This document discusses several topics in mathematics: - Euclid's division lemma and algorithm for finding the highest common factor of two integers. It provides an example applying these steps to find the HCF of 4052 and 12576. - The fundamental theorem of arithmetic, which states that every composite number can be uniquely expressed as a product of prime numbers. - A brief revisiting of irrational numbers, which cannot be written as integers divided by integers where the denominator is not equal to zero.

1. MATHS PROJECT
BY = PRIYA DARSHINI

2. REAL NUMBERS
• INTRODUCTION

3. EUCLID’S DIVISION LEMMA
Positive integer ‘A’and ‘B’ there exist unique integer ‘Q’and’R’ satisfying
‘a=bq+R,O<r<b
A lemma is a proven statement used for proving another statement
‘if counted in pairs ,one will remain’
‘if counted in three, two will remain’
‘if counted in fore, three will remain ‘
‘if counted in five ,fore will remain’
‘if counted in six, five will remain

4. EUCLID’S DIVISION ALGORITHM
• To obtain the HCF of two positive integer ,say ‘c’ and ‘d’ , with c > d,
Follow the steps
1. Apply the division lemma to find ‘q’ and ‘r’ where a=bq + r, o< r< b
2. If r=o ;, the HCF is b, if r is not equal to ‘o’appiy Euclid lemma to ‘d’
and ‘r’
3. Continue the process till the remainder is ‘zero’ . The divison at this
stage will be the required HCF

5. EX;-
HCF of 4052 and 12576
=12576=420*3+420
=4052=420*9+272
=420=272*1+148
=272=148*1+124
=148=124*1+24
=124=24*5+4
=24=4*6+0
HCF is 4

6. THE FUNDAMENTAL THEOREM OF
ARITHMIC
• Every composite number can be expressed as a product of
primes, and this factorization is unique ,apart from the order in with
yhe prime factor cccur
• In general , given a composite number ‘X’ , x=p1,p2……p where
p1,p2……, pn i.e p1<p2<….<pn

7. REVISITING IRRATIONAL NUMBER
• Recall ,a number ‘s’ is called irrational it cannot be written in form p/q
Where p and q are integer and’ q ‘is not equal to’ o’