Class IX(Chapter 1.Numbers System)
1.Natural Number (N) : All number s used in counting are known as natural numbers such as
1,2,3,….., is called natural number.Denoted by N.N={1,2,3,……,n}.
2.Whole Number(W): All natural numbers including zero , is called Whole numbers.Denoted by W.
W={0,1,2,3,…..n}.
3.Integers(I):- All number with +ve and –ve sign including 0, is known as Integers.Denoted by I.
I={-n,…,-2,-1,0,1,2,….n}
4. Rational number(Z):-A number ‘r’ is called a rational number, if it can be written in the form p/q,where p and q are integers and q ¹ 0. Such as 2/5,/9 etc
Fact 1:-Pythagoras discovered irrational numbers around 400 BC. These numbers are called irrational numbers(irrationals), because they cannot be written in the form ofa ratio of integers.
Fact2:- Hippacus has an unfortunate end, either for discovering that 2 is irrational
Fact 3:- Historical Fact Z comes from theGerman word “zahlen”, which means“to count”
Fact4: The Greek genius Archimedes was the first to compute digits in the decimal expansion of p. He showed 3.140845< p < 3.142857. Aryabhatta (476 – 550 C.E.), the great Indian mathematician and astronomer, found the value of p correct to four decimal places (3.1416).
Example: Find five rational numbers between 1 and 2.
Solution: Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1,
i.e., 1 =6/6 and 2 =12/6
. Then We can check that 7/6,8/6,9/6,10/6 and 11/6 are all rational
numbers between 1 and 2. So, the five numbers are 7/6,8/6,9/6,10/6 and 11/6.
5. Irrational number :-A number ‘s’ is called irrational, if it cannot be written in the form p/q
, where pand q are integers and q ¹ 0. Example:- 2, 3, 15,, p, 0.10110111011110…
6.Real numbers(R):- the collectionof all rational numbers and irrational numbers together
make up what we call the collection of real numbers,
which is denoted by R. Therefore, a real number is either rational or irrational.
We can say that every real number is represented by a unique point on the number
line. Also, every point on the number line represents a unique real number.This is why we call the number line, the real number line.