Key Types of Numbers and Properties Explained for Class 9
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:
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.
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.
FAQs on Number System for Class 9 Maths: Complete Guide
1. What are real numbers in the context of the Class 9 syllabus?
Real numbers are the set of numbers that includes both rational and irrational numbers. Essentially, any number that can be plotted on a number line is a real number. This includes positive and negative numbers, fractions, and decimals. For Class 9, the focus is on understanding this entire collection and the properties of its subsets, as per the CBSE 2025-26 syllabus.
2. What is the main difference between rational and irrational numbers?
The primary difference lies in their decimal representation and fractional form. A rational number can be written as a fraction p/q, where p and q are integers and q is not zero. Its decimal form is either terminating (e.g., 0.5) or non-terminating but recurring (e.g., 0.333...). An irrational number cannot be written as a simple fraction, and its decimal form is both non-terminating and non-recurring (e.g., the value of π or √2).
3. Is 0 a rational number? Explain why.
Yes, 0 is a rational number. The definition of a rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. We can write 0 as 0/1, 0/2, or 0 divided by any non-zero integer. Since it fits the definition perfectly, it is classified as a rational number.
4. How does decimal expansion help identify if a number is rational or irrational?
The nature of a number's decimal expansion is a key identifier.
- If the decimal expansion terminates (e.g., 2.25) or if it is non-terminating but has a repeating pattern of digits (e.g., 1.090909...), the number is rational.
- If the decimal expansion is non-terminating and shows no repeating pattern (e.g., 3.14159265...), the number is irrational.
5. What is the practical importance of rationalising the denominator?
Rationalising the denominator is a process used to remove an irrational number (like a square root) from the bottom of a fraction. The main importance is to standardise the form of an expression. This makes it easier to perform further calculations, like addition or subtraction with other fractions, and simplifies the process of approximating the final value of the expression.
6. Why is 'pi' (π) considered an irrational number, but 22/7 is rational?
This addresses a common misconception. The value 22/7 is a rational approximation of π, not its exact value. The actual decimal representation of π is non-terminating and non-recurring (3.14159...). Because it cannot be written as an exact fraction of two integers, π is an irrational number. In contrast, 22/7 is clearly in the p/q form, making it a rational number.
7. What are the main laws of exponents for real numbers that are important for Class 9?
For positive real numbers 'a' and 'b' and rational exponents 'p' and 'q', the key laws are:
- ap ⋅ aq = ap+q (Product Rule)
- (ap)q = apq (Power of a Power Rule)
- ap / aq = ap-q (Quotient Rule)
- ap ⋅ bp = (ab)p (Power of a Product Rule)
- a0 = 1 (Zero Exponent)
- a-p = 1/ap (Negative Exponent)
8. Can all integers be considered rational numbers? Explain with an example.
Yes, all integers are rational numbers. This is because any integer 'n' can be written in the p/q form by setting the denominator 'q' to 1. For example, the integer 5 can be written as 5/1, the integer -3 can be written as -3/1, and 0 can be written as 0/1. Since they all satisfy the condition for rational numbers, the set of integers is a subset of the set of rational numbers.
9. Give an example of how to represent an irrational number, like √3, on the number line.
To represent √3 on the number line, you can use the Pythagoras theorem. First, represent √2 by constructing a right-angled triangle with a base of 1 unit and height of 1 unit. The hypotenuse is √2. Then, using this length as the base and a perpendicular of 1 unit, construct a new triangle. The new hypotenuse will be √( (√2)² + 1² ) = √(2 + 1) = √3. This length can then be marked on the number line using a compass.
10. What is the difference between whole numbers and natural numbers?
The only difference between whole numbers and natural numbers is the inclusion of the number zero.
- Natural numbers are the counting numbers, starting from 1: {1, 2, 3, 4, ...}.
- Whole numbers include all the natural numbers plus zero: {0, 1, 2, 3, ...}.
