Course Code: MATH

Course Title: HISTORY OF MATHEMATICS

MODULE
(Prelim)
TOPIC OUTLINE

Module 1

a. Introduction
b. Simple interest and Simple discount
c. Ordinary and Exact Interest
d. Actual and Exact time
e. Approximate time and Exact time

Name of Student:
____________________________________________

Course/Year: _____________
 OUTCOMES

At the end of the lesson, the students should be able to:

a. Define and understand the key terms,

OVERVIEWS

There are many excellent reasons to study the history of mathematics. It helps students
develop a deeper understanding of the mathematics they have already studied by seeing how
it was developed over time and in various places. It encourages creative and flexible thinking
by allowing students to see historical evidence that there are different and perfectly valid
ways to view concepts and to carry out computations. Ideally, a History of Mathematics
course should be a part of every mathematics major program.

LESSON PROPER

During their first 300,000 years of existence, human beings explained the phenomena that
surrounded them (rains, death, harvest) by turning to the idea of magic and the influence
of the gods. And so it was until around the sixth century B.C. when a revolution began in
Ancient Greece in search of the basic principles to understand the observable world.

Rational thinking began to weaken the grip of mythological ideas; it was the awakening
of science. Pythagoreanism was one of the pillars of that cultural eruption and also the
first philosophical current in which women participated and stood out.

Pythagoras of Samos (569 BC – 475 B.C. approx.) and his followers sought to decipher
the foundations of reality through numbers, and in doing so they created mathematical
abstraction. Prior mathematics, developed empirically by the Egyptians and
Mesopotamians, was a collection of rules for practical questions, such as dividing up a
piece of land. It is possible that Pythagoras learned from them in his travels, but he took
geometry and arithmetic much further. He was the first to observe that there is a set of
axioms from which all other reasoning can be deduced—through demonstration, which
the Pythagoreans established as a basic tool for constructing the framework of
mathematics.

Mathematics stopped being a means, to become an end in itself. For the Pythagoreans,
the search for knowledge was the way to fully realize this. Around this idea they created
the Pythagorean community, which was also guided by ethical and moral principles,
translated into a series of rules dictated by its leader. “The original Pythagoreans lived
together in close-knit communities and abided by a strict discipline, which extended to
dietary matters, wearing apparel, and the proper times to engage in sexual intercourse,”
says the classical historian Sara Pomeroy in her book Pythagorean Women: Their History
and Writing.
THE FIRST WOMAN MATHEMATICIAN IN HISTORY

In addition to the mysterious figure of Pythagoras, founder of the group, many other
people laid the foundations of this current of thought. Among them were women like
Theano, considered by some authors to be the first female mathematician in history.
“Pythagoras was the first Greek philosopher to include women among his disciples,”
notes Pomeroy. The Neo-Platonic philosopher Iamblichus (3rd century A.D.) lists 17
women among the 235 names of known Pythagoreans. “They were not “muted” like their
respectable Athenian contemporaries in old Greece,” says Pomeroy. “Some of their witty
and prudent remarks were quoted by later authors,” she adds. Of all of them, Theano was
the most mentioned and influential Pythagorean woman.

Despite her indisputable notoriety, there is some confusion regarding her biography.
Theano is usually considered to be the wife of Pythagoras and the daughter of Brontino
(another member of the Pythagorean school); however, other sources identify her as a
disciple of Pythagoras and the wife of Brontino. Some authors affirm that Theano was
one of the survivors of the bloody attack on the sect in which Pythagoras died—
according to different versions and legends—after which she dedicated herself to
spreading the doctrine of the master.

Beyond this work, it is possible that Theano wrote some of her own texts that were not
published. In addition to some moral reflections, she also researched cosmology,
medicine and mathematics. She was particularly interested in the theory of the golden
ratio and regular polyhedrons, also known as Platonic solids. One of them, the
dodecahedron, had a prominent place in the arithmetical mysticism of the Pythagoreans.

There
are only five regular polyhedra: the tetrahedron (composed of 4 equilateral triangles), the cube (6
squares), the octahedron (8 equilateral triangles), the dodecahedron (12 regular pentagons) and
the icosahedron (20 equilateral triangles). Credit: Максим Пе
DEVOTION TO NUMBERS

For the Pythagoreans, the number was the essential material of all things. From their own
experiments with the monochord they found that numbers described the theory of musical
sounds and also explained the movement of the planets, as the Babylonian
mathematicians already knew.
Guided by this devotion to numbers, they studied and classified them in many different
ways: to start, they differentiated between odd and even numbers; then they classified
them into prime numbers (those that can only be divided by one and themselves) and
compound numbers (those that have more divisors); and, surprisingly, they managed to
find perfect numbers and even pairs of friendly numbers. Perfect numbers are those that
are equal to the sum of their divisors (e.g. 6 = 1 + 2 + 3), and one number is a friend of
another if adding the divisors of one obtains the other, and vice versa. For example, 220
and 284:

– The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284.

– The divisors of 284 are 1, 2, 4, 71 and 142, which add up to 220.

Pythago
reans found that numbers described the theory of musical sounds. Credit: Melchoir
In addition, the Pythagoreans attributed mystical characteristics to numbers. The number
one represented the origin, odd numbers were masculine and even ones feminine. They
believed that all nature and objects were characterized by numbers, which were their
primary material and the cause of their modifications and permanent states.

It was precisely that vision, which turned from numbers back towards magic, that was the
beginning of the end for the Pythagoreans. They discovered that reality could not be
measured with whole numbers (or quotients of them), but that more complicated values
were needed: irrational numbers, with their infinite decimal figures that are not repeated
periodically (like the number π). It was one of the most surprising discoveries made by
the Pythagoreans, possibly by applying the Pythagorean theorem to something as
simple as calculating the diagonal of a square of side 1. These incommensurable numbers
destroyed their vision and showed that mathematics, and the world, were much more
complex than they expected.