Class 10th Maths Notes of Ch 1 Real Numbers | NCERT Class 10

1. Real Numbers

Set of rational and irrational numbers.

2. Rational Numbers

These numbers can be represented in the form of p/q where, q = 0 and both p and q are rational numbers.

(i) Natural Numbers = All numbers from 1 to infinity.
For Ex = 1, 5, 6, 8

(ii) Whole Numbers = All numbers from 0 to infinity.
For Ex = 0, 1, 2, 4, 7, 8

(iii) Negative Numbers = All numebers from -1 to infinity(negative)
For Ex = -1, -2, -4, -6

(iv) Integers = Integers is a set of all whole numbers and negative numbers.
For Ex = -2, -1, 0, 1, 4, 6

(v) Fractions = These are the numbers which are in the form of p/q.
For Ex = 3/4, 6/5, 10/9, 50/3

(vi) Decimal Numbers = Numbers with a decimal point.

• (a) Terminating numbers = Numbers which are completely divisible and hence terminate.For Ex = 4.78, 2.5, 2.678, 6.587
• (b) Non terminating repeating numbers = Numbers that do not terminate but repeats a pattern of digits.For Ex = 4.14141414..., 5.767767767767..., 98.1232323..., 6.12345345345345...

3. Irrational Numbers

Form of non-terminating and non-repeating decimal numbers.

4. Prime Numbers

Numbers who have exactly 2 factors first itself and second 1.
For Ex => 2, 3, 13, 23

5. Composite Numbers 

Numbers who have more than two factors.
For Ex => 4, 6, 8, 50

6. Euclid's Division Lemma

For any two integers a and b there exixt two integers q and r so that a = bq + r, 0 <= r < b.
Here, a = dividend, b = divisor, q = quotient and r = remainder.
For Ex => 6 = 3 * 2 + 0
We can also say that,
dividend = divisor * quotient + remainder.

7. Fundamental Theorem of Arithmetics

Every composite number can be expressed as the product of a prime number and this type of factorization is unique.
For Ex =>
8 = 2 * 2 * 2 *1
15 = 3 * 5 * 1

8. HCF

HCF is the highest common factor of a given number a. HCF is also known as GCD.
It can be found by = 

• Factorization Method.
• Division Method.
• Euclid's Division Algorithm.

9. LCM

LCM is the lowest common multiple. It can be found by the Factorisation Method.

10. Euclid's Division Algorithm

In the euclids division algorthim we use euclid's division lemma until we get the remainder zero.
a = bq + r, 0 <= r < b.

For Ex =>
225 = 135 * 1 + 90
135 = 90 * 1 + 45
90 = 45 * 2 + 0
r = 0 therefore the process stops.
The divisor at the last stage is the HCF.
HCF(225, 135) = 45

11. Factorization Method

In the factorization method, we just take the LCM of the given numbers.

For ex =>
6 = 3 * 2 * 1
10 = 5 * 2 * 1
The product of the common factors is HCF.
HCF(6, 10) = 2
The product of all factors taking common factors as one is LCM.
LCM(6, 10) = 2 * 3 * 5 = 30

Conclusion

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