These comprehensive RBSE Class 10 Maths Notes Chapter 1 Real Numbers will give a brief overview of all the concepts.

## RBSE Class 10 Maths Chapter 1 Notes Real Numbers

Important Points

1. Euclid’s Division Algorithm — Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.

2. Euclid’s Division Lemma—Given two positive integers a and b we can find out the whole numbers q and r satisfying a = bq + r, 0 < r < b, i.e., such numbers exist. Euclid was the first Greek mathematician who put forth a new ideology for the study of plane Geometry. According to him on dividing a positive integer by some other positive integer, we obtain the quotient q and the remainder r and the remainder r is either zero or smaller than the divisor b i.e, 0 < r < b.

In simple words

Dividend (a) = Divisor (b) x Quotient (q) + Remainder (r)

3. Euclid’s Division Algorithm is based on Euclid’s Division Lemma. Using this technique we can compute the Highest Common Factor (HCF) of two given positive integers a and b (a > b). Following the steps given below:

Step I : Apply Euclid’s Division Lemma to find q and r where a = bq + r, 0 ≤ r < b.

Step II : If r = 0, then HCF = b. If r ≠ 0, then we apply Euclid’s Division Lemma on b and r.

Step III : Continue this process till the remainder is zero. The division at this stage is HCF (a, b). Also,

HCF (a, b) = HCF (b, r)

With the help of this main relation, other relations may also be written as given below :

4. Fundamental Theorem of Arithmetic – Every composite number can be expressed (factorised) as a product of prime numbers and this factorisation is unique.

For any two positive integers a and b, H.C.F. (a, b) x L.C.M. (a, b) = a * b.

5. Important Points –

(i) Euclid’s Division Algorithm is not only useful in computing the H.C.F. of large numbers but also important for the reason that this is one of more algorithms that were first of all used as a program in a computer.

(ii) Euclid’s Division Lemma and Euclid’s Division Algorithm are so closely interlinked that people often call Euclid’s Division Lemma as Euclid’s Division Algorithm.

(iii) Euclid’s Division Lemma/Algorithm is stated only for positive integers. However it can be applied for all integers (except zero, i.e., b ≠ 0).

6. If p is a prime number and if p divides a^{2}, then p will divide a. where a is a positive integer.

7. √2, √3, √5 and in general, √p are irrational numbers, where p is a prime number.

8. Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form \(\frac{p}{q}\) where p and q are co prime, and the prime factorisation of q is of the form 2^{n} 5^{m}, where n, m are non-negative integers.

9. Let x = \(\frac{p}{q}\) be a rational number, such that the prime factorisation of q is of the form 2^{n} 5^{m}, where n, m are non-negative integers. Then, x has a decimal expansion which terminates.

10. Let x = \(\frac{p}{q}\) be a rational number, such that the prime factorisation of q is not of the form

2^{n} 5^{m}, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

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