The number system is the first chapter that has been given here for students to get a reference for the same. Here you will learn about the Number System with its definition and types of numbers. Also, learn the definition of all the types along with their properties. Students who are preparing for the exam could use these materials as a source of learning and have revision at the time of the final exams.
Introduction to Number System Class 9
The collection of numbers is called the number system. These numbers are of different types such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Let us see the table below to understand the examples.
| Natural Numbers | N | 1, 2, 3, 4, 5, … |
| Whole Numbers | W | 0,1, 2, 3, 4, 5… |
| Integers | Z | …., -3, -2, -1, 0, 1, 2, 3, … |
| Rational Numbers | Q | p/q form, where p and q are integers and q is not zero. |
| Irrational Numbers | I | Which can’t be represented as rational numbers |
Natural Numbers
All the numbers that start from 1 and end till infinity ( ∞ ) are Natural Numbers Such as 1,2,3,4,5,6,7,8,9,..... ∞ (infinity).
This set of numbers lie on the right side of the number line and they are positive number (+ve).
(Note - All Natural Numbers are Whole Numbers.)
Whole Numbers
All the numbers that start from 0 till ∞ (infinity) are called Whole Numbers. They are just 0 added to the beginning of natural numbers.
Example - 0,1,2,3,4,5,.......∞(infinity)
(Note - All the Whole Numbers are Integers.)
Integers
Integers are numbers that are whole numbers that are positive (+1), negative(-1), or maybe zero (0).
(Note - All Integers are Rational Numbers.)
Example:- 1,-2, 0,-14 all are integers.
Rational Numbers
The numbers can be represented in the form of p/q, where q ≠ 0 are known as rational numbers.
Example:- 13/49, 234/3
(Note - All Rational Numbers are Real Numbers.)
Irrational Numbers
A number is called an irrational number if it cannot be represented in the form of p/q.
Example:- √3, √5, √11, etc.
(Note - All Irrational Numbers are Real Numbers.)
Real Numbers
The collection of Rational Numbers & Irrational numbers are called Real Numbers.
Real Numbers are denoted by capital (R).
Every real number is a unique position on the number line and also every position on the number line represents a unique real number.
Difference between Terminating and Recurring Decimals
| Terminating Decimals | Repeating Decimals |
| If the decimal expression of a/b terminates. i.e. comes to an end, then the decimal so obtained is called Terminating decimals. | A decimal in which a digit or a set of digits repeats repeatedly periodically is called a repeating decimal. |
| Example: ¼ =0.25 | Example: ⅔ = 0.666… |
Some special Features of Rational Numbers.
- Every Rational number is expressible either as a terminating decimal or as a repeating decimal.
- Every terminating decimal is a rational number.
- Every repeating decimal is a rational number.
Irrational Numbers
- The non-terminating, non-repeating decimals are irrational numbers.
Example: 0.0100100001001…
- Similarly, if m is a positive number which is not a perfect square, then √m is irrational.
Example: √3
- If m is a positive integer which is not a perfect cube, then 3√m is irrational.
Example: 3√2
Properties of Irrational Numbers
- These satisfy the commutative, associative and distributive laws for addition and multiplication.
- Sum of two irrationals need not be irrational.
Example: (2 + √3) + (4 – √3) = 6
- Difference of two irrationals need not be irrational.
Example: (5 + √2) – (3 + √2) = 2
- Product of two irrationals need not be irrational.
Example: √3 x √3 = 3
- The quotient of two irrationals need not be irrational.
2√3/√3 = 2
- Sum of rational and irrational is irrational.
- The difference of rational and irrational number is irrational.
- Product of rational and irrational is irrational.
- Quotient of rational and irrational is irrational.
Real Numbers
A number whose square is non-negative is called a real number.
- Real numbers follow Closure property, associative law, commutative law, the existence of an additive identity, existence of additive inverse for Addition.
- Real numbers follow Closure property, associative law, commutative law, the existence of a multiplicative identity, existence of multiplicative inverse, Distributive laws of multiplication over Addition for Multiplication.
Rationalisation
If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number is called rationalization.
Example:
3/√2 = (3/ √2) x (√2/√2) = 3 √2/2
Laws of Radicals
Let a>0 be a real number, and let p and q be rational numbers, then we have:
i) (ap).aq = a(p+q)
ii) (ap)q = apq
iii)ap/aq = a(p-q)
iv) ap x bp = (ab)p
Example: Simplify (36)½
Solution: (62)½ = 6(2 x ½) = 61 = 6
Some Practise Question for You to do.
Q1) The simplest form of 1.6~ is?
Q2) An irrational number between √2and √3 is?
Q3) Give an example of two irrational numbers whose sum as well as the product is rational.
More questions with answers at:- WoW Study India
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