Number Systems

Table of Contents 


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What is a Number System?

  • A number system is a system of representing numbers.
  • It is also called “the system of numeration”.
  • It defines a set of values to represent a quantity.

For example, the system of numbers we use to represent the numbers in daily life use the \(10\) digits \(0,1,2,3,4,5,6,7,8,\) and \(9\), and this system is known as the “Decimal Number System”.

Decimal number system is in terms of ones, tens, hundreds, thousands, etc


Types of Number Systems in Maths

We have different types of number systems in Maths. But there are \(4\) main types of number systems:

  1. Binary number system (Base - \(2\)) 
  2. Octal number system  (Base - \(8\)) 
  3. Decimal number system  (Base - \(10\)) 
  4. Hexadecimal number system  (Base - \(16\)) 

We will study each of these systems one by one in detail.

Binary Number System

  • The binary number system uses only two digits: \(0\) and \(1\).
     
  • Thus the numbers in this system have a base \(2\).
     
  • \(0\) and \(1\) are called bits.
     
  • \(8\) bits together make a byte.
     
  • The data in computers is stored in terms of bits and bytes.

Binary number system has a base 2 and uses only 0 and 1

Examples:

\(10001_2, 111101_2, 1010101_2 \) are some examples of numbers in the binary number system.

Octal Number System

  • The octal number system uses eight digits: \(0,1,2,3,4,5,6\) and \(7\).
     
  • Thus the numbers in this system have a base \(8\).
     
  • The advantage of this system is it has less number of digits when compared to several other systems and hence there would be fewer computational errors.

octal number system has base 8 and uses only 0-7

Examples:

\(35_{8}, 923_{8}, 141_{8}\) are some examples of numbers in the octal number system.

Decimal Number System

  • The decimal number system uses ten digits: \(0,1,2,3,4,5,6,7,8\) and \(9\).
     
  • Thus the numbers in this system have a base \(10\).
     
  • If we see any number without any base, it means that its base is \(10\).
     
  • This is the system that we generally use to represent the numbers in real life.

decimal number system has a base 2 and uses 0-9

Examples:

\(723_{10}, 32_{10}, 4257_{10}\) are some examples of numbers in the decimal number system.

Hexadecimal Number System

  • The hexadecimal number system uses sixteen digits/alphabets: The digits \(0,1,2,3,4,5,6,7,8, 9\) and the alphabets \(A,B,C,D, E, F\).
     
  • Thus the numbers in this system have a base \(16\).
     
  • Here, \(A-F\) of the hexadecimal system means the numbers \(10-15\) of the decimal number system respectively.
     
  • This system is used in computers to reduce the large-sized strings of the binary system.

hexadecimal system has a base 16 and uses 0-9 and A-F

Examples:

\(7B3_{16}, 6F_{16}, 4B2A_{16}\) are some examples of numbers in the hexadecimal number system.


Difference Between Decimal and Binary Number Systems

The differences between the decimal and binary number systems are as follows:

Decimal Number System Binary Number System
It uses \(10\) digits:\(0,1,2,3,4,5,6,7,8\) and \(9\) It uses \(2\) digits: \(0\) and \(1\).
Its base is \(10\) and it doesn’t need to be written always to show that the number is in the decimal number system. Its base is \(2\) and it has to be written to show that the number is in the binary number system.
It is the most general system used in daily life. It is the system used by the computers to store and process the data.

Number Systems Chart

The information about number systems is summarised in the following chart:

Number systems chart showing binary number system, octal number system, decimal number system and hexadecimal number system.

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Number Systems Conversions

A number can be converted from one number system to another number system. 

We will see how to do this.

Binary / Octal / Hexadecimal System to Decimal System

To convert a number from binary / octal / hexadecimal system to the decimal system, we use the following steps.

The steps are shown by an example of a number in the binary system.

Example:

Convert \(100111_2\) into the decimal system.

Solution:

Step 1: Identify the base of the given number.

Here, the base of \(100111_2\) is \(2\).

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.

The exponents should start with \(0\) and increase by \(1\) every time as we move from right to left.

Since the base here is \(2\), we multiply the digits of the given number by \(2^0, 2^1, 2^2,...\) from right to left.

number system conversion example of binary number 100111 into decimal number system

Step 3: We just simplify each of the above products and add them.

number system conversion example of the binary number 100111 into decimal number system

Here, the sum is the equivalent number in the decimal number system of the given number.

Or, we can use the following steps to make this process simplified. 

\[ \begin{align}
100111 &= 1 \times 2^5 + 0 \times  2^4 + 0 \times  2^3+ 1 \times 2^2 + 1 \times 2^1 + 1 \times  2^0 \\[0.3cm]
&= 1 \times 32 + 0 \times  16 + 0 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times  1 \\[0.3cm]
&= 32 +0+0+4+2+1\\[0.3cm]
&= 39
\end{align} \]
Thus,

\(\therefore 100111_2 = 39_{10}\)

Decimal to Binary / Octal / Hexadecimal System

To convert a number from decimal system to binary / octal / hexadecimal system, we use the following steps.

The steps are shown on how to convert a number from the decimal system to the octal system.

Example:

Convert \(4320_{10}\) into the octal system.

Solution:

Step 1: Identify the base of the required number.

Since we have to convert the given number into the octal system, the base of the required number is \(8\).

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.

Repeat this process (dividing the quotient again by the base) until we get the quotient to be less than the base.

number system conversion of the decimal number 4320 into octal number system.

Step 3: The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Number system conversion of the decimal number 4320 into octal number system

Thus,

\(\therefore 4320_{10} = 10340_{8} \)

One System to Another System

To convert a number from one of the binary / octal / hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example:

Convert \(1010111100_2\) to the hexadecimal system.

Solution:

Step 1: Convert this number to the decimal number system as explained in the above process.

number system conversion of binary number 1010111100 into hexadecimal number system

Thus,

\[1010111100_2 = 700_{10} \rightarrow (1) \]

Step 2: Convert the above number (which is in the decimal system), into the required number system.

Here, we have to convert \(700_{10}\) into the hexadecimal system using the above-mentioned process.

number system conversion of binary number 1010111100 into hexadecimal number system

Thus,

\[ 700_{10} = 2BC_{16}  \rightarrow (2) \]

From the equations (1) and (2), 

\(1010111100_2 =  2BC_{16}\)
 
important notes to remember
Important Notes
  1. To convert a number from binary/octal/hexadecimal system to decimal system:

    Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base, where the exponents start from \(0\) and increase by \(1\) every time. Then just add the above products.

  2. To convert a number from decimal system to binary/octal/hexadecimal system:

    Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form repetitively until we get the quotient to be less than the base. The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

    Math Notes

Number Systems Conversion Calculator

You can enter a number in one of the desired systems and the following calculator will show the equivalent number in the other systems.

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Solved Problems

Example 1

 

 

Convert \(300_{10}\) into the binary system (base- \(2\)).

Solution:

\(300_{10}\) is in the decimal system.

We divide \(300\) by \(2\) and note down the quotient and the remainder.

We will repeat this process for every quotient until we get a quotient which is less than \(2\).

number system conversion of decimal number 300 into binary number system

The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.

Thus,

 \(300_{10} = 100101100_2\)
Example 2

 

 

Convert \(5BC_{16}\) into the decimal system.

Solution:

\(5BC_{16}\) is in the hexadecimal system.

We know that \(B=11\) and \(C= 12\) in the hexadecimal system.

So we get the equivalent number in the decimal system using the following process:

Number system conversion of hexadecimal number 5BC into decimal number system

Thus,

\(5BC_{16}= 1468_{10}\)
Example 3

 

 

Convert \(144_8\) into the hexadecimal system.

Solution:

The base of \(144_8\) is \(8\).

First, we will convert this number into the decimal system as follows:

number system conversion of octal number 144 into hexadecimal number system

Thus, 

\[144_8= 100_{10}  \rightarrow (1) \]

Now we will convert this into the hexadecimal system as follows:

Converting octal into hexadecimal system

Thus, 

\[100_{10} = 64_{16}  \rightarrow (2) \]

From the equations (1) and (2), we can conclude that:

\(144_8=  64_{16}\)
 
Challenge your math skills
Challenging Questions
  1. Convert \(111000_2\) into the decimal system.
  2. Convert \(348_8\) into the decimal system.
  3. Convert \(3AF_{16}\) into the binary system.
  4. Convert \(300_{10}\) into the binary system.

Practice Questions

Here are a few problems related to number systems.

Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 
 
 

Important Topics

Given below are the list of topics that are closely connected to number systems and place values. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

You can also download free worksheets to practice this concept better.


Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What are the 4 types of number systems?

There are \(4\) main types of number systems:

  • Binary number system (Base - \(2\)) 
  • Octal number system  (Base - \(8\)) 
  • Decimal number system  (Base - \(10\)) 
  • Hexadecimal number system  (Base - \(16\)) 

2. What is a number system with examples?

A number system is a system of representing the numbers. Some examples of numbers in different number systems are \[10010_2, 234_8, 428_{10}, 4BA_{16}\]

For more information, you can go through the “Types of Number Systems” section of this page.

3. What are the rules of a number system?

To convert a number from binary/octal/hexadecimal system to decimal system:

  1. Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
  2. The exponents should start with \(0\) and increase by \(1\) every time as we move from right to left.
  3. We just simplify each of the above products and add them.

To convert a number from decimal system to binary/octal/hexadecimal system:

  1. Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.
  2. Repeat this process (dividing the quotient again by the base) until we get the quotient to be less than the base.
  3. The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems:

  1. We first convert it into the decimal system.
  2. Then we convert it to the required system.

4. Why are the number systems used?

There are different purposes of each number system, such as:

  • The binary number system is used to store the data in computers.
  • The advantage of the octal number system is it has less number of digits when compared to several other systems and hence, there would be fewer computational errors.
  • The decimal number system is the system that we use in daily life.
  • The hexadecimal number system is used in computers to reduce the large-sized strings of the binary system.
  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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