Counting numbers like 1, 2, 3, 4, … are called natural numbers. We denote the collection of natural numbers by ‘N’.
All natural numbers together with zero are called whole numbers. We denote the group of whole numbers by the symbol ‘W’.
The negative numbers, the positive numbers, and zero together form the set of integers. The set (collection) of all integers is denoted by the symbol ‘Z’.
Note: The symbol Z is taken from the German word “Zahlen”, which means “to count”.
A number which can be represented in the form of p/q is called a rational number. Rational numbers include natural numbers, integers, and all the (negative and positive) fractions. The collection of rational numbers is denoted by the symbol ‘Q’.
Example: 1/2, 4/5, 26/8, etc.
Note: The symbol Q comes from the word ‘quotient’ and the word ‘rational’ comes from ‘ratio’.
➢ Rational Numbers Between Two Given Rational Numbers
(i) Let ‘x’ and ‘y’ be two rational numbers.
A rational number lying between ‘x’ and ‘y’ = (x + y)/2.
(ii) Let ‘x’ and ‘y’ be two rational numbers such that x < y. For finding ‘n’ rational numbers between ‘x’ and ‘y’, let
Then ‘n’ rational numbers between ‘x’ and ‘y’ are:
(x + d); (x + 2d), (x + 3d), … (x + nd).
Note: The Result II is known as the method of finding rational numbers in one step.
A number which cannot be written in the form of p/q, (where p and q are integers and q ≠ 0), is called an irrational number. We denote an irrational number by ‘S’.
Examples: √2, √3, √5, √6, √11, π, 0.10110111011110...., etc., are not rational numbers, so they are irrational numbers.
(i) An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating 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.
➢ Properties of Irrational Numbers
The collection of all rational numbers and irrational numbers together make up a collection of real numbers. The collection of real numbers is denoted by ‘R’. The square of a real number is always non-negative(≥0).
(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 “number-line” as “real-line” also. All numbers positive and negative, integers and rational numbers, square-roots, cube-roots, 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.
All rational numbers can be expressed as a decimal number. A rational number in the form of p/q, (q ≠ 0), on division of ‘p’ by ‘q’, either the remainder becomes zero or never becomes zero.
Look at the following examples:
Remainder = 0
Quotient = 0.375
Here, the decimal expansion ends (terminates) after a finite number of steps, such numbers are called terminating.
Note: On expressing a rational-number as its decimal expansion, if remainder becomes zero after certain stage, then the decimal-expansion is terminating. If the remainder never becomes zero but repeats after certain stage, then the decimal expansion is non-terminating recurring.
Remainder: Every time we are getting the remainder 2, which will never end.
Quotient: 0.666…, i.e. the digit 6 is repeating every time.
Here, the decimal expansion does not end (does not terminate) even after an infinite number of steps. Such numbers are called non-terminating.
(iii) Let us consider one more example of a non-terminating rational number.
Express 14/11 as a decimal number
Let us divide 14 by 11.
Remainder: We are getting the remainder as 3 and 8 alternatively.
Quotient: The digits after the decimal point are repeating in a block of two digits, i.e. ‘2’ and ‘7’.
In such cases, the decimal expansion go on forever and we get a repeating block of digits in the quotient. We call such numbers as non-terminating repeating (or non-terminating recurring).
(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 non-terminating recurring (repeating).
The decimal expansion of an irrational number is non-terminating and non-recurring (non-repeating).
(ii) √3 = 1.732050807...
(iii) √7 = 2.645751311 ...
(iv) π = 3.14159265358 ....
(v) √2 = 1.41421356237 ....
Note: Generally we take π = 22/7 which is not true. In fact π is an irrational number having a value which is non-terminating and non-repeating, i.e. π = 3.1415926535897932384 …
whereas 22/7 is rational number whose value is non-terminating repeating, i.e. 22/7 = 3.142857142857…
Thus, π and 22/7 are same only up to two places of decimal.
➢ Special Properties of Rational And Irrational Numbers
(i) If ‘m’ is a positive integer but not a perfect square, then m is an irrational number.
∴ √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
(ii) 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
We know that there is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line. The process of successive magnification helps us to locate a real number on the number line.
Note: The process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification.
➢ Rationalizing Factor (RF)
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