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 *p*and *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*.