NOTE CLASS 10 MATHEMATICS CHAPTER-1

REAL NUMBERS

                 TOPICS

1. EUCLID'S DIVISION LEMMA 

2. EUCLID'S DIVISION ALGORITHM

3. FUNDAMENTAL THEOREM OF ARITHMETIC

4. DECIMAL REPRESENTATION OF RATIONAL NUMBERS

5.UNIT DIGIT OF ANY POWER

EUCLID'S DIVISION LEMMA

Given two positive integers a and b, there exist unique integers q and r satisfying a= bq + r where 

0 ≤ r < b

For eg,

 11 = 2 × 5 + 1

Here, a = 11 , b = 2 , q = 5 , r = 1

Here, 'a' is the dividend, 'q' is the quotient, 'b' is the divisor and 'r' is the remainder.Here 'q' and 'r' can be zero.

 

For eg,

       

     15 = 17 × 0 + 15 (here q is zero)

      10 = 2 × 5 + 0 (here r is zero)

EUCLID'S DIVISION ALGORITHM

Euclid's division algorithm is a technique to find HCF (highest common factor) of 2 positive integers. 

Euclid's division algorithm can be stated as follows :

HCF of 2 positive integers c and d (c > d) can be obtained by the following steps :

Step-1: Apply Euclid's division lemma to c and d. So, we find 'q' and 'r' such that

  c = qd + r    0 ≤ r < d

Step-2: if r = 0, then d is the HCF of 'c' and 'd'. If 'r' is not equal to '0', then apply division lemma to 'd' and 'r'.

Step-3: continue the process til the remainder is zero. The divisor at this stage will be required HCF.

THE FUNDAMENTAL THEORY OF ARITHMETIC

The theorem states that every composite number can be expressed as a product of prime numbers, and their factorization is unique, apart from the order in which the prime factor occur.

i.e., given any composite number can be expressed as a product of prime numbers, and their factorization is unique, apart from the order in which the prime factors occur.

i.e., given any composite number, there is one and only one way to write it as a product of primes. 

for eg: 315 = 3 × 3 × 7 × 5

Decimal representation of Rational numbers

Any rational number can be represented either in terminating decimal or non - terminating recurring decimals. In case of terminating decimal representation, the representation becomes zero.

It can be seen that any rational number, whose decimal expression terminates can be expressed as a rational number whose denominator is a multiple of 10.

for eg: 0.78, 0.1986

The only factorization of 10 is 5 × 2 i.e, the denominator is always in the form of 2m × 5n 

where 'a' and 'b' are non negative integers.

THE UNIT DIGIT OF ANY POWER

The unit digit of any power of 0 , 1 , 5 and 6 is  0 , 1 , 5 and 6 respectively.