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