[Maths Class Notes] on Number System For Class 9 Pdf for Exam

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.

Rational Numbers

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.

Irrational 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 

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

Decimal Expansion of Real Numbers 

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

1. 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. 

2. 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…

3. 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.

Rational Numbers on the Number Line 

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.