of Number Systems?

There are various types of number systems and some of them are,

- Binary Number System
- Octal Number System
- Decimal Number System
- Hexadecimal Number System

### How to Calculate Binary Numbers?

Binary numbers are calculated from dicmal numbers by dividing the decimal number with 2 and writing the remainder. Then we arrange all the remainders from newest to oldest to get the binary number.

### How to Add Binary Numbers?

Binary numbers are added by using the formulas written below,

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (carry 1)

Next Article

Number System in Maths