real numbers ppt for class 10th the best i made it i got full marlks

1. DONE BY – RISHAB JAIN
10TH C
ROLL.NO.- 34

2. History of real numbers
• In mathematics, a real number is a
value that represents a quantity along
a continuous line
• The real numbers include all
the rational ,
irrational,integers,whole,natural
numbers

3. About the mathematicians
• Mathematicians related with real numbers are Carl Friedrich
Gauss, Muhammed ibn musa al , Adrien-Marie Legendre, Évariste
Galois, Charles Hermite and many other mathematicians.
• Carl Friedrich Gauss is often referred to as the ‘Prince of
Mathematicians’ and is considered one of the three
greatest mathematicians of all time, along with Archimedes
and Newton. He has made fundamental contributions to
both mathematics and science.

4. Euclid’s lemma and its remarks
• Euclid’s lemma is a lemma that captures a fundamental
property of prime numbers If a prime divides the
product of two numbers, it must divide at least one of
those numbers. It is also called Euclid's first theorem.
• This property is the key in the proof of
the fundamental theorem of arithmetic. It is used to
define prime elements, a generalization of prime
numbers to arbitrary commutative rings. The lemma is
not true for composite numbers

5. Comparison of division algorithm
• A division algorithm is an algorithm which, given two
integers N and D,computes
their quotient and/or remainder, the result of division.
• Most used division method is Euclid’s division lemma in
real numbers. The method used in algebra and
polynomial for division is long polynomial division

6. Important applications of euclid’s lemma
` euclid’s division lemma to compute hcf of two positive integers .
To obtain the HCF of two positive integers, say c and d, with 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.

7. Application 1
Example : 35 and 225
We have 225 > 135,
step1 : So, we apply the division lemma to 225 and 135 to obtain 225 =
135 × 1 + 90 Here remainder 90 ≠ 0, we apply the division
lemma again to 135 and 90 to obtain
135 = 90 × 1 + 45 We consider the new divisor 90 and new remainder
45 ≠ 0.
step 2 : apply the division lemma to obtain
90 = 2 × 45 + 0
step 3 : Since that time the remainder is zero, the process get stops.
The divisor at this stage is 45
Therefore, the HCF of 135 and 225 is 45.

8. Application 2
Example : Show that every positive even integer is of the form 2q,
and that every
positive odd integer is of the form 2q + 1, where q is some integer.
Solution : Let a be any positive integer and b = 2. Then, by Euclid’s
algorithm,
a = 2q + r, for some integer q ≥ 0, and r = 0 or r = 1, because 0 ≤ r < 2. So,
a = 2q or 2q + 1.
If a is of the form 2q, then a is an even integer. Also, a positive integer
can be
either even or odd. Therefore, any positive odd integer is of the form 2q +
1.

9. Finding HCF and LCM and relationship
• HCF of two or more numbers = product of the smallest power of
each common prime factor involved in the numbers .
• LCM of two or more numbers = product of the greatest power of
each prime factor involved in the numbers
• for any two positive integers a and b , HCF(a,b)*LCM(a,b) = a*b
HCF(a,b) = a*b⁄HCF(a,b)
LCM(a,b) = a*b⁄HCF(a,b)
• for any three positive integers a,b,c
HCF(a,b,c) = a*b*c*LCM(a,b,c)⁄LCM(a,b)*LCM(b,c)*lcm(a,c)
LCM(a,b,c) = a*b*c*HCF(a,b,c)⁄HCF(a,b)*HCF(b,c)*HCF(a,c)

10. finding HCFand LCM in word problems
we are given that, 105 goats, 140 donkeys and 175 cows. There is only
one boat which will have to make many /y/ trips in order to do so. The
lazy boatman has his own conditions for transporting them. He insists
that he will take the same number of animals in every trip and they have
to be of the same kind. He will naturally like to take the largest
possible number each time. We need to tell the number of animals that
went in each trip.
Given that
Number of goats = 105
Number of donkeys = 140
Number of cows = 175.

11. Therefore, the largest number of animals in 1 trip = H.C.F. of 105, 140
and 175.
First we consider 105 and 140.
By applying Euclid’s division lemma
Therefore, H.C.F. of 105 and 140 = 35
Now, we consider 35 and 175.
By applying Euclid’s division lemma
Therefore, H.C.F. of 105, 140 and 175 = 35
Hence, the number of animals went in each trip is.

12. Prime numbers
• A prime number (or a prime) is a
natural number greater than 1 that has no
positive divisors other than 1 and itself.
• Eg; the number 17 can only be divided by 17
and 1.
• It is used in cryptography and Prime numbers
are also used for hash tables and pseudorandom
number generator

13. Composite numbers
• A composite number is a positive integer
that has at least one positive divisor other than
one or the number itself.
• Eg : 4 ,it is divisible by 1,2 and 4.
• They are used to explore various
applications in science ,economics ,iit and many
important sections.

14. proofs of irrationality
Assume that √2 is a rational number, meaning that there exists a pair of integers whose ratio is √2.
If the two integers have a common factor, it can be eliminated using the Euclidean algorithm. Then √2 can be
written as an irreducible fraction such that and are co prime integers (having no common factor).