Types of Number System in Class 9 with Properties and Solved Examples
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 Complete Concept Guide
1. What is a number system in Class 9 Maths?
A number system is a way of representing and classifying numbers based on their properties and forms. In Class 9 Maths, the number system mainly includes:
- Natural numbers (N): 1, 2, 3, ...
- Whole numbers (W): 0, 1, 2, 3, ...
- Integers (Z): ..., -2, -1, 0, 1, 2, ...
- Rational numbers (Q): Numbers of the form p/q, where q ≠ 0
- Irrational numbers: Numbers that cannot be written as p/q
- Real numbers (R): All rational and irrational numbers together
2. What are real numbers in the number system?
The real numbers (R) include all rational and irrational numbers on the number line. Real numbers consist of:
- Rational numbers: Fractions like 3/4, -2, 0.5
- Irrational numbers: √2, π, √5
3. What is the difference between rational and irrational numbers?
The main difference is that rational numbers can be written in the form p/q (q ≠ 0), while irrational numbers cannot be expressed as a fraction. Key differences:
- Rational numbers have terminating or repeating decimals (e.g., 0.75, 0.333...)
- Irrational numbers have non-terminating, non-repeating decimals (e.g., √2 = 1.4142...)
- Example of rational: 7/8
- Example of irrational: π
4. How do you convert a decimal into a rational number?
A terminating or repeating decimal can be converted into a rational number by expressing it in the form p/q. Steps for a terminating decimal (example: 0.25):
- Write 0.25 = 25/100
- Simplify: 25/100 = 1/4
- Let x = 0.333...
- 10x = 3.333...
- Subtract: 10x − x = 3
- 9x = 3 ⇒ x = 1/3
5. What is Euclid’s Division Lemma?
The Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here:
- a = dividend
- b = divisor
- q = quotient
- r = remainder
6. How do you find the HCF using Euclid’s Division Algorithm?
The HCF using Euclid’s Division Algorithm is found by repeatedly applying a = bq + r until the remainder becomes 0. Steps (example: find HCF of 48 and 18):
- 48 = 18 × 2 + 12
- 18 = 12 × 1 + 6
- 12 = 6 × 2 + 0
7. What is an irrational number with an example?
An irrational number is a number that cannot be written in the form p/q and has a non-terminating, non-repeating decimal expansion. Examples:
- √2 = 1.414213...
- π = 3.14159...
8. How do you represent √2 on the number line?
The value of √2 can be represented on the number line using the Pythagoras theorem. Steps:
- Draw a line segment OA = 1 unit.
- At A, draw AB ⟂ OA with AB = 1 unit.
- Join OB. Then OB = √2.
- With center O and radius OB, cut the number line to mark √2.
9. What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique (except for order). Example:
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
10. When does a rational number have a terminating decimal expansion?
A rational number p/q (in lowest form) has a terminating decimal expansion if the prime factorization of q contains only 2’s and/or 5’s. In other words, q must be of the form 2ⁿ × 5ᵐ. Examples:
- 1/8 → 8 = 2³ → Terminating (0.125)
- 1/20 → 20 = 2² × 5 → Terminating (0.05)
- 1/3 → 3 is not 2 or 5 → Non-terminating repeating
