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For all you know ‘mathematics’ is entirely based on numbers. Without numbers, we would not have been able to study the concept of trigonometry, vectors, calculus, and algebra or in fact any aspect of life. So, whenever we talk about mathematics we are thankful for the discovery of the number system.

A Number system is a way to group numbers that are similar. Just like we learnt to separate letters into consonants and vowels, we can divide numbers into various groups. All those groups of numbers having similar characteristics are called number systems.This article follows the CBSE syllabus for class 9.

The numbers which can be expressed as the ratio of two integers are called rational numbers. In saying so, we must realise all integers are rational numbers. It is defined as numbers that can be written in the form of p/q, where p and q are integers and q is not equal to 0.

e.g. 5 is an integer. But it can be expressed as 5/1, which is a ratio of two integers 5 and 1.

All decimal numbers are not rational, as some of you might think.

e.g. √2 = 1.414…. it has infinite numbers after the decimal and thus, cannot be expressed as a ratio.

1.5 = 3/2, 0.5 = 1/2, etc are examples of decimal numbers that are also rational numbers.

It is defined as numbers that cannot be written in the form of p/q, where p and q are integers and q is not equal to 0.

e.g. √2=1.414 ,√15=

Real numbers constitute all the rational and irrational numbers. So, a real number can either be a rational number or an irrational number. Thus, every point on the number line actually signifies a real number. Therefore, √2 is a real number as well as 0.5, 3, etc

When we expand the real numbers into their decimal forms, we get three types of numbers:

### Terminating Decimal Numbers

The numbers, which on expanding into the decimal form, give the remainder as zero (0), is called terminating decimal numbers.

e.g. 7/8 = 0.875 , here the remainder on dividing 7 by 8 is zero. Thus, the decimal expansion of 7/8 is terminating.

### Non-Terminating Decimal Numbers

The numbers, which on expansion to the decimal form, never give zero as a remainder are called non terminating decimal expansions.

e.g. √2 = 1.414…

### Recurring Decimal Numbers

The numbers, which on expanding in the decimal form have repeating digits in the quotient are called recurring decimal numbers. Recurring means to re-occur.

e.g. 1/3 =0.3333... Here, 3 is recurring

3/7 = 0.428571428571.... Here, 428571 is recurring.

1/3 is a rational number as well as 7/8. So, from here, we can conclude that rational numbers are either terminating (e.g. 7/8) or non-terminating recurring (e.g. 1/3). Therefore, the decimal expansion of an irrational number is non terminating and non-recurring.

As mentioned earlier, each point on the number line is a real number. That means, each point is either rational or irrational. If we want to locate a number on the number on the number line how will we do it? Suppose, we want to locate 0.5 on the number line.

0 < 0.5 < 1 or, 0.5 = 0.5 1/2,

We can clearly understand that 0.5 lies exactly at the line joining 0 and 1. So, we can bisect the line to locate 0.5 on the number line.

FAQ (Frequently Asked Questions)

1. What do we understand about the octal number system?

An octal system is what we call as the base eight systems. The octal maths typically consists of digits like 0,1,2,3,4,5,6,7.The value of “!” is mathematically expressed as eight and 0 as ones or 108 .

2. What do we understand about the Hexadecimal number system?

It is a mathematical number system that has a base value of16. There are in total 16 ways to represent numbers via this number system. They are represented as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Here A, B, C, D, E and F are single bit representations of decimal value 10, 11, 12, 13, 14 and 15. It needs only 4 bits in order to represent the value of any digit.

3. What are the methods of Number system conversion?

There are various techniques of the number system conversion. These are as follows:

Decimal to Other Base System

Binary to Octal

Binary to Hexadecimal

Octal to Binary

Hexadecimal to Binary

4. Give one example of Binary to Hexadecimal conversion?

For the purpose of this type of conversion, we need to first divide the binary digits into sets of four. Make alterations in every set of four binary digits to symbol of one hexadecimal.

An example of the same is as below:

Binary − 101012 = Hexadecimal − 1516