**Table of Contents**

1. | Introduction |

2. | What is the Number System? |

3. | Types of Number Systems |

4. | Number System Conversions and Charts |

5. | Number Systems in Computers and Math |

6. | Summary |

7. | Frequently Asked Questions (FAQs) |

25^{th} September 2020

Reading Time: 6 Minutes

**Introduction**

The Number System is the system of representing numbers. The various types of Number Systems used in Maths and other applications are binary, decimal, octal, hexadecimal, etc. This article would like to cover the Number System’s entire concepts with their types, conversions, and some practice questions.

So Let’s Begin.

**What is the Number System?**

**Definition :**

A Number System is defined as a system of writing to express numbers that can be easily understood by the person it was intended for. It is the mathematical notation for representing numbers of a given set by using various other digits along with other symbols in a predefined structured manner. It hence proves to provide a unique representation for every number and represents the arithmetic and algebraic structure of the figures, thereby allowing us to operate arithmetic operations like addition, subtraction, multiplication, and division.

The value of any digit in a number can be determined by:

- The digit
- Its position in the number
- The base of the number system

**Types of Number System (International, Decimal, Octal, Indian, etc.)**

There are various types of number systems in mathematics and computer science. The four most common number system types are:

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

**Decimal Number System (Base 10 Number System)**

The decimal number system has base 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions starting from the rightmost digit moving towards left represent units, tens, hundreds, thousands, and so on, respectively. This system is called the decimal number system.

Every position shows a particular power of the base (10). For example, the decimal number 1547 consists of the digit 7 in the units position, 4 in the tens place, 5 in the hundreds position, and 1 in the thousands place whose value can be written as

(1×1000) + (5×100) + (4×10) + (7×1)

1000 + 500 + 40 + 7

1547

**Binary Number System (Base 2 Number System)**

The Base-2 number system is also famously known as the Binary Number System wherein only two binary digits exist, i.e., 0 and 1. The figures described under this system are known as binary numbers, which are the combination of 0 and 1. For example, 110110 is a binary number.

We can convert any system into binary and vice versa.

Example |

Write (14)_{10} as a binary number.

**Solution:**

∴ (14)_{10} = (1110)_{2}. |

**Octal Number System (Base 8 Number System)**

In the Octal Number System, the base is 8 as it uses numbers from 0 to 7 to represent all numbers used in Mathematics. Octal numbers are also commonly encountered in computer applications.

Example |

Convert 2268 into decimal.

**Solution:**

2268 = 2 × 8^{3} + 2 × 8^{2} + 6 × 8^{1} + 8 × 8^{0}

= 2 × 512 + 2 × 64 + 6 × 8 + 8

= 1024 + 128 + 40 + 8

= 1200 |

**Hexadecimal Number System (Base 16 Number System)**

In the hexadecimal system, numbers are represented with base 16 i.e. using the digits 0 to 9 and alphabets A to F. The below-given table shows the representation of numbers in the Hexadecimal Number System

Hexadecimal | 0 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |

Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

**International Numeral System**

The place values of digits go in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousands, Millions, Ten Million, and so on, in the International Numeral System. In the number 98,765,432 the place values of each digit are:

- 2 – Ones
- 3 – Tens
- 4 – Hundreds
- 5 – Thousands
- 6 – Ten Thousand
- 7 – Hundred Thousands
- 8 – Millions
- 9 – Ten Million

The relations between them are:

- 1 hundred = 10 tens
- 1 thousand = 10 hundreds = 100 tens
- 1 million = 1000 thousand
- 1 billion = 1000 millions

International Numeral System

Comparison between Indian and International Numeral System

Comparing the two numeral systems we observe that:

- 100 thousand = 1 lakh
- 1 million = 10 lakhs
- 10 millions = 1 crore
- 100 millions= 10 crores

**Placement of Comma**

Commas are placed in large numbers to help us read and write them easily. In Indian and the International systems, the commas are placed at different positions.

As per the Indian numeral system, the first comma is placed after the hundreds place post, which they then placed after every two digits. E.g., 9,87,65,43,210

### As per the international numeral system, the first comma is placed after the hundreds place, post which they are placed after every three digits. E.g., 9,876,543,210

## **Number System Conversions and Charts**

As you know, decimal, binary, octal and hexadecimal number systems are positional value number systems. To convert binary, octal, and hexadecimal to decimal numbers, we just need to add the product of each digit with its positional value. Here we’re going to learn other conversions among these number systems.

**Decimal to Binary**

Decimal numbers can be converted to binary by repeated division of the number by 2 while recording the remainder. Let’s take an example to see how this happens.

The remainders are to be read from bottom to top to obtain the binary equivalent.

43_{10} = 101011_{2}

**Decimal to Octal**

Decimal numbers can be converted to octal by repeated division of the number by 8 while recording the remainder. Let’s take an example to see how this happens.

Reading the remainders from bottom to top,

473_{10} = 731_{8}

**Decimal to Hexadecimal**

Decimal numbers can be converted to octal by repeated division of the number by 16 while recording the remainder. Let’s take an example to see how this happens.

Reading the remainders from bottom to top we get,

423_{10} = 1A7_{16}

### To convert a binary number to octal number, these steps are followed −**Binary to Octal and Vice Versa**

- Starting from the least significant bit, make groups of three bits.
- If there are one or two bits less in making the groups, 0s can be added after the most significant bit
- Convert each group into its equivalent octal number.

Let’s take an example to understand this.

10110010101_{2} = 2625_{8}

To convert an octal number to binary, each octal digit is converted its 3-bit binary equivalent according to this table.

Octal Digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Binary Equivalent | 000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 |

54673_{8} = 101100110111011_{2}

**Binary to Hexadecimal**

To convert a Binary number to a Hexadecimal number, steps that need to be followed −

- Starting from the least significant bit, make groups of four bits.
- If there are one or two bits less in making the groups, 0s can be added after the most significant bit.
- Convert each group into its equivalent hexadecimal number.

Let’s take an example to understand this.

10110110101_{2} = DB5_{16}

To convert an octal number to binary, each octal digit is converted to its 3-bit binary equivalent.

**Roman to Arabic Conversion:**

Roman Numbers are nothing but a unique way of writing numbers. Roman Numerals is a number system that was introduced in ancient Rome. The purpose of Roman Numbers was to price various goods and services. The Roman Numbers were used in the Roman Empire and by the people of Rome in their day to day life. These Roman Numbers were also used in Europe.

Look at some important Roman Numbers and how they are represented in Hindu Arabic Numerical System:

**Number Systems in Computers and Math**

In Computers, we use binary systems with base 2 as Computers understand the binary language of 0’s and 1’s. In Maths, we use different types of numbers System like decimal, binary, octal, and hexadecimal.

**SUMMARY**

So here we come to the end of this article; almost everything about the Number System and its Conversion and different types of Number System is taken care of. We hope you all got the relevant information; please place your valuable feedback and likes.

**Frequently Asked Questions (FAQs)**

**The cumulative addition of the four binary bits ( 1 + 1 + 1 + 1) gives? **

Ans: 100 = 4_{10}

**The binary addition 1 + 1 + 1 gives**

Ans: 11 = 3_{10}

**The digital systems usually operate on ........systems.**

Ans: Binary

**One hex digit is sometimes referred to as a(n):**

Ans: Nibble

**Convert into decimal: (214)**_{8} =?

_{8}=?

Ans: (140)_{10}

**The hexadecimal equivalent of (170)**_{10}

_{10}

Ans: (AA)_{16}

**The decimal equivalent of (LM17) is given by**

Ans: 491

**The two digits hexadecimal number which has the largest value is ___, which corresponds to _____**

Ans: FF, 255 decimal

**The octal number 645 in the power of 8 is equal to**

Ans: 421

**If the decimal number is a fraction, then its binary equivalent is obtained by ________ the number continuously by 2. **

Ans: Multiplying