# Video Lecture &Short Notes:Real Numbers 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 p/q form ### 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 # 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 2a5b, 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 2a5b, 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 2a5b where a and b are positive integers, then the decimal expansion of the rational number p q does not terminate and is recurring.