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Binary Number System
Binary number system is used to define a number in binary system. Binary system is used to represent a number in terms of two numbers only, 0 and 1. The binary number system is used commonly by computer languages like Java, C++. As the computer only understands binary language that is 0 or 1, all inputs given to a computer are decoded by it into series of 0's or 1's to process it further. In this lesson we will learn how to convert a decimal number to its binary number and the conversion of binary number to decimal number.
1.  What is Binary Number System? 
2.  Binary Number System Chart 
3.  Binary to Decimal Conversion 
4.  Decimal to Binary Conversion 
5.  Operations on Binary Numbers 
6.  FAQs on Binary Number System 
What is Binary Number System?
"Bi" in Binary means "two". Hence, this draws back the line to the representation of a number in terms of 0 and 1 only. It is possible to express decimal numbers in terms of a binary number system easily. Decimal numbers and binary numbers have different notations. A decimal number is represented with a base of 10 while a binary number is represented with a base of 2. For example, 2 in decimal notation is represented as \((2)_{10}\). The binary number for 2 is represented as \((10)_{2}\). Hence, 10 is the binary number representation for the number 2.
Binary Number System Chart
The numbers from 1 to 10 can be expressed in a binary number system in the following way:
Binary to Decimal Conversion
The binary number can be converted to a decimal number by expressing each digit as a product of the given number 1 or 0 to the respective power of 2. If a binary number has n digits, B = \(a_{n1}...a_{3}a_{2}a_{1}a_{0}\), the decimal number for it is given as, D = (a_{0}×2^{0}) + (a_{1}×2^{1}) + (a_{2}×2^{2}) + ...
Let us understand this using an example.
We can convert 10101 to the decimal number form in the following way:
The binary number 10101 is expressed as \((10101)_{2}\) = (1 × 2^{4}) + (0 × 2^{3}) + (1 × 2^{2}) + (0 × 2^{1}) + (1 × 2^{0}) = \((21)_{10}\). Thus, the binary number 10101 is expressed as \((21)_{10}\).
Decimal to Binary Conversion
The decimal number can be converted to a binary number by dividing the given number by 2 until we obtain the quotient as 1. The numbers are written from downwards to upwards.
Let us understand this using an example.
We can convert 30 to a binary number form in the following way:
The decimal number 30 is expressed as \((30)_{10}\) = \((11110)_{2}\).
Operations on Binary Numbers
Binary Addition
We add the binary numbers digit by digit and obtain the answer to the addition. It is important to remember the table below while adding the two binary numbers.
Binary Subtraction
The binary numbers are subtracted digit by digit and the answer is obtained. The table below is kept in mind while subtracting two binary numbers.
Binary Multiplication
The rules to multiply any two binary numbers is given as:
1's and 2's complement of a Binary Number
 1's complement of a binary number is given by inverting the digits of the binary number. For example, 1's complement of \((101)_{2}\) is \((010)_{2}\).
 2's complement of a binary number is given by inverting the digits of the binary number and adding 1 to the least significant bit. For example, 2's complement of \((111)_{2}\) is \((001)_{2}\) which is obtained by taking 1's complement of \((111)_{2}\) and adding 1 to the least significant bit.
Tips to Remember
Here are some important points to remember on binary number system:
 A binary number consists of two numbers 0s and 1s.
 Binary numbers are represented with 2 at their base. For example, \((101)_{2}\).
 Each digit in a binary number is referred to as a bit. For example, \((111)_{2}\) is a threebit binary system.
 Binary addition is also referred to as the "AND" operation.
 Binary multiplication is also called an "OR" operation.
 The binary subtraction can be done by taking 1's and 2's complement of the binary number.
 The most significant digit in a binary number represents the sign of a binary number which is used to perform signed binary operations. 1 represents a negative sign while 0 represents a positive sign.
Topics Related to Binary Number System
Examples on Binary Number System

Example 1: Convert the decimal number \((162)_{10}\) in binary.
Solution: In order to obtain the binary number for 162, we can divide it continuously by 2.
Quotient Remainder 81 0 40 1 20 0 10 0 5 0 2 1 1 0 0 1
\(\therefore\) The binary number for \((162)_{10}\) is \((10100010)_{2}\). 
Example 2: Convert the binary number \((100101)_{2}\) to decimal number.
Solution: The binary number \((100101)_{2}\)_{ }= (1 × 2^{0}) + (0 × 2^{1}) + (1 × 2^{2}) + (0 × 2^{3}) + (0 × 2^{4}) + (1 × 2^{5}) = \((37)_{10}\)
\(\therefore\) The binary number \((100101)_{2}\) is \((37)_{10}\). 
Example 3: Prove that binary number \((1000100)_{2}\) can be converted to the decimal number \((68)_{10}\).
Solution: The binary number \((1000100)_{2}\) = (0 × 2^{0}) + (0 × 2^{1}) + (1 × 2^{2}) + (0 × 2^{3}) + (0 × 2^{4}) + (0 × 2^{5}) + (1 × 2^{6}) = 64 + 4 = \((68)_{10}\)
\(\therefore\) The binary number \((1000100)_{2}\)_{ }can be converted to the binary number \((68)_{10}\).
FAQs on Binary Number System
What is Binary Number System?
The system of representation in which a number can be expressed in terms of only two digits (0 and 1) with base 2 is known binary number system.
Why Binary Number System is Used in Computers?
Computer systems always process the given instructions using 0 or 1 as they either exist in on state or off state. This makes it easier for them to process the information faster.
What Does 10101 mean in Binary Number System?
10101 means 21 in a binary number system.
How Do You Convert a Decimal Number to Binary Number System?
A decimal number can be converted to a binary number system by dividing the given number by 2 until we obtain the quotient as 1. The numbers are written from downwards to upwards.
How to Convert a Binary Number to a Decimal Number?
We can convert a binary number to a decimal number expressing each digit as a product of the given number 1 or 0 to the respective power of 2. If a binary number has n digits, B = \((a)_{n1}\).. \((a)_{3}\) \((a)_{2}\) \((a)_{1}\) \((a)_{0}\), the decimal number for it is given as, D = ( \((a)_{0}\)×2^{0}) + ( \((a)_{1}\)×2^{1}) + ( \((a)_{2}\)×2^{2}) + ...
What Does 1011 mean in Binary Number System?
1011 means 11 in binary number system as 1 × 2^{0 }+ 1 × 2^{1}+ 0 × 2^{2 }+ 1 × 2^{3 }= 1+ 2+ 0 + 8 = 11.
How Do You Write 13 in Binary Number System?
13 means 1101 in the binary number system. We continuously divide 13 by 2 until the quotient is 1. In this case, the steps that would be followed are:
 13/2 give 6 as a quotient and 1 as a remainder.
 6/2 give 3 as a quotient and 0 as a remainder.
 3/2 give 1 as a quotient and 1 as a remainder.
 Now, the number is written from the bottom to the top as \((13)_{10}\) = \((1101)_{2}\).
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