This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.

1. MATHS PROJECT
REAL NUMBERS
- Farhan Alam
X – B
12

2. 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 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)

3. HISTORY OF REAL NUMBERS
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. The concept of irrationality was
implicitly accepted by early Indian mathematicians since Manava (c. 750–690
BC), who were aware that the square roots of certain numbers such as 2 and
61 could not be exactly determined. 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

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. Hippasus
Hippasus of Metapontum
was a Pythagorean philosopher. Little is
known about his life or his beliefs, but he is
sometimes credited with the discovery of the
existence of irrational numbers. The discovery
of irrational numbers is said to have been
shocking to the phythagoreans, and Hippasus
is supposed to have drowned at sea,
apparently as a punishment from the gods, for
divulging this. However, the few ancient
sources which describe this story either do not
mention Hippasus by name or alternatively tell
that Hippasus drowned because he revealed
how to construct a dodecahedron inside
a sphere. The discovery of irrationality is not
specifically ascribed to Hippasus by any
ancient writer. Some modern scholars though
have suggested that he discovered the
irrationality of √2, which it is believed was
discovered around the time that he lived.

6. EUCLID”S LEMMA
In number theory, Euclid's lemma (also called Euclid's first theorem)
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.

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

8. FUNDAMENTAL THEOREM OF ARITHMETIC
In number theory, the fundamental theorem of arithmetic, also called
the unique factorization theorem or the unique-prime-factorization
theorem, states that every integer greater than 1 either 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. PRIME FACTORISATION
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. REVISITING IRRATIONAL NUMBERS
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. REVISITING RATIONAL NUMBERS AND THEIR DECIMAL
EXPANSIONS
Rational numbers are 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 decimal numeral that does not end in an infinite sequence of zeros
(contrasted with terminating decimal ).