**Introduction**

Counting as well as measuring objects is usually done using numbers. Numbers are also integral for performing various arithmetic calculations. We have learnt about different sets of numbers such as natural numbers, whole numbers real numbers etc. Similar to the various sets of numbers, we also have different number systems that have unique characteristics and are used for various purposes. Let us learn about different number systems.

**Definition**

A number system is a system that is used to represent numbers. Another name that is used for number systems is system of numeration. In other words, a number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. What the most common number systems thatare in use? Let us find out.

**Types of Number Systems**

The most common number systems that are used are –

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

Let us learn more about these number systems one by one.

**Decimal Number System**

The decimal number system is composed of 10 numerals or symbols. The word “ Deca “ means 10, which is why the system is known as the decimal number system. What are these 10 numerals that comprise the decimal number system? These numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Using these numerals, any number or quantity can be represented. The decimal number system uses the base 10. This means that 654_{10}, 125_{10} are examples of numbers represented in the decimal number system. Therefore, every position in the decimal number system shows a particular power of the base ( 10 ).

Let us understand it through an example.

**Example**

Represent the number 2512 in the decimal number system.

**Solution**

We have been given the number 2512 and we are required to represent it in the decimal number system. The number 2512, in the decimal number system will be equal to

2512 = 2 x 10^{3} + 5 x 10^{2} + 1 x 10^{1} + 2 x 10^{0}

Let us consider another example.

**Example**

Represent the number 4168.23 in the decimal number system.

**Solution**

We have been given the number 4168.23 and we are required to represent it in the decimal number system. It is important to understand that the decimal point separates the positive powers of 10 from the negative powers of 10. The number 4168.23, in the decimal number system will therefore be equal to

4 x 10^{3} + 1 x 10^{2} + 6 x 10^{1} + 8 x 10^{0} + 2 x 10^{-1} + 3 x 10^{-2}

**Binary Number System**

In the binary number system, there are only two possible digit values, 0 and 1. This is the reason, Binary number system is also known as base – 2 number system. It is also a positional value system where each binary digit has its own value or weight expressed as a power of 2. In fact the digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. For example, 101101_{2} and 1011_{2} are examples of the binary number system. Let us understand it through an example.

**Example**

Represent the binary number 1110_{2} as a decimal equivalent.

**Solution**

We have been given the binary number 1110_{2} and we are required to find its decimal equivalent. The decimal equivalent of the binary number 1110_{2} will be –

1110_{2} = ( 1 x 2^{3} ) + ( 1 x 2^{2} ) + ( 1 x 2^{1} ) + 1 x 2^{0} )

⇒ 1110_{2} = 8 + 4 + 2 + 1

⇒ 1110_{2} = 14

**Hence, the decimal equivalent of 1110**_{2}** will be ( 14 ) **_{10}

**Octal Number System**

The octal number system has a base of 8, meaning that it has 8 unique symbols. These 8 symbols are 0, 1, 2, 3, 4, 5, 6 and 7. Thus each digit of an octal number can have any value from 0 to 7. Digits like 8 and 9 are not included in the octal number system.

The octal number system is also a positional value system where each digit has its own value or weight expressed as a power of 8. The places to the left of the octal point are positive powers of 8 and places to the right are negative powers of 8. Let us understand it through an example. Just as the binary, the octal number system is used in minicomputers but with digits from 0 to 7.

Let us understand it through an example.

**Example**

Represent the octal number 3721.2406_{8} as a decimal equivalent.

**Solution**

We have been given the octal number 3721.2406_{8} and we are required to find its decimal equivalent. The decimal equivalent of the octal number 3721.2406_{8} will be –

3721.2406_{8} = ( 3 x 8^{3} ) + ( 7 x 8^{2} ) + ( 1 x 8^{1} ) + ( 2 x 8^{-1} ) + ( 4 x 8^{-2} ) + (0 x 8^{-3} ) + ( 6 x 10^{-4} )

⇒ 3721.2406_{8} = 3 x 512 + 7 x 64 + 1 x 8 + 2 x 0.125 + 4 x 0.015625 + 0 + 6 x 0.000244

⇒ 3721.2406_{8} = 1536 + 448 + 16 + 1 + 0.25 + 0.0625 + 0 + 0.001464

⇒ 3721.2406_{8} = 2001.313964_{10}

**Hence, the decimal equivalent of the octal number 3721.2406**_{8}** will be 2001.313964**_{10}

**Hexadecimal Number System**

The hexadecimal number system uses the base 16. Thus it has 16 possible digit symbols. It uses the digits 0 to 9 along with letters, A, B, C, D, E, and F as the 16 digit symbols. Similar to the octal number system, the hexadecimal number system is also a positional value system where each digit has its own value or weight expressed as a power of 16. The places to the left of the hexadecimal point are positive powers of 16 and places to the right are negative powers of 16.

The hexadecimal number system is used in computers to specify memory addresses which are 16 – bit and 32 – bit long. For example, memory address 110101101010111 is a big binary address but with hex it is D6F which is easier to remember. Hexadecimal number system is also used to represent colour codes , for example, ( FF, FF, FF ) represents White in RGB value and ( 80, 80, 80 ) represents Grey in RGB value.

**Relationship between Various Number Systems**

The following tables demonstrates the relationship between the binary, decimal, octal and hexagonal number systems –

Hexadecimal Number System | Octal Number System | Decimal Number System | Binary Number System |

0 | 0 | 0 | 0 0 0 0 |

1 | 1 | 1 | 0 0 0 1 |

2 | 2 | 2 | 0 0 1 0 |

3 | 3 | 3 | 0 0 1 1 |

4 | 4 | 4 | 0 1 0 0 |

5 | 5 | 5 | 0 1 0 1 |

6 | 6 | 6 | 0 1 1 0 |

7 | 7 | 7 | 0 1 1 1 |

8 | 1 0 | 8 | 1 0 0 0 |

9 | 1 1 | 9 | 1 0 0 1 |

A | 1 2 | 1 0 | 1 0 1 0 |

B | 1 3 | 1 1 | 1 0 1 1 |

C | 1 4 | 1 2 | 1 1 0 0 |

D | 1 5 | 1 3 | 1 1 0 1 |

E | 1 6 | 1 4 | 1 1 1 0 |

F | 1 7 | 1 5 | 1 1 1 1 |

Now that we know the relationship between different number systems, how do we convert a number from number system to another? Let us find out.

**Conversion of Decimal Number System to Binary Number System**

The common method of converting decimal to binary is repeated division method. In this method, the number is successively divided by 2 and its remainders recorded. The final binary result is obtained by assembling all the remainders with the last remainder being the most significant bit ( MSB ).

Let us understand it through an example.

Suppose we want to find the binary equivalent of the decimal number 17. We shall first divide 17 by 2 as follows –

Next, we will write the remainders in the last to first order as shown below –

Thus this means that 17 of the decimal number system is equivalent to 10001 in the binary system. Therefore,

**5**_{10}** = 10001**_{2}

**Conversion of Decimal Number System to Octal Number System**

A decimal integer can be converted to the octal number system by using the repeated division method just in the manner in which a decimal number is converted to a binary number. However, the only difference here is that we use the division factor 8 instead of 2. Let us understand it using an example.

**Example**

Convert the decimal number 266 into an octal number.

**Solution**

We have been given the decimal number 266 and we need to find its equivalent octal number. We will have,

Note that the first remainder becomes the least significant digit ( L S B ) of the octal number and the last remainder becomes the most significant digit ( M S B ) of the octal number. Thus,

**The decimal number 266**_{2}** will be equal to 142**_{8}** in the octal number system.**

**Conversion of Decimal Number System to Hexadecimal Number System**

A decimal integer can be converted to the Hexadecimal number system by using the repeated division method just in the manner in which a decimal number is converted to a binary number. However, the only difference here is that we use the division factor 16 instead of 2. Let us understand it using an example.

**Example**

Convert the decimal number 423_{2} to the hexadecimal number

Solution

We have been given the decimal number 423_{2 }and we are required to find its hexadecimal equivalent. We will have

Note that the first remainder becomes the least significant digit ( L S B ) of the hexadecimal number and the last remainder becomes the most significant digit ( M S B ) of the hexadecimal number. Thus,

**The decimal number 423**_{2}** will be equal to 1A7**_{16}** in the hexadecimal number system.**

**Conversion of Binary Number System to Decimal Number System**

We know that the binary number system is a positional number system where each binary digit ( bit ) carries a certain weight. Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain it. Let us understand it through an example.

**Example**

Find the decimal equivalent of the binary number 11011_{2}

**Solution**

We have been given the binary number11011_{2} and we are required to find its decimal equivalent. The given binary number can be written as

1 | 1 | 0 | 1 | 1 |

Let us add the position values of each digit of the given binary number. We will have,

11011_{2} = 1 x 2 ^{4} + 1 x 2 ^{3} + 0 x 2 ^{2} + 1 x 2 ^{1} + 1 x 2 ^{0}

⇒11011_{2} = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1

⇒11011_{2} = 16 + 8 + 0 + 2 +1

⇒11011_{2} = 27

**Hence, the decimal equivalent of the binary number 11011**_{2}** will be 27**_{10}

**Conversion of Binary Number System to Octal Number System**

For converting binary integers to octal integers the bits of the binary integer are grouped into groups of three bits starting at the least significant bit ( L S B ). Then each group is converted into its octal equivalent. Let us understand it using an example.

**Example**

Convert the binary number 100111010_{2} to an octal number

**Solution**

We need to convert the binary number 100111010_{2} to an octal number

**Hence, the binary number 100111010**_{2}** in the octal number system will be 472**_{8}

**Conversion of Binary Number System to Hexadecimal Number System**

Binary numbers can be converted to hexadecimal numbers by grouping them to four starting at the binary point.

Let us understand it using an example.

**Example**

Convert the binary number 10110110101_{2} to hexadecimal number

**Solution**

We need to convert the binary number 10110110101_{2} to a hexadecimal number. We will have,

**Hence, the binary number 10110110101**_{2}** in the hexadecimal number system will be DB**_{16}

**Conversion of Octal Number System to Decimal Number System**

An octal number can be easily converted to its decimal equivalent by multiplying each octal digit by its positional weight.

Let us understand it using an example.

**Example**

Convert the octal number 372_{8} to binary number

**Solution**

We need to convert the binary number 372_{8} to binary number. We will have,

372_{8} = 3 x 8^{2} + 7 x 8^{1} + 2 x 8^{0}

⇒ 372_{8} = 3 x 64 + 7 x 8 + 2 x 1

⇒ 372_{8} = 250_{10}

**Hence, the octal number 372**_{8}** in the decimal number system will be 250**_{10}**. **

**Key Facts and Summary**

- A number system is a system that is used to represent numbers.
- The decimal number system is composed of 10 numerals or symbols. These numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
- Every position in the decimal number system shows a particular power of the base ( 10 ).
- In the binary number system, there are only two possible digit values, 0 and 1. This is the reason, Binary number system is also known as base – 2 number system.
- In the binary number system, there are only two possible digit values, 0 and 1.
- The octal number system has a base of 8, meaning that it has 8 unique symbols. These 8 symbols are 0, 1, 2, 3, 4, 5, 6 and 7.
- The hexadecimal number system uses the base 16. Thus it has 16 possible digit symbols. It uses the digits 0 to 9 along with letters, A, B, C, D, E, and F as the 16 digit symbols.
- The common method of converting decimal to binary is the repeated division method. In this method, the number is successively divided by 2 and its remainders recorded. The final binary result is obtained by assembling all the remainders with the last remainder being the most significant bit ( MSB ).
- Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain it.
- A decimal integer can be converted to the octal number system by using the repeated division method just in the manner in which a decimal number is converted to a binary number. However, the only difference here is that we use the division factor 8 instead of 2.
- A decimal integer can be converted to the Hexadecimal number system by using the repeated division method just in the manner in which a decimal number is converted to a binary number. However, the only difference here is that we use the division factor 16 instead of 2.
- For converting binary integers to octal integers the bits of the binary integer are grouped into groups of three bits starting at the least significant bit ( L S B ). Then each group is converted into its octal equivalent.

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