The document presents various theorems and concepts related to real numbers, including Euclid's division lemma, the fundamental theorem of arithmetic, and properties of rational numbers. It explains the unique factorization of composite numbers and demonstrates how to find the highest common factor (HCF) using both division and prime factorization methods. Additionally, it discusses the irrationality of √2 and the conditions under which a rational number has a terminating or non-terminating decimal expansion.

1. WELCOME

2. REAL
NUMBERS
TOPIC:

3. Presenting By-
SHRADDHA MANTRI-27
(Xth B)

6. RATIONAL NUMBERS

9. Theorem 1.1 (Euclid’s Division Lemma) :
Given positive integers a and b,there exist unique integers
q and r satisfying a = bq + r, 0 ≤ r < b.
• Example :
Prove that every positive even integer is of the form
2m and every positive odd integer is of the form 2m + 1,
where m is any integer.
Solution:
Let a be any positive integer and let b = 2.
According to Euclid’s division lemma, there exist two unique
integers m and r such that
a = bm + r = 2m + r, where 0 ≤ r < 2.
Thus, r = 0 or 1 If r = 0, i.e.,
if a = 2m, then the expression is divisible by 2. Thus, it is an
even number .
If r = 1, i.e., if a = 2m + 1, then the expression is not
divisible by 2.
Thus, it is an odd number . Thus, every positive even integer
is of the form 2m and every positive odd integer is of the
form 2m + 1.

10. Euclid’s division algorithm
To obtain the HCF of two positive integers, say c and
d, where c > d, follow the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So,
we find whole numbers, q and
r such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0,
apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero.
The divisor at this stage will
be the required HCF.

11. Example : Find the HCF of 48 and 88.
Solution : Take a = 88, b = 48
Applying Euclid’s Algorithm, we get
88 = 48 × 1 + 40 (Here, 0 ≤ 40 < 48)
48 = 40 × 1 + 8 (Here, 0 ≤ 8 < 40)
40 = 8 × 5 + 0 (Here, r = 0)
Therefore , HCF (48, 88) = 8
For any positive integer a, b, HCF (a, b) × LCM (a, b) = a × b

12. Theorem 1.2 (Fundamental Theorem of Arithmetic) :
Every composite number can be expressed (
factorised ) as a product of primes, and this
factorisation is unique, apart from the order in which
the prime factors occur.
• The Fundamental Theorem of Arithmetic says that
every composite number can be factorised as a
product of primes.
• It also says that any composite number it can be
factorised as a product of prime numbers in a ‘unique’
way, except for the order in which the primes occur.
• That is, given any composite number there is one and
only one way to write it as a product of primes , as
long as we are not particular about the order in
which the primes occur.

13. Example:
Find the HCF of 300, 360 and 240 by the
prime factorisation method.
Solution:
300 = 22 × 3 × 52
360 = 23 × 32 × 5
240 = 24 × 3 × 5
Therefore,HCF (300, 360, 240) = 22 × 3 × 5 = 60

14. Video….

15. Theorem 1.3 : Let p be a prime number. If p divides
a₂, then p divides a, where a is a positive integer.
*Proof :
Let the prime factorisation of a be as follows :
a = p₁p₂ . . . pn, where p₁,p₂, . . ., pn are primes, not
necessarily distinct.
Therefore, a² = (p1p₂ . . . pn)(p₁p₂ . . . pn) = p²₁ p²₂. . . p²n.
Now, we are given that p divides a ₂.
Therefore, from the Fundamental Theorem of Arithmetic, it
follows that p is one of the prime factors of a².
However, using the uniqueness part of the Fundamental
Theorem of Arithmetic, we realise that the only prime factors
of a₂ are p₁, p₂, . . ., pn.
So p is one of p₁, p₂ , . . ., pn. Now, since a = p₁ p₂ . . . pn, p
divides a.

16. Theorem 1.4 : √2 is irrational.
• Proof :
Let us assume, to the contrary, that √2 is rational.
So, we can find integers r and s (≠ 0) such that √ 2 =r/s.
Suppose r and s have a common factor other than 1. Then, we
divide by the common factor to get √ 2 a/b,where a and b are
coprime.
So, b √ 2 = a.
Squaring on both sides and rearranging, we get 2b² = a².
Therefore, 2 divides a².
Now, by Theorem 1.3, it follows that 2 divides a.
So, we can write a = 2c for some integer c.
Substituting for a, we get 2b² = 4c², that is, b² = 2c²
This means that 2 divides b², and so 2 divides b (again using
Theorem 1.3 with p = 2).
Therefore, a and b have at least 2 as a common factor.
But this contradicts the fact that a and b have no common
factors other than 1.
This contradiction has arisen because of our incorrect assumption
that √ 2 is rational.
So, we conclude that √ 2 is irrational.

17. Theorem 1.5 :
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 factorization
of q is of the form 2n5m, where n, m are
non-negative integers.
• Example

18. Theorem 1.6 :
Let x = p/q be a rational number, such
that the prime factorization of q is of the
form 2n5m, where n, m are non-negative
integers. Then x has a decimal expansion
which terminates.

19. • Example

20. Theorem 1.7 :
Let x =p/q be a rational number, such that
the prime factorization of q is not of the
form 2n5m, where n, m are non-negative
integers. Then, x has a decimal expansion
which is non-terminating repeating
(recurring).
Thus, we can conclude that the decimal
expansion of every rational number is
either terminating or non-terminating
repeating.

21. THANK
YOU