ppt

1. REAL NUMBERS
1.The fundamental theorem of Arithmetic.
2. Revisiting irrational numbers.
3.Revisiting rational numbers and their decimal representation.

2. THE FUNDAMENTAL THEOREM OF
ARITHMETIC
 Every composite number can be expressed (factorised) as a product of primes, and this
factorization is unique.
 The method of finding the HCF and LCM of two positive numbers by the prime factorization
method.
 Example: Find HCF and LCM of 108 and 150 108 =2² X 3³ and 150 =2 X 3 X 5² HCF(108,150) =2 X3
= Product of SMALLEST power of each common prime factor in the numbers.
 LCM(108,150)= 2² X 3³ X5² = Product of GREATEST power of each common prime factor in the
numbers. Notice that HCF(108,150) X LCM(108,150)= 108 X150

3. REVISITING IRRATIONAL NUMBERS
 In this section, we will prove that numbers of the form √p are irrational where p is a prime.
 Example: Prove √2 is irrational. Proof: Assume √2 is rational. Then √2 =a/b ,where a and b are co-
prime and b≠ 0.
 Squaring both sides, we get 2b² = a² , i.e. 2 divides a²,implies 2 divides a. Let a=2c.Then ,
substituting for a, we get 2b²=4c² i.e.b² = 2c² This means that 2 divides b²,and so divides b.
 Therefore, a and b have at least 2 as a common factor. This contradicts the fact a and b have no
common factors other than 1.
 So, we conclude that √2 is irrational. Similarly, we can prove that √3 ,√5 etc are irrational.

4. Proving Irrational Numbers
Example: Show that 3 − √5 is irrational.
Proof: Assume 3 − √5 is rational. Then 3 − √5=a/b, where and b are co-prime,b≠0.
Rearranging the equation, we get √5=3 − (a/b) = (3b − a)/b Since a and b are integers (3b − a)/b is
rational, and so , √5 is rational.
This contradicts the fact that √5 is irrational. Therefore, our assumption is wrong

5. REVISITING RATIONAL NUMBERS AND THEIIR
DECIMAL REPRESENTATION - I
 Theorem 1: 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 in the
form 2 n5m where n and m are non-negative integers.
 Example : 0.107 = 107/1000= 107/(2³ x 5³ )
 Example: 7.28 = 728/100 = 728 / 10²

6. REVISITING RATIONAL NUMBERS AND
THEIR DECIMAL REPRESENTATION - II
 Theorem 2 : Let x = p/q be a rational number such that prime factorisation of q is of the form 2
n5m where n and m are non-negative integers. Then x has a decimal representation that
terminates.
 Example: 3/8 = 3/2³ = 0.375
 Example: 13/250=13/ 2 x 5³ = 0.052

7. REVISITING RATIONAL NUMBERS
AND THEIR DECIMAL REPRESENTATION - III
 Theorem 3 :
 Let x=p/q ,where p and q are coprimes be a rational number such that prime factorisation of q is
not of the form 2 n5m ,where n and m are non-negative integers.
 Then ,x has a decimal expansion which is nonterminating repeating.