1.Number System

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.

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