# A detailed guide on Number System in Mathematics | 2021

## Important topic based on the recent pattern of MCQ based on Number System

The number system or the numeral system is the system of identifying and expressing numbers. The number system presents a unique representation of each number and signifies the arithmetic and algebraic structure of the figures. It enables us to perform different arithmetic operations like addition, subtraction, and division.

The different kinds of number systems in mathematics are:-

- Binary
- Octal
- Decimal
- Hexadecimal

This article includes the whole concept of the number system with its kinds, conversions, and examples.

The number of radix or base ten is known as decimals. It’s the first number system in mathematics, where all the modern and ancient calculations are done. Other number systems based on this number is also derived from it. It’s generated using the combined use of digits ranging from zero to nine.

The number system uses digits like 0 and 1, which are the very common digits in the system, and thus it is used to express binary numbers. Whereas on the other side, 0 to 9 digits are used for different types of number systems. Let us discuss the different kinds of number systems.

## Decimal Number System (having Base 10)

The decimal number system includes a base of 10 since it practices ten digits from 0 to 9. The positions succeeding to the left of the decimal point express units, tens, hundreds, thousands, etc. The place value is described from right to left. The units have the position value as 100, tens have the position value as 101, hundreds as 102, thousands as 103, and carry on.

For example, the decimal number system 1747 comprises of the digit 7 in the units place, 4 in the tens place, 7 in the hundreds place, and 1 in the thousands position whose value can be addressed as

**1747**

**Explanation**:- The unit digit 7 with base 10 is multiplied by 10 with power 0.

The tens place digit 4 with base 10 is multiplied by 10 with power 1.

The hundredth place digit 7 with base 10 is multiplied by 10 with power 2.

The thousand place digit 1 with base 10 is multiplied by 10 with power 3.

Here is a description of it:

(1×10^{3}) + (7×10^{2}) + (4×10^{1}) + (7×10^{0})

=> (1×1000) + (7×100) + (4×10) + (7×1)

=> 1000 + 700 + 40 + 7

=> 1747

**10675 has place values as**

(1 × 10^{4}) + (0 × 10^{3}) + (6 × 10^{2}) + (7× 10^{1}) + (5 × 10^{0})

=> 1 × 10000 + 0 × 1000 + 6 × 100 + 7 × 10 + 5 × 1

=> 10000 + 0 + 600 + 70 + 5

=> 10675

## Binary Number System (having Base 2)

The Binary number system has a base of 2. The number of digits existing here is two, i.e., 0 and 1. Mostly, the common base-2 is a radix of 2. The figures represented in this system are called binary numbers, which are 0 and 1.

For example, 100101 is binary. The binary number system is beneficial in electronic devices and computer systems since it can be quickly implemented using just two states, ON and OFF, i.e., 0 and 1.

Using the concept of 8421, he Decimal Numbers 0-9 are expressed in binary as:

- 0= 8421

[0000]

- 1= 8421

[0001]

- 2 = 8421

[0010]

- 3 = 8421

[0011]

- 4 = 8421

[0100]

- 5 = 8421

[0101]

- 6 = 8421

[0110]

- 7 = 8421

[0111]

- 8 = 8421

[1000]

- 9 = 8421

[1001]

We can describe 0 in binary representation as 0000 because the sum of 8,4,2,1 none can form a 0. Similarly, for 9, the representation is 1001 because one 8 and one 1 can sum up to forming 9 from the given sequence of 8421. You can transform any system into binary and vice versa.

## Octal Number System (the ones having base 8)

The octal Number System has a base value of 8. It uses the number of digits from 0-7 for the conception of Octal Numbers. We can convert octal numbers to Decimal value by multiplying every digit with the place value and then summing the result. The octal numbers are beneficial for the description of UTF8 Numbers. The octal numbers are usually used in computer applications.

**Example: **Convert 217 (octal number representation) into decimal.

**Solution:**

217 = (2 × 8^{2}) + (1 × 8^{1}) + (7 × 8^{1})

=> (2 × 64) + (1 × 8) + (7 × 1)

=> 128 + 8 + 7

=> (143)_{10}

**Explanation**: The unit digit 7 with base 8 is multiplied by 8 with power 0.

The digit 1 with base 8 is multiplied by 8 with power 1.

The digit 2 with base 8 is multiplied by 8 with power 2.

## Hexadecimal Number System (having base 16)

The Hexadecimal Number System has a base value of 16. It uses the number of 16 digits i.e. from 0 to 15, to produce its numbers. The digits from 0 to 9 are taken just the same as the digits in the decimal number system except for the integers from 10 to 15 are represented as A to F i.e.

10 is expressed as A

11 = B

12 = C

13 = D

14 = E

15 = F

The hexadecimal Numbers are beneficial for managing memory address locations.

## Number System Conversion

The number system conversion is a pretty simple task. Any of the number from any number system can be transformed into different number systems with the use of some techniques that are mentioned below:

### Conversion of Decimal Number System to Other:

The change of a number system means the transition from one base to different.

#### A. Decimal to Binary number system Conversion:

For decimal to binary, the two steps needed to perform the conversion are mentioned below:-

- Do the division process on the integer and the progressive quotient with the base of binary(2).
- Later, do the multiplication on the integer and the progressive quotient with the binary(base 2).

**Example: **Convert 19 with base 10 to binary:

Division by 2 | Quotient | Remainder |

19/2 | 9 | 1 |

9/2 | 4 | 1 |

4/2 | 2 | 0 |

2/2 | 1 | 0 |

And, the result goes from bottom to up, including the quotient of the last division. So, the result is (10011)_{2}.

#### B. Decimal to Octal number system Conversion

For converting decimal to octal, the two steps needed to perform are mentioned below:

- Do the division process on the integer and the next quotient with the octal base (base 8).
- After this, do the multiplication process on the integer and the next quotient with the octal base (base 8).

**Example 1: **(160.25)_{10}

**Step 1:**

Divide 160 and its consecutive quotients with base 8.

Operation | Quotient | Remainder |

160/8 | 20 | 0 |

20/8 | 8 | 4 |

Operation | Answer | Carry |

0.25*8 | 0 | 2 |

As a result goes from bottom to up, including the quotient, at last, the answer will be (240.2)_{8}

#### C. Decimal to hexadecimal conversion

For converting decimal to hexadecimal, the two steps needed to perform are mentioned below:

- Do the division process on the integer and the next quotient with the base of hexadecimal (16).
- Later, do the multiplication process on the integer and the continuous quotient with the hexadecimal base (16).

**Example 1: **(152)_{10}

**Step 1:**

Divide 152 and its continuous quotients with base 8.

Operation | Quotient | Remainder |

152/16 | 9 | 8 |

9/16 | 0 | 9 |

As a result goes from bottom to up, including the last quotient value, the answer will be (98)_{16}.

### Conversion of Binary to other Number Systems:

The three conversions feasible for binary numbers are binary to decimal, binary to octal, and binary to hexadecimal.

#### A. Binary to Decimal Conversion

The method of converting binary number systems to decimals is quite easy. The method begins by multiplying every bit of binary number with its identical positional weights. Finally, add all the products.

**Example 1: **(11110.001)_{2}

Multiply every bit of (10110.001)_{2} with its corresponding positional weight, and in the last, add all the products of each bit with its weight.

**(11110.001)**** _{2 }**= (1×2

^{4}) + (1×2

^{3}) + (1×2

^{2}) + (1×2

^{1}) + (0×2

^{0}) + (0×2

^{-1}) + (0×2

^{-2}) + (1×2

^{-3})

**=> **(1×16) + (1×8) + (1×4) + (1×2) + (0×1) + (0×1⁄2) + (0×1⁄4) + (1×1⁄8)

**=> **16 + 8 + 4 + 2 + 0 + 0 + 0 + 0.125

**=> **(30.125 )_{10}

#### B. Binary to Octal Conversion

The binary and octal base numbers are 2 and 8, respectively. In a binary number, the combination of three bits is similar to one octal digit. The two steps to convert a binary number into an octal number are mentioned below:

- You need to secure the pairs of three bits on both sides of the binary point. If anyone or two bits are left in a set of three bits pairs, we attach the demanded number of zeros on the last sides.
- Then, write the octal digits resembling every pair.

**Example 1: **(111111101011)_{2}

1. We create pairs of three bits on the binary point, as follows:

111 111 101 011

Next, write the octal digits which resemble every pair.

**(111110101011)**_{2}** = **(7753)_{8}

#### C. Binary to Hexadecimal Conversion

The binary and hexadecimal bases are 2 and 16, respectively. In a binary number system, the combination of four bits is similar to one hexadecimal digit. The two steps to change a binary number system into a hexadecimal number are mentioned below:

- Make the pairs of four-four bits on each side of the binary point. If anyone, two, or three bits are still left in a combination of four bits, append the demanded number of zeros on needed sides.
- Write the hexadecimal digits resembling every pair.

**Example 1: **(11110101111.0011)_{2}

1. Create pairs of four-four bits on each side of the binary point. Like,

111 1010 1111.0011

On the left side, the first pair only holds three bits. Present a total set of four bits, attach one zero on the left side.

0111 1010 1111.0011

2. Write the hexadecimal digits that resemble every pair.

**(011110101111.0011)**_{2}** = **(7AF.3)_{16}

### Octal number system to other Number System

The octal number can be converted into different number systems. The method of changing octal to decimal differs from the other process.

#### A. Conversion of Octal to Decimal number system

The method of changing octal to decimal is just like binary to decimal. The method begins by multiplying the numbers of octal with their identical positional weights. Then, add all products.

**Example 1: **(149.25)_{8}

**Step 1:**

Multiply every digit of 149.25 with its corresponding positional weight, and then add products of each bit with its weight.

**(149.25)8 **= (1×8^{2}) + (4×8^{1}) + (9×8^{0}) + (2×8^{-1}) + (5×8^{-2})

**(149.25)8 **= 64 + 32 + 9 + (2×1⁄8) + (5×1⁄64)

**(149.25)8 **= 64 + 32 + 9 + 0.25 + 0.078125

**(149.25)8 **= 105.328125

The conversion of a decimal number of the octal 149.25 is **105.328125**

#### B. Octal to Binary Conversion

The method of changing octal to binary is just the opposite method of binary to octal. Write the three-three bits of binary code for each octal number digit.

**Example 1: **(105.25)_{8}

Write the three-bit binary code for 1, 0,5 2, and 5.

**(108.25)**_{8}** **= (001 000 101.010 101)_{2}

The binary conversion of the octal number 105.25 is **(001000101.010101)**_{2}

#### C. Conversion of Octal to hexadecimal number system

To convert octal to hexadecimal, the two steps needed to do, are mentioned below:-

- Find the binary equivalent of
**25**. - Create the sets of four bits on each side of the binary point. If any one, two, or three bits are still left in a set of four bits combinations, Add the demanded number of zeros on the utmost sides. Write hexadecimal digits resembling every pair.

**Example 1: **(142.25)_{8}

**Step 1:**

**W**rite the three-bit binary code.

**(142.25)**_{8}** **= (001100010.010101)_{2}

The binary number of the octal number is **(001100010.010101)**_{2}

**Step 2:**

1. Create sets of four bits on each side of the binary point.

0 0110 0010.0101 01

On the left side, the first set has one digit only, and on the right side, the end set contains two-digit only. Create perfect pairs of four bits, append zeros on needed sides.

0000 0110 0010.0101 0100

2. Write the hexadecimal code, which resembles every pair.

**(0000 0110 0010.0101 0100)**_{2}** **= **(62.54)**_{16}

### Conversion of Hexa-decimal to other Number System:

Hexadecimal numbers can easily be transformed into different number systems. The method of changing hexadecimal to decimal differs from others.

#### A. Conversion of Hexa-decimal to Decimal

The method of changing hexadecimal to decimal is just like binary to decimal. The method begins by multiplying the numbers of hexadecimal digits with their similar positional weights. Then, add all products.

**Example 1: **(142A.25)_{16}

**Step 1:**

Multiply every number of **142A.25** with its corresponding positional weight, then add the products of every bit with its weight.

**(142A.25)**** _{16 }**= (1×16

^{3}) + (4×16

^{2}) + (2×16

^{1}) + (A×16

^{0}) + (2×16

^{-1}) + (5×16

^{-2})

**(142A.25)**** _{16 }**= (1×4096 )+ (4×256) +(2×16) + (10×1) + (2×16-1) + (5×16-2)

**(142A.25)**** _{16 }**= 4096 + 1024 + 32 + 10 + (2×1⁄16) + (5×1⁄256)

**(142A.25)**** _{16 }**= 5162 + 0.125 + 0.125

**(142A.25)**** _{16 }**= 5162.14453125

The decimal number of the hexadecimal number is **5162.14453125**

#### B. Conversion of Hexadecimal to Binary

The method of changing hexadecimal to binary is the opposite method of binary to hexadecimal. Write the four-four bits binary code of every hexadecimal number.

**Example 1: **(142A.25)_{16}

Write the four-bit binary code.

(142A.25)_{16 }= (0001 0100 0010 1010.0010 0101)_{2}

The binary number of the hexadecimal number is **(1010000101010.00100101)**_{2}**.**

#### C. Conversion of Hexadecimal to Octal

To convert hexadecimal to octal, the two steps needed to do, are mentioned below

- Find the binary equivalent of the hexadecimal digit.
- Then, create the sets of three-three bits on each side. If one or two bits are still left in a combination of three bits pairs. Add the expected amount of zeros on needed sides. Compose the octal digits resembling every pair.

**Example 1: **(AF.2B)_{16}

**Step 1:**

Use binary as a mediator.

Write the four-bit binary code.

The binary code of (AF.2B)_{16} is (10101111.00101011).

**Step 2:**

Create sets of three-three bits on each side of the binary point.

010 101 111.001 010 110

Write the octal code, which resembles every pair.

The octal number of the hexadecimal AF.2B is (257.126)_{8. }

## Study Notes Based on Other Topics

Unit-I Teaching Aptitude Based Question

Unit-II Research Aptitude Based Question

Unit-III Comprehension Based Question

Unit-IV Communication Based Question

Unit-V Mathematical Reasoning and Aptitude Based Question

Unit-VI Logical Reasoning Based Question

Unit-VII Data Interpretation Based Question

Unit-VIII Information and Communication Technology (ICT) Based Question

Unit-IX People, Development and Environment Based Question

Unit-X Higher Education System Based Question

References