This document discusses different types of real numbers including rational and irrational numbers. It defines rational numbers as numbers that can be expressed as p/q where p and q are integers. Irrational numbers are defined as numbers with a non-terminating and non-repeating decimal representation. Euclid's division lemma and algorithm are explained as ways to find the highest common factor of two numbers. The fundamental theorem of arithmetic states that every composite number can be expressed as a product of primes. Relationships between the highest common factor and lowest common multiple of numbers are also covered. Finally, the document defines terminating, repeating, and non-terminating decimals.

2. REALNUMBERS
 We have studied in earlier classes that rational and irrational number
taken together form a set of real numbers. It is denoted by ‘R’.
 This set of real numbers have the following subsets
• Natural numbers
• Whole numbers
• Negative numbers
• Integers
• Fractional/decimal numbers

3. RATIONALNUMBERS
Rational numbers are numbers that can be expressed in the
form of p/q where q≠0 and p, q are integers.
They can be expressed in both terminating repeating
rational number and non terminating repeating rational
number.
They are denoted by Q.

4. IRRATIONALNUMBER
A number is an irrational number, if it has a non
terminating and non repeating decimal
representation.

5. EUCLID’S DIVISION LEMMA
Lemma- a lemma is proven statement used for proving
another statement.
Euclid’s division lemma states that – For any two positive
integers a and b, there exist unique integer q and r
satisfying a=bq+r , 0 ≤ r < b.
Example : For a=15 and b=3
15=3(5)+0 where q=5 and r=0

6. EUCLID’S DIVISION ALGORITHM
 Algorithm- An algorithm is series of well defined steps which gives a procedure for solving
a type of problem.
 Euclid’s division algorithm can be used to find the H.C.F. of two numbers. It can also be
used to find some common properties of numbers.
 To obtain the H.C.F. of two positive integers, say c and d, with c>d, we have to follow
the steps below:
 Step1: Apply Euclid’s division lemma to c and d. So, we find whole numbers, q and r
such that c=dq+r
 Step2 : If r=0, d is the H.C.F. of c and d. If r does not equal to 0, apply the division
lemma to d and r.
 Step3 : Continue the process till the remainder is zero. The divisor at this stage will be the
required H.C.F .

7. THEFUNDAMENTALTHEOREMOF ARITHMETIC
Every composite number can be expressed
as a product of primes. This representation
is called prime factors occur.

8. 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.
The LCM of two numbers id equal to the product
of the greatest powers of all prime factors of the
two numbers
H.C.F.
(Highest
Common
Factor)
L.C.M.
(Lowest
Common
Multiple)

9. RELATIONSHIPBETWEENH.C.F. ANDL.C.M.
 For any two integers a and b
HCF(a,b) x LCM(a,b) = a x b
Example :
If a=3 and b=6
HCF(3,6) X LCM(3,6)=3 X 6
Hence verified

10. DECIMALEXPANSION
 Terminating Decimals
 Terminating decimals are
decimals that contain a finite
number of digits.
 Examples
 36.8
 0.125
 4.5
Repeating Decimals
 Repeating decimals are
decimals that contains a finite
number of digits.
 Examples:
 0.912121212
 0.666666666
Non Terminating Decimals
While expressing a fraction into a decimal
by the division method, if the remainder is
never obtained the such a decimal is called
Non Terminating non repeating decimal

11. Made By: Aditya
Chowdhary
Class: X