1. Topic
Real Numbers

2. Real Numbers
Real numbers consist of all the rational and irrational numbers.
The real number system has many subsets:
Natural Numbers
Whole Numbers
Integers

3. Natural Numbers
Natural numbers are the set of counting numbers which starts from 1.
They are denoted by N
Example : {1, 2, 3,…}

4. Whole Numbers
Whole numbers are the set of numbers that include 0 plus the set of natural numbers.
Example : {0, 1, 2, 3, 4, 5,…}

5. Integers
An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
It is denoted by Z .
Example : Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

6. Rational Numbers
Rational numbers are any numbers that can be expressed in the form of a/b , where a and b are integers, and b ≠ 0.
They can always be expressed by using terminating decimals or repeating decimals.
Example : 2/3, 6/7,1

7. Non Terminating Decimals
Repeating Decimals
Terminating Decimals
Terminating decimals are decimals that contain a finite number of digits.
Examples:
36.8
0.125
4.5
Repeating decimals are decimals that contain a infinite number of digits.
Examples:
0.333… …
Non Terminating Decimals
While expressing a fraction into a decimal by  the division method, if the division process  continues indefinitely, and zero remainder is  never obtained then such a decimal  is called Non-Terminating Decimal OR
A non-terminating decimal is a decimal never repeats. Example : ,   

8. Euclid's Division Lemma
Euclid's division lemma states that " For any two positive integers a and b, there exist integers q and r such that a=bq+r , 0≤ r< b
Lemma: A lemma is a proven statement used for proving another statement
Example : For a= 15,b=3 it is observed that
15=3(5)+0
where q=5 and r=0

9. Euclid Division Algorithm
Algorithm: An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
Euclid division algorithm can be used to find the HCF of two numbers. It can also be used to find some common properties of numbers.
To obtain the HCF of two positive integers,say c and d, with c>d , we have to follow the steps below:
STEP 1: Apply euclid division lemma, to c and d. So, we find whole numbers,q and r such that c=dq+r
STEP 2: If r=0,d is the HCF of c and d. If r does not equal to 0 , apply the division lemma to d and r.
STEP 3: Continue the process till the remainder is zero. The divisior at this stage will be the required HCF.

10. Question : Use Euclid algorithm to find the HCF of 455 and 42 ?
Solution : Since 455>42, we apply division lemma to 455 and 42 , to get
455= 42 × 10+35
Remainder is not zero therefore we apply lemma to 42 and 35,
42=35 ×1+7
Again remainder is not zero therefore we apply lemma to 35 and 7
35=7 ×5+0
The remainder has become zero , and we cannot proceed any further ,therefore the HCF of 455 and 42 is the divisor at this stage , i,e 7

11. The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes. This representation is called prime factorisation of the number. This factorisation is unique, apart from the order in which the prime factors occur.

12. Highest Common Factor(HCF): Lowest Common Multiple(LCM):
The HCF of two numbers is equal to the product of the terms containing the least powers of common prime factors of the two numbers.
Highest Common Factor(HCF):
The LCM of two numbers is equal to the product of the terms containing the greatest powers of all prime factors of the two numbers.
Lowest Common Multiple(LCM):

13. Relationship between HCF and LCM
For any two positive
integers a and b,
  HCF (a, b) × LCM (a, b)
= a × b
Example : if a=3 and b=6
HCF(3,6) × LCM(3,6) = 3× 6
3 × 6 = 18
18 = 18
Hence verified……..

14. Question : Given that HCF(306,657)=9,
find LCM(306,657)
Solution : We know that the product of the HCF and the LCM of two
numbers is equal to the product of the given numbers .
Therefore HCF(306,657) LCM(306,657) =
9 LCM(306,657) =
LCM(306,657) = /9
LCM =22338 × × × × ×

15. Irrational Numbers
Irrational numbers are any numbers that cannot be expressed as a/b .
They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern.
Examples of irrational numbers: … … Pi

16. repeating or recurring.
Things to Remember
. Let p be a prime number. If p divides a2, then p also divides a, where a is a positive integer.
. When prime factorisation of q is of the form 2m5n.Then x has a decimal expansion which terminates and when q is not of the form 2m5n, then x has a decimal expansion which is non-terminating
repeating or recurring.

17. . Is 17/8 has a terminating decimal expansion?
Sol : We have 17/8
=17/23×52
So , the denominator 8 of 17/8 is of the form 2m×5n therefore it has a terminating decimal expansion.
.Is 29/343 has a terminating decimal expansion?
Sol : We have 29/343
=29/35
Clearly 343 is not of the form 2m×5n
therefore it has a non terminating decimal expansion.

18. Made by Group 1
(Roll no 1-5)
Abhishek Jasrotia
Abhishek Bandral
Akhil
Amandeep
Anamika