The document discusses the history and development of real numbers. It states that simple fractions were used by the Egyptians as early as 1000 BC, while the Greeks around 500 BC realized the need for irrational numbers like the square root of 2. The Middle Ages saw the acceptance of zero, negative numbers, and the treatment of irrational numbers as algebraic objects. Real numbers represent quantities on a continuous line and include rational numbers like integers and fractions as well as irrational numbers like square roots and pi.

1. REAL NUMBERS
made by :-
Utkarsh srivastava
10th - f

2. HISTORYOF REALNUMBERS
Simple fractions have been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of
chords") in, c. 600 BC, include what may be the first "use" of irrational numbers .
Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in
particular the irrationality of the square root of 2.
The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first
by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat
irrational numbers as algebraic objects, which was made possible by the development of algebra. Arabic
mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real
numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept
irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of
square roots, cube roots and fourth roots

3. REALNUMBERS
In mathematics, a real number is a value that represents a quantity along a
continuous line. The real numbers include all the rational numbers, such as
the integer −5 and the fraction 4/3, and all the irrational numbers such
as √2 (1.41421356…, the square root of two, an irrational algebraic number)
and π (3.14159265…, atranscendental number)

4. EUCLID
Euclid , also known as Euclid of
Alexandria, was a Greek mathematician,
often referred to as the "Father of
Geometry". He was active
in Alexandria during the reign
of Ptolemy I (323–283 BC). His Elements is
one of the most influential works in
the history of mathematics, serving as the
main textbook for
teaching mathematics (especially geometry)
from the time of its publication until the late
19th or early 20th century. In the Elements,
Euclid deduced the principles of what is now
called Euclidean geometry from a small set
of axioms. Euclid also wrote works
on perspective, conic sections, spherical
geometry, number theory and rigor.

5. EUCLID”S LEMMA
In number theory, Euclid'slemma (also called Euclid'sfirsttheorem) is a lemma that captures a
fundamental property of prime numbers, namely: If a prime divides the product of two numbers, it
must divide at least one of those numbers. For example since133 × 143 = 19019 is divisible by 19, one or
both of 133 or 143 must be as well. In fact, 19 × 7 = 133.
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. For example, 4 does not divide 6 and 4 does not divide
10, yet 4 does divide their product, 60.

6. REMARKS
2) Although Euclid’s divison algorithm is stated for only positive integers, it
can be extended for all integers except zero ,i.e. 0.
1) Euclid’s divison lemma and algorithm are so closely interlinked that people
often Call former as the divison algorithm also.

7. USEEUCLID’SALGORITHMTOFINDTHE
HIGHESTCOMMONFACTOROFTHETWO
NUMBERS606AND91.?
 606 ÷ 91 = 6 R 60
91 ÷ 60 = 1 R 31
60 ÷ 31 = 1 R 29
31 ÷ 29 = 1 R 2
29 ÷ 2 = 14 R 1
2 ÷ 1 = 2 r 0
so 1 is the Hcf (The last non-zero remainder is the Hcf)

8. FUNDAMENTALTHEOREMOFARITHMETIC
In number theory, the fundamentaltheoremof arithmetic, also called the uniquefactorizationtheoremor
the unique-prime-factorizationtheorem, states that every integer greater than 1either is prime itself or is
the product of prime numbers, and that, although the order of the primes in the second case is
arbitrary, the primes themselves are not. For example,
1200 = 24 × 31 × 52 = 3 × 2× 2× 2× 2 × 5 × 5 = 5 × 2× 3× 2× 5 × 2 × 2 = etc.
The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second,
no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the
product.
The requirement that the factors be prime is necessary: factorizations containing composite
numbers may not be unique (e.g. 12 = 2 × 6 = 3 × 4).

9. PRIMEFACTORISATION
Find the HCF and LCM of 24 and 40
24 = 2 x 3 x 2 x 2 and 40 = 2 x 2 x 2 x 5
HCF: The common factors of 24 and 40 are 2 x 2 x 2 = 8. So the HCF and LCM of 24 and 40 = 8
LCM: We take the prime factors of the smaller number (24), and they are 2, 3, 2, and 2. The only
prime factor from the larger number (40) not in this list is 5.
So the LCM of 24 and 40 is 2 x 3 x 2 x 2 x 5 = 120

10. REVISITINGIRRATIONALNUMBERS
Show that 5 – √3 is irrational.
That is, we can find coprime a and b (b ≠ 0) such that
Therefore,
Rearranging this equation, we get
Since a and b are integers, we get is rational, and so √3 is rational.
But this contradicts the fact that √3 is irrational.
This contradiction has arisen because of our incorrect assumption that 5–√3 is rational.
So, we conclude that 5 − √3 is irrational.
Let us assume, to the contrary, that 5 – √3 is rational.

11. REVISITINGRATIONALNUMBERSAND
THEIRDECIMALEXPANSIONSRationalnumbersare of two types depending on whether their decimal form is terminating or non terminating
A decimal number that has digits that do not go on forever.
Examples:
0.25 (it has two decimal digits)
3.0375 (it has four decimal digits)
In contrast a Recurring Decimal has digits that go on forever
Example: 1/3 = 0.333... (the 3 repeats forever) is a Recurring Decimal, not a Terminating Decimal
Terminating
Non - Terminating
a decimalnumeral that does not end in an infinite sequence of zeros (contrasted with terminating decimal ).