CBSE Class 10 Real Numbers Important Formulas and concepts for exams

CBSE Class 10 Real Numbers Important Formulas and concepts for exams. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

EUCLID’S DIVISION LEMMA

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where0 £ <r b .

Here we call ‘a’ as dividend, ‘b’ as divisor, ‘q’ as quotient and ‘r’ as remainder.

Dividend = (Divisor x Quotient) + Remainder

If in Euclid’s lemma r = 0 then b would be HCF of ‘a’ and ‘b’.

NATURAL NUMBERS

Counting numbers are called natural numbers i.e. 1, 2, 3, 4, 5, ……………. are natural numbers.

WHOLE NUMBERS

All counting numbers/natural numbers along with 0 are called whole numbers i.e. 0, 1, 2, 3, 4, 5……………. are whole numbers.

INTEGERS

All natural numbers, negative of natural numbers and 0, together are called integers. i.e.………. – 3, – 2, – 1, 0, 1, 2, 3, 4, ………….. are integers.

ALGORITHM

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

LEMMA

A lemma is a proven statement used for proving another statement.

EUCLID’S DIVISION ALGORITHM

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

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 ¹ 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.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.

The Fundamental Theorem of Arithmetic

Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The prime factorisation of a natural number is unique, except for the order of its factors.

* HCF is the highest common factor also known as GCD i.e. greatest common divisor.

* LCM of two numbers is their least common multiple.

* Property of HCF and LCM of two positive integers ‘a’ and ‘b’: 

Ø HCF(a, b) ´ LCM (a, b) = a´x b

Ø LCM (a, b) = a x b / HCF (a, b) 

Ø HCF (a, b) = a x b /LCM (a, b)

PRIME FACTORISATION METHOD TO FIND HCF AND LCM

HCF(a, b) = Product of the smallest power of each common prime factor in the numbers. LCM(a, b) = Product of the greatest power of each prime factor, involved in the numbers.

RATIONAL NUMBERS

The number in the form of p/q

where ‘p’ and ‘q’ are integers and q ¹ 0 , e.g. 2/3 , 3/5 , 5/7 , .......

Every rational number can be expressed in decimal form and the decimal form will be either terminating or non-terminating repeating. e.g. 5/2 = 2.5 (Terminating), 2/3 = 0.66666.... or 0.6 (Non- terminating repeating).

IRRATIONAL NUMBERS

The numbers which are not rational are called irrational numbers. e.g. √2, √3, √5, etc .

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

v If p is a positive integer which is not a perfect square, then √m is an irrational, e.g.

√2 , √5, √6 , √8,...etc .

v If p is prime, then √p is also an irrational.

RATIONAL NUMBERS AND THEIR DECIMAL EXPANSIONS

Ø Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p /q

where p and q are coprime, and the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

Ø Let x = p /q be a rational number, such that the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Ø Let x = p / q be a rational number, such that the prime factorisation of q is not of the form 2ⁿ5^m,

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

v The decimal form of irrational numbers is non-terminating and non-repeating.

v Those decimals which are non-terminating and non-repeating will be irrational numbers. e.g. 0.20200200020002……. is a non-terminating and non-repeating decimal, so it irrational.

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