Table of contents  
Introduction  
1. Natural Numbers  
2. Whole Numbers  
3. Integers  
4. Rational Numbers  
5. Irrational Numbers  
6. Real Numbers 
Note: The symbol Z is taken from the German word “Zählen”, which means “to count”.
Note: The symbol Q comes from the word ‘quotient’ and the word ‘rational’ comes from ‘ratio’.
Then ‘n’ rational numbers between ‘x’ and ‘y’ are: (x + d); (x + 2d), (x + 3d), … (x + nd)
This is known as the method of finding rational numbers in one step.
Note:
(i) An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless nonrepeating digits to the right of the decimal point i.e., an irrational number is an infinite decimal.
(ii) Irrational numbers are rarely used in daily life, but they do exist on the number line. In fact, on the number line, between 0 and 1, there are an infinite number of irrational numbers.
If ‘m’ is a positive integer but not a perfect square, then √m is an irrational number. i.e. √1 = 1, rational number;
√2, irrational number
√3, irrational number
√4 = 2, rational number
√5, irrational number
√6, √7, √8 irrational number
√9 = 3 rational number
If ‘m’ is a positive integer but not a perfect cube, then ∛m is an irrational number. So ∛2, ∛3, ∛4, ∛5, ∛6, ∛7,.. are irrational number
But ∛1 = 1, rational number
∛8 = 2, rational number
∛27 = 3 , rational number
∛64 = 4 , rational number
Remember
(i) Every real number is either rational or irrational.
(ii) Between any two real numbers there exists an infinite number of real numbers.
(iii) On the number line each point corresponds to a unique real number. On the other hand, every real number can be represented by a unique point on the number line. That is why we call a “numberline” as “realline” also. All numbers positive and negative, integers and rational numbers, squareroots, cuberoots, π (pi) are present on a number line
(iv) 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.
(a) The Decimal Representation of Rational Numbers
Remember
(i) The number of digits in the repeating block is always less than the divisor.
(ii) The decimal expansion of a rational number is either terminating or nonterminating recurring (repeating).
(b) The Decimal Representation of Irrational Numbers
Note: Generally we take π = 22/7 which is not true. In fact π is an irrational number having a value which is nonterminating and nonrepeating, i.e. π = 3.1415926535897932384… whereas 22/7 is rational number whose value is nonterminating repeating, i.e. 22/7 = 3.142857142857…
Thus, π and 22/7 are same only up to two places of decimal.
If the product of two rational numbers is rational, then each one is called the rationalizing factor of the other.
(i) If ‘a’ and ‘b’ are integers, then
(a + √b) and (a  √b) are RF of each other.
(ii) If ‘x’ and ‘y’ are natural numbers, then
(√x+ √y) and (√x  √y) are RF of each other.
(iii) If ‘a’ and ‘b’ are integers and ‘x’ and ‘y’ are natural numbers, then
(a +b√x) and (a b√x) are RF of each other.
If a > 0 and b > 0 be real numbers, ‘m’ and ‘n’ be rational numbers, then we have
Note: The process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification.
48 videos378 docs65 tests

1. What are natural numbers? 
2. What are whole numbers? 
3. What are integers? 
4. What are rational numbers? 
5. What are irrational numbers? 
48 videos378 docs65 tests


Explore Courses for Class 9 exam
