**Short Notes** & **Video Lectures** on **Real Numbers for Class 10 Board Exam** (Padhte Chalo, Badhte Chalo- Eckovation)Join Group Code101010

Numbers which can be represented on a** number line** are called *Real Numbers.*

Real numbers consist of

(i) **Rational Numbers**: Numbers which can be represented by* p/q *form

(ii)

**Irrational Numbers**: Numbers which

**cannot**be represented by

*form*

**p/q**### Video Lecture To Understand The Introduction Of Real Numbers

### Euclid’s Division Lemma

Let ‘a’ and ‘b’ be any two positive integers. Then there exist unique integers ‘q’ and ‘r’ such that

a = bq + r, 0 ≤ r ≤ b

# Steps To Find H.C.F By Using Euclid’s Division Lemma

**Step 1 : **Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.

**Step 2 : **If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

**Step 3 : **Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

# Prime & Composite Numbers

Prime Numbers : The numbers which are only divisible by 1 or itself

Composite Numbers: The numbers which are divisible by numbers other than 1 or itself

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# Fundamental Theorem Of Arithmetic

Every **composite number** can be written as a product of prime numbers and this factorization is unique.

# Theorems on Rational and Irrational Numbers

**Irrational Numbers**

**Theorem 1**-Let p be a **prime number.** If p divides a2, then p divides a, where a is a positive integer

**Rational Numbers**

**Theorem 1** : If p q is a rational number, such that the prime factorization of q is of the form 2^{a}5^{b}, where a and b are positive integers, then the decimal expansion of the rational number p q terminates.

**Theorem 2** :If a rational number is a terminating decimal, it can be written in the form p q , where p and q are co prime and the prime factorization of q is of the form 2^{a}5^{b}, where a and b are positive integers

**Theorem 3** : If p q is a rational number such that the prime factorization of q is not of the form 2^{a}5^{b} where a and b are positive integers, then the decimal expansion of the rational number p q does not terminate and is recurring.

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