Exploring Real Numbers and Number Systems

 
TOPIC
TOPIC
 REAL NUMBERS
 REAL NUMBERS
 
 
Sub Topics
Sub Topics
 
Revisiting the Number System
Definition of Real Number
Euclid’s Division Lemma
Euclid’s Division Algorithm to find HCF of 2 given positive
Integers.
Activity to find HCF
Prime Factorization of Composite Numbers
Fundamental Theorem of Arithmetic.
Application of Fundamental Theorem of Arithmetic (HCF and LCM
of two or more numbers and some other problems)
 Revisiting Irrational Numbers
Decimal Expansion of Rational Numbers
Art Integration
 
Learning Objectives
Learning Objectives
 
To express the division of numbers as dividend = (divisor x quotient ) +
remainder , and generalise this relation for any positive integers ‘a’ and ‘b’.
To understand Euclid’s Division Lemma
To understand the difference between Euclid’s Division Lemma and
Euclid’s Division Algorithm
To be able to find out HCF of two given numbers using Euclid’s Division
Algorithm
To be able to find HCF and LCM using Prime Factorization
To be able to use the formula HCF x LCM = Product of two Numbers
To be able to understand Fundamental Theorem of Arithmetic
To be able to prove the irrationality of a given Number
To be able to define Rational Numbers on the basis of their decimal
expansions
To be able to decide whether the given Rational Number is  terminating or
non-terminating decimal by looking at its denominator
 
REVISITING THE NUMBER SYSTEM
REVISITING THE NUMBER SYSTEM
 
 
DEFINITION OF REAL NUMBERS
DEFINITION OF REAL NUMBERS
 
    Real numbers can be defined as the union of
both the rational and irrational numbers. They
can be both positive and negative and are
denoted by the symbol “R”. All the natural
numbers, whole numbers, integers, rational
and irrational numbers come under this
category.
 
Examples : 0, 5 ,-10 ,¾, -¼ , 0.123, 2. 34 ,
5.010110111....... , √ 5
 
QUIZ TIME
QUIZ TIME
 
Is every integer a rational number ?
 
Is every whole number  a natural number ?
 
Is every whole number an integer ?
 
Is every rational number a real number ?
 
Do real numbers include irrational numbers ?
 
Lemma and Algorithm
Lemma and Algorithm
 
A 
Lemma
 is a proven statement used for
proving another statement.
 
An 
Algorithm 
is a series of well defined steps
which gives a procedure for solving a type of
problem.
 
Statement of Euclid’s Division Lemma
 
If a and b be two given positive integers, then
there exist unique integers q and r such that
 
a = b x q + r , 0 ≤ r 
˂
 b .
Note
 :
1.
Although Euclid’s Division Lemma is stated for
only positive integers a and b, but it is also true
when a and b are any kind of integers , but b ≠
0.
2.
If b divides a, then r = 0 and q is the quotient .
3.
If b > a , then q = 0 and r = a .
 
 
 
 
 
 
HCF of two positive integers using Euclid’s Division
HCF of two positive integers using Euclid’s Division
Algorithm
Algorithm
 
To Find HCF of two given
numbers using Euclid’s Division
Algorithm
 
Example
 : 
Find HCF of
66 and 420 by using
Euclid’s Division
Algorithm
Sol
 : As 420 > 66
 
 so, a = 420 and b = 66
 
420 = 66 x 6 + 24
 
  66 = 24 x 2 +18
 
  24 = 18 x 1 + 6
 
  18 = 
6 
x 3 + 0
Hence ,  HCF = 6
 
Activity to find HCF
Activity to find HCF
 
METHOD
 – Paper folding
MATERIALS REQUIRED 
– Different sheet of colored papers, pair of
scissors, glue, marker and ruler.
PROCEDURE
1. Take any two positive integer a and b (a>b). Example :- a=15 , b= 6.
2. Cut a rectangular sheet of length 15cm and breadth 6cm.
 
 
 
3. The maximum length of the square that can be fitted in the given
rectangular sheet is 6cm. Cut a square sheet of each side 6cm from a
different color paper.
 
Activity continued...
4. Paste this square on rectangle and we can find two such squares can be fit.
 
 
 
 
5. After pasting two square , a rectangular shape is left of dimension
3cm x 6cm.
6. We can write a mathematical expression for the shape obtained as
15 = 6 ×2+3.
7. Now consider the rectangle of dimension 3cm by 6cm. Repeat the same
procedure in this rectangle .
8. We find two squares of each side 3cm is being fit
      into the rectangle.
 
9. Keep on filling the rectangle with squares till the initial rectangle is
completely covered.
 
 
 
 
 
 
 
 
 
 
 
Activity continued...
 
10. The length of last square is the HCF of given positive integers.
 
OBSERVATION
 – So we observe that :
 1. In mathematical form,
                15 = 6 x 2 +3
                  6 = 3 x 2 +0
 2. Here the length of the last square is 3cm . So HCF(15, 6) = 3
 
CONCLUSION
 – By paper cutting and pasting , we can find the HCF of two
given positive integers by applying   
Euclid’s Division Algorithm
.
 
httpshttps://youtu.be/E26ek8HwbJU:
 
WORKSHEET -1
( Euclid’s Division Lemma)
 
1.
Find the HCF of the following numbers
a) 280 , 12
b) 288 , 120
c) 867 , 254
d) 135 , 225
e) 441 , 567 , 693
2.  Use Euclid’s Division Lemma to show
that one and only one out of n , n+1 and
n+2 is divisible by 3, where n is any
positive integer.
 
Prime Factorization of
Prime Factorization of
Composite Numbers
Composite Numbers
 
The natural numbers like 2, 3, 5, 7, 11 …. are 
prime
 
numbers
 as they have exactly two
factors.
The natural numbers like 4, 6, 8, 9, 10…are 
composite
 
numbers
 as they have more than
two factors.
These composite numbers can be written as product of primes. We can find the prime
factorization of a composite number in two different ways as shown below.
 
 
 
 
 
 
 
 
 
Division method                                                        Tree method
   Hence , Prime Factorization of  48 = 2x2x2x2x3 =  
2
4
x3
 
 
 
 
 
 
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WORKSHEET – 2
(PRIME FACTORIZATION 
)
 
1.
 
 Find the prime factorization of the
following numbers by division
method.
 
a) 156
 
b) 5005
 
c) 7429
2.
 
 
Find the prime factorization of the
following numbers by factor tree
method.
 
a)  275
 
b) 120
 
c) 4284
 
 
3.
From the given fig. Find the
value of x ,y and z.
 
 
Fundamental Theorem of
Fundamental Theorem of
Arithmetic
Arithmetic
 
Statement 
: Every composite number can be
expressed as a product of primes and this
factorization is unique, apart from the order in
which the prime factors occur.
Example
 : 56 = 2 x 2 x 2 x 7
   
    = 7 x 2 x 2 x 2
   
    = 2 x 7 x 2 x 2
Hence the prime factorization of a number can be
written in any order, but writing in ascending
order is advisable.
 
 
 
Application of Fundamental Theorem of Arithmetic
Application of Fundamental Theorem of Arithmetic
 
1.  Finding HCF of two or more numbers.
 Find the prime factorization of the given numbers.
 Express the prime factorization in the exponential form.
 Find the product of common factors with lowest power.
2.   Finding LCM of two or more numbers.
 Find the prime factorization of the given numbers.
 Express the prime factorization in the exponential form.
 Find the product of all the factors involved with highest power.
Example : 
Find the  HCF and LCM of 12, 15 and 21 by using  Fundamental
Theorem of  Arithmetic .
Sol
.  12 = 2² x 3
  
 15 = 3 x 5
  
 21 = 3 x 7
Hence, HCF = 3
  
      LCM = 2² x 3 x 5 x 7 = 420
 
Application of Fundamental Theorem of
Application of Fundamental Theorem of
Arithmetic
Arithmetic
 
3.
The product of LCM and HCF of two numbers = Product of the two numbers.
4.
Show that 7 x 11 x 13 + 13 is  a composite number.
 
 Sol
.    Let the given number be denoted by n.
  
So, n = 7 x 11 x 13 + 13
  
          = 13 ( 7 x 11 + 1 )
  
          = 13 ( 77 +1 )
  
          = 13 x 78
 This means that n has a factor 13 other than 1 and n . Hence n is a composite number.
5.    Check whether 6ⁿ can end with the digit 0 for any natural number n .
Sol
.   6 = 2 x 3
  
 6ⁿ = ( 2 x 3 )ⁿ
  
      = 2ⁿ x 3ⁿ
If the number 6ⁿ, for any n, were to end with the digit 0 then , it would
have the factors as 2 and 5. but here 5 is not a factor of 6ⁿ.
Hence,  6ⁿ can never end with 0 for any n.
 
 
WORKSHEET -3
(
Fundamental Theorem of Arithmetic
)
 
1.
Find the HCF and LCM of the following numbers
 
a) 24, 60, 150
 
b) 6, 72, 120
2.
Show that 2 x 3 x 7 x 11 x 17 +11 is a composite
number.
3.
Check whether 12ⁿ can end with the digit 0 for
any natural number n .
4.
Find the LCM and HCF of 26 and 91 , and verify
that LCM x HCF = Product of the two numbers.
 
Proof of irrationality of some
Proof of irrationality of some
numbers
numbers
 
 
Question
 : Prove that √2 is an irrational number.
Solution :
 
Let √2 be a rational number
 
Therefore, √2= p/q       [ p and q are in their least terms i.e., HCF of (p , q )=1 and q ≠ 0]
 
 On squaring both sides,  we get
        p²= 2q²          ...(1)
 
Clearly, 2 is a factor of 2q²
 
 2 is a factor of p²      [since, 2q²=p²]
 
 2 is a factor of p
 
Let p =2 m for all m ( where  m is a positive integer)
 
Squaring both sides, we get
 
  p²= 4 m²       ...(2)
 
 From (1) and (2), we get
        2q² = 4m²
 
      q²= 2m²
 
 Clearly, 2 is a factor of 2m²
 
 
       2 is a factor of q²   [since, q² = 2m²]
 
       2 is a factor of q
 
 Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1
 
Therefore, Our supposition is wrong
 
Hence √2 is not a rational number i.e., irrational number
 
SOME PROBLEMS BASED ON IRRATIONAL
SOME PROBLEMS BASED ON IRRATIONAL
NUMBERS
NUMBERS
EXAMPLE :
EXAMPLE :
 Prove that 3 + 2
 Prove that 3 + 2
√ 5
√ 5
 is an irrational number , where 
 is an irrational number , where 
√ 5 is an irrational
√ 5 is an irrational
number.
number.
 
IMPORTANT POINTS
IMPORTANT POINTS
 
Every rational number can be expressed as p/q , where p and q
are integers having no common factor (other than 1 ) and q ≠
0.
If p and q are integers q  ≠ 0 then p/q is a rational number.
If p be a prime , then √p is an irrational number.
If a prime p divides “a² ” , then p divides “a” where a is an
integer  ( positive  integer) .
If HCF (p , q) = 1 , then p and q are called co-prime  or
relatively prime integers.
Positive integers p and q are co-prime , if 1 is the only
common factor of p and q.
 
WORKSHEET - 4
 
1.
 Answer the following in one word, one sentence or
as per the exact requirement :
 
(a) Let p and q be two distinct prime numbers. Write
HCF (p , q).
 
 
(b) Fill in the blank space in the statement given
below :
 
i) If p be a prime, then √p is _______number.
 
ii) If p/q is a rational number then p and q are _____ .
 
(c) √27 is an irrational number. Justify the statement.
2.
Prove that the following numbers are irrational :
  
(a) 2√3 – 4
 
(b)  1/√2
  
(c) √3 + √5
 
DECIMAL EXPANSION OF RATIONAL
DECIMAL EXPANSION OF RATIONAL
NUMBERS
NUMBERS
 
Let us express the following decimal numbers in  form p/q where p and q are co-
prime numbers and q ≠ 0
 
i) 
    
 
 
ii)
 In the above examples the terminating decimal expansion reduces to a rational
number of the form p/q where p and q are co-prime numbers and q is of the form
 
( n and m are non negative integers).
Let us express the following rational numbers in form of decimals.
 
i) 
     
ii)
 
So these examples  show us how we can convert a rational number of the form p/q
where p and q are co-prime numbers and q is of the form 2
n
×5
m 
to a terminating
decimal.
note : If in a rational number of p/q form q is not in the form of 2
n
×5
m
 ,where n and
m are non negative integers then its decimal expansion is non-terminating
repeating.
 
 
 
 
 
 
 
 
 
 
 
 
WORKSHEET - 5
 
1.
 Without actually performing the long division, state
whether the following rational numbers will have a
terminating or non-terminating repeating decimal
expansion:
 
(a) 17/8 
 
(b) 64/455
 
(c)77/210
 
(d)35/50
 2.  The following real numbers have decimal expansions as
given below. In each case decide whether they are rational
or not. If they are rational and of the form p/q write prime
factors of q so that p and q are co-primes.
 
(a) 43.123
 
(b) 0.120120012000......
  
(c)43.123
3.   Express 0.234234234......in the form p/q where p and q are
co-prime integers also find the prime factorisation of q .
 
SUMMARY
SUMMARY
 
A lemma is a proven statement used for proving another
statement.
An algorithm is a series of well-defined steps which gives a
procedure for solving a type of problems.
EUCLID’S DIVISION LEMMA – If a and b be two given positive
integers and a>b  then there exist unique integers q and r such that
a = bq + r where              0 ≤ r < b.
We can find HCF of two or more numbers by using Euclid’s
division algorithm (lemma).
FUNDAMENTAL THEOREM OF ARITHMETIC- Every composite
number can be expressed as  product of primes in a unique way
apart from the order in which the prime factors occur.
We can find both HCF and LCM of two or more numbers by using
fundamental theorem of arithmetic.
 
Art Integration
 
Art Integration
 
Steps to find HCF and LCM of given numbers by using
Fundamental Theorem of Arithmetic-
Find the prime factorization of the given numbers and express in
exponential form.
HCF = product of only common factors with least power.
LCM = product of all the factors involved with highest power.
If a and b are two positive numbers, then
HCF(a, b) x LCM(a, b) = a x b
If 
p
1
 and p
2 
 are two distinct prime numbers, then
HCF(p
1
 , p
2
 ) = 1
The HCF of two co-prime numbers = 1 and
     LCM = product of the given  co-prime numbers
If p is a prime number and p divides 
a
2 
 then p divides a ,
where ‘a’ is a positive integer.
 
 
If p be a prime, then √p is an irrational number.
Every rational number can be expressed as p/q, where p and q
are co-prime integers and q≠0.
If a = p/q, where p and q are co-prime and q = 
2
n
x 5
m 
 (n and
m are whole numbers), then the rational number ‘a’ has
terminating decimal expansion.
If a = p/q, where p and q are co-prime and q can not be written
as 
2
n
x 5
m 
 (n and m are whole numbers), then the decimal
expansion of ‘a’ is non-terminating recurring.
Decimal representation of every rational number is either
terminating or non-terminating recurring.
If the decimal representation of a real number is
non-terminating  non-recurring , then it is an irrational
number.
 
 
 
 
CONCEPT MAPPING
CONCEPT MAPPING
 
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Dive into the world of real numbers with topics ranging from the definition of real numbers to Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic. Discover the properties of rational and irrational numbers, learn to find the highest common factor (HCF) and least common multiple (LCM), and explore the decimal expansion of rational numbers. Test your knowledge with quizzes on integers, rational numbers, and more.


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  1. TOPIC REAL NUMBERS

  2. Sub Topics Revisiting the Number System Definition of Real Number Euclid s Division Lemma Euclid s Division Algorithm to find HCF of 2 given positive Integers. Activity to find HCF Prime Factorization of Composite Numbers Fundamental Theorem of Arithmetic. Application of Fundamental Theorem of Arithmetic (HCF and LCM of two or more numbers and some other problems) Revisiting Irrational Numbers Decimal Expansion of Rational Numbers Art Integration

  3. Learning Objectives To express the division of numbers as dividend = (divisor x quotient ) + remainder , and generalise this relation for any positive integers a and b . To understand Euclid s Division Lemma To understand the difference between Euclid s Division Lemma and Euclid s Division Algorithm To be able to find out HCF of two given numbers using Euclid s Division Algorithm To be able to find HCF and LCM using Prime Factorization To be able to use the formula HCF x LCM = Product of two Numbers To be able to understand Fundamental Theorem of Arithmetic To be able to prove the irrationality of a given Number To be able to define Rational Numbers on the basis of their decimal expansions To be able to decide whether the given Rational Number is terminating or non-terminating decimal by looking at its denominator

  4. REVISITING THE NUMBER SYSTEM

  5. DEFINITION OF REAL NUMBERS Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive and negative and are denoted by the symbol R . All the natural numbers, whole numbers, integers, rational and irrational numbers come under this category. Examples : 0, 5 ,-10 , , - , 0.123, 2. 34 , 5.010110111....... , 5

  6. QUIZ TIME Is every integer a rational number ? Is every whole number a natural number ? Is every whole number an integer ? Is every rational number a real number ? Do real numbers include irrational numbers ?

  7. Lemma and Algorithm A Lemma is a proven statement used for proving another statement. An Algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

  8. Statement of Euclids Division Lemma If a and b be two given positive integers, then there exist unique integers q and r such that a = b x q + r , 0 r b . Note : 1. Although Euclid s Division Lemma is stated for only positive integers a and b, but it is also true when a and b are any kind of integers , but b 0. 2. If b divides a, then r = 0 and q is the quotient . 3. If b > a , then q = 0 and r = a .

  9. HCF of two positive integers using Euclids Division Algorithm

  10. To Find HCF of two given numbers using Euclid s Division Algorithm Example : Find HCF of 66 and 420 by using Euclid s Division Algorithm Sol : As 420 > 66 so, a = 420 and b = 66 420 = 66 x 6 + 24 66 = 24 x 2 +18 24 = 18 x 1 + 6 18 = 6 x 3 + 0 Hence , HCF = 6

  11. Activity to find HCF METHOD Paper folding MATERIALS REQUIRED Different sheet of colored papers, pair of scissors, glue, marker and ruler. PROCEDURE 1. Take any two positive integer a and b (a>b). Example :- a=15 , b= 6. 2. Cut a rectangular sheet of length 15cm and breadth 6cm. 3. The maximum length of the square that can be fitted in the given rectangular sheet is 6cm. Cut a square sheet of each side 6cm from a different color paper.

  12. Activity continued... 4. Paste this square on rectangle and we can find two such squares can be fit. 5. After pasting two square , a rectangular shape is left of dimension 3cm x 6cm. 6. We can write a mathematical expression for the shape obtained as 15 = 6 2+3. 7. Now consider the rectangle of dimension 3cm by 6cm. Repeat the same procedure in this rectangle . 8. We find two squares of each side 3cm is being fit into the rectangle. 9. Keep on filling the rectangle with squares till the initial rectangle is completely covered.

  13. Activity continued... 10. The length of last square is the HCF of given positive integers. OBSERVATION So we observe that : 1. In mathematical form, 15 = 6 x 2 +3 6 = 3 x 2 +0 2. Here the length of the last square is 3cm . So HCF(15, 6) = 3 CONCLUSION By paper cutting and pasting , we can find the HCF of two given positive integers by applying Euclid s Division Algorithm.

  14. httpshttps://youtu.be/E26ek8HwbJU:

  15. WORKSHEET -1 ( Euclid s Division Lemma) 1. Find the HCF of the following numbers a) 280 , 12 b) 288 , 120 c) 867 , 254 d) 135 , 225 e) 441 , 567 , 693 2. Use Euclid s Division Lemma to show that one and only one out of n , n+1 and n+2 is divisible by 3, where n is any positive integer.

  16. Prime Factorization of Composite Numbers The natural numbers like 2, 3, 5, 7, 11 . are prime numbers as they have exactly two factors. The natural numbers like 4, 6, 8, 9, 10 are composite numbers as they have more than two factors. These composite numbers can be written as product of primes. We can find the prime factorization of a composite number in two different ways as shown below. Division method Tree method Hence , Prime Factorization of 48 = 2x2x2x2x3 = 24x3

  17. WORKSHEET 2 (PRIME FACTORIZATION ) From the given fig. Find the value of x ,y and z. 1. Find the prime factorization of the following numbers by division method. a) 156 b) 5005 c) 7429 Find the prime factorization of the following numbers by factor tree method. a) 275 b) 120 c) 4284 3. 2.

  18. Fundamental Theorem of Arithmetic Statement : Every composite number can be expressed as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. Example : 56 = 2 x 2 x 2 x 7 = 7 x 2 x 2 x 2 = 2 x 7 x 2 x 2 Hence the prime factorization of a number can be written in any order, but writing in ascending order is advisable.

  19. Application of Fundamental Theorem of Arithmetic 1. Finding HCF of two or more numbers. Find the prime factorization of the given numbers. Express the prime factorization in the exponential form. Find the product of common factors with lowest power. 2. Finding LCM of two or more numbers. Find the prime factorization of the given numbers. Express the prime factorization in the exponential form. Find the product of all the factors involved with highest power. Example : Find the HCF and LCM of 12, 15 and 21 by using Fundamental Theorem of Arithmetic . Sol. 12 = 2 x 3 15 = 3 x 5 21 = 3 x 7 Hence, HCF = 3 LCM = 2 x 3 x 5 x 7 = 420

  20. Application of Fundamental Theorem of Arithmetic 3. The product of LCM and HCF of two numbers = Product of the two numbers. 4. Show that 7 x 11 x 13 + 13 is a composite number. Sol. Let the given number be denoted by n. So, n = 7 x 11 x 13 + 13 = 13 ( 7 x 11 + 1 ) = 13 ( 77 +1 ) = 13 x 78 This means that n has a factor 13 other than 1 and n . Hence n is a composite number. 5. Check whether 6 can end with the digit 0 for any natural number n . Sol. 6 = 2 x 3 6 = ( 2 x 3 ) = 2 x 3 If the number 6 , for any n, were to end with the digit 0 then , it would have the factors as 2 and 5. but here 5 is not a factor of 6 . Hence, 6 can never end with 0 for any n.

  21. WORKSHEET -3 (Fundamental Theorem of Arithmetic) 1. Find the HCF and LCM of the following numbers a) 24, 60, 150 b) 6, 72, 120 Show that 2 x 3 x 7 x 11 x 17 +11 is a composite number. Check whether 12 can end with the digit 0 for any natural number n . Find the LCM and HCF of 26 and 91 , and verify that LCM x HCF = Product of the two numbers. 2. 3. 4.

  22. Proof of irrationality of some numbers Question : Prove that 2 is an irrational number. Solution : Let 2 be a rational number Therefore, 2= p/q On squaring both sides, we get p = 2q ...(1) Clearly, 2 is a factor of 2q 2 is a factor of p 2 is a factor of p [ p and q are in their least terms i.e., HCF of (p , q )=1 and q 0] [since, 2q =p ] Let p =2 m for all m ( where m is a positive integer) Squaring both sides, we get p = 4 m ...(2) From (1) and (2), we get 2q = 4m q = 2m Clearly, 2 is a factor of 2m 2 is a factor of q [since, q = 2m ] 2 is a factor of q Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1 Therefore, Our supposition is wrong Hence 2 is not a rational number i.e., irrational number

  23. SOME PROBLEMS BASED ON IRRATIONAL NUMBERS EXAMPLE : Prove that 3 + 2 5 is an irrational number , where 5 is an irrational number.

  24. IMPORTANT POINTS Every rational number can be expressed as p/q , where p and q are integers having no common factor (other than 1 ) and q 0. If p and q are integers q 0 then p/q is a rational number. If p be a prime , then p is an irrational number. If a prime p divides a , then p divides a where a is an integer ( positive integer) . If HCF (p , q) = 1 , then p and q are called co-prime or relatively prime integers. Positive integers p and q are co-prime , if 1 is the only common factor of p and q.

  25. WORKSHEET - 4 Answer the following in one word, one sentence or as per the exact requirement : (a) Let p and q be two distinct prime numbers. Write HCF (p , q). (b) Fill in the blank space in the statement given below : i) If p be a prime, then p is _______number. ii) If p/q is a rational number then p and q are _____ . (c) 27 is an irrational number. Justify the statement. Prove that the following numbers are irrational : (a) 2 3 4 (b) 1/ 2 1. 2. (c) 3 + 5

  26. DECIMAL EXPANSION OF RATIONAL NUMBERS Let us express the following decimal numbers in form p/q where p and q are co- prime numbers and q 0 i) ii) In the above examples the terminating decimal expansion reduces to a rational number of the form p/q where p and q are co-prime numbers and q is of the form ( n and m are non negative integers). Let us express the following rational numbers in form of decimals. i) ii) So these examples show us how we can convert a rational number of the form p/q where p and q are co-prime numbers and q is of the form 2n 5m to a terminating decimal. note : If in a rational number of p/q form q is not in the form of 2n 5m,where n and m are non negative integers then its decimal expansion is non-terminating repeating.

  27. WORKSHEET - 5 Without actually performing the long division, state whether the following rational numbers will have a terminating or non-terminating repeating decimal expansion: (a) 17/8 (b) 64/455 2. The following real numbers have decimal expansions as given below. In each case decide whether they are rational or not. If they are rational and of the form p/q write prime factors of q so that p and q are co-primes. (a) 43.123 (b) 0.120120012000...... 3. Express 0.234234234......in the form p/q where p and q are co-prime integers also find the prime factorisation of q . 1. (c)77/210 (d)35/50 (c)43.123

  28. SUMMARY A lemma is a proven statement used for proving another statement. An algorithm is a series of well-defined steps which gives a procedure for solving a type of problems. EUCLID S DIVISION LEMMA If a and b be two given positive integers and a>b then there exist unique integers q and r such that a = bq + r where 0 r < b. We can find HCF of two or more numbers by using Euclid s division algorithm (lemma). FUNDAMENTAL THEOREM OF ARITHMETIC- Every composite number can be expressed as product of primes in a unique way apart from the order in which the prime factors occur. We can find both HCF and LCM of two or more numbers by using fundamental theorem of arithmetic.

  29. Art Integration

  30. Art Integration

  31. Steps to find HCF and LCM of given numbers by using Fundamental Theorem of Arithmetic- Find the prime factorization of the given numbers and express in exponential form. HCF = product of only common factors with least power. LCM = product of all the factors involved with highest power. If a and b are two positive numbers, then HCF(a, b) x LCM(a, b) = a x b If p1and p2 are two distinct prime numbers, then HCF(p1, p2) = 1 The HCF of two co-prime numbers = 1 and LCM = product of the given co-prime numbers If p is a prime number and p divides a2 then p divides a , where a is a positive integer.

  32. If p be a prime, then p is an irrational number. Every rational number can be expressed as p/q, where p and q are co-prime integers and q 0. If a = p/q, where p and q are co-prime and q = 2nx 5m (n and m are whole numbers), then the rational number a has terminating decimal expansion. If a = p/q, where p and q are co-prime and q can not be written as 2nx 5m (n and m are whole numbers), then the decimal expansion of a is non-terminating recurring. Decimal representation of every rational number is either terminating or non-terminating recurring. If the decimal representation of a real number is non-terminating non-recurring , then it is an irrational number.

  33. CONCEPT MAPPING

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