class 9 maths REAL NUMBERS We have learnt about various types of numbers in our earlier classes. Let us review them and know more about numbers. Natural numbers: Counting numbers are known as natural numbers. We always start counting anything from 1 always. So natural numbers starts from 1. Thus, 1, 2, 3, 4, 5, 6, etc., are all natural numbers. Whole numbers: All natural numbers together with 0 form the collection of all whole numbers. Thus, 0, 1, 2, 3, 4, 5, 6, etc., are all whole numbers. ØEvery natural number is a whole number. Ø0 is a whole number which is not a natural number Integers: All natural numbers, 0 and negative of natural numbers form the collection of all integers. Rational numbers whose denominator is 1 is called Integer. Thus, –4, –3, –2, –1, 0, 1, 2, 3, 4, etc. are all integers. ØEvery natural number is an integer. ØEvery whole number is an integer. p q
Rational numbers: Any number of the form , where p and q are integers and q¹ 0, are known as rational numbers. Operations on rational numbers: For any two rational numbers p/q and m/n, we define Addition + = Subtraction
- =
Multiplication
× =
Division
÷ = × =
,
where m≠0, n≠0
SIMPLEST FORM OF A RATIONAL NUMBER: A rational number
p is said to be in simplest form, if p and q are integers having no common factor q
other than 1 (i.e. both p and q are relatively co-prime). Here q¹ 0 2 3 4 5 6 1 , , , etc. is . 4 6 8 10 12 2
Thus, the simplest form of each of , Similarly, the simplest form of
6 2 76 4 is and that of is . 9 3 133 7