Republic of the Philippines
Tarlac State University
COLLEGE OF ENGINEERING AND TECHNOLOGY
Tarlac City
G113
MATHEMATICS IN THE MODERN WORLD
I
The Nature of Mathematics
Prepared by
CID LAPUZ
AUGUST 2018
I
The Nature of Mathematics
MATHEMATICS – science of structure, order, and relation that has evolved from elemental practices of counting,
measuring, and describing the shapes of objects.
BRIEF HISTORY OF MATHEMATICS
3000 BC - Mesopotamian civilization together with Ancient Egypt and Ebla began
using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field
of astronomy and to formulate calendars and record time.
The most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322 (Babylonian c.
1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 2000–1800 BC) and the Moscow Mathematical
Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples and so, by inference,
the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic
arithmetic and geometry.
6th century BC - the term "mathematics" was coined from the ancient Greek (mathema), meaning "subject of instruction".
- the ancient Romans used applied mathematics in surveying, structural engineering, mechanical
engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts.
- Chinese mathematics made early contributions, including a place value system and the first use of negative
numbers.
- The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world
today evolved over the course of the first millennium AD in India and were transmitted to the Western
world via Islamic mathematics.
- Contemporaneous with but independent of these traditions were the mathematics developed by the Maya
civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya
numerals.
12th century - Many Greek and Arabic texts on mathematics were translated into Latin, leading to further development of
mathematics in Medieval Europe.
19th century - the International Congress of Mathematicians was founded and continues to spearhead advances in the
field.
MATHEMATICIANS
Albert Einstein (1879-1955)
Nationality: German, American
Famous For: E=m*c2
Albert Einstein excelled in mathematics early in his childhood. He liked to study math on his own.
He was once quoted as saying, “I never failed in mathematics…before I was fifteen I had mastered
differential integral calculus.”
THEORY OF SPECIAL RELATIVITY - explains how space and time are linked for objects that are moving at a
consistent speed in a straight line. One of its most famous aspects concerns objects moving at the speed of light.
Isaac Newton (1642-1727)
Nationality: English
Famous For: Mathematical Principles of Natural Philosophy
The book of Sir Isaac Newton, Mathematical Principles of Natural Philosophy, became the catalyst to
understanding mechanics. He is also the person credited for the development of the binomial theorem.
Leonardo Pisano Bigollo (1170-1250)
Nationality: Italian
Famous For: Fibonacci sequence
Heralded as “the most talented western mathematician of the middle ages,” Leonardo Pisano Bigollo is
better known as Fibonacci. He introduced the Arabic-Hindu number system to the western world. In his
book, Liber Abaci(Book of Calculation), he included a sequence of numbers that are known today as
“Fibonacci numbers.”
FIBONACCI SEQUENCE - is a set of numbers that starts with zero and one and proceeds based on the rule that each
number is equal to the sum of the preceding two numbers.
Thales (c. 624 – c.547/546 BC)
Nationality: Greek
Famous For: Father of science & Thales’ theorem
Thales used principles of mathematics, specifically geometry, to solve everyday problems. He is
considered as the “first true mathematician”. His deductive reasoning principles are applied in
geometry that is a product of “Thales’ Theorem.”
THALES' THEOREM - The diameter of a circle always subtends a right angle to any point on the circle
Pythagoras (c. 570 – c. 495 BC)
Nationality: Greek
Famous For: Pythagorean theorem
Pythagoras is best known in mathematics for the Pythagorean Theorem.
PYTHAGOREAN THEOREM – states that the area of the square built upon the hypotenuse of a right triangle is equal
to the sum of the areas of the squares upon the remaining sides.
René Descartes (1596-1650)
Nationality: French
Famous For: Cartesian coordinate system
The “Cartesian coordinate system” in mathematics is named after Rene Descartes. As a mathematician,
he is seen as the father of analytical geometry in addition to explaining “infinitesimal calculus and
analysis.”
CARTESIAN COORDINATE SYSTEM - a coordinate system that specifies each point uniquely in a plane by a pair
of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines,
measured in the same unit of length.
Archimedes (c. 287 – c. 212 BC)
Nationality: Greek
Famous For: Greatest mathematician of antiquity
Archimedes provided principles and methods used in mathematics today. He provided the exact numerical
value of pi, developed a system for large numbers to be expressed, and the method of exhaustion.
John Forbes Nash, Jr. (1928)
Nationality: American
Famous For: Nash embedding theorem
The work of American mathematician John Nash includes studies in differential geometry, game
theory, and partial differential equations. He is best known for the Nash embedding theorem. His work
in algebraic geometry is also seen as milestone in mathematics.
NASH EMBEDDING THEOREM - states that every Riemannian manifold can be isometrically embedded into
some Euclidean space. Isometric means preserving the length of every path. For instance, bending without
stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because
curves drawn on the page retain the same arclength however the page is bent.
Blaise Pascal (1623-1662)
Nationality: French
Famous For: Pascal’s Triangle
Pascal is recognized for two mathematical areas of study, projective geometry and probability theory.
He describes in his paper, Treatise on the Arithmetical Triangle, an easy to understand table of
“binomial coefficients” known as Pascal’s Triangle
PASCAL’S TRIANGLE - To build the triangle, start with "1" at the top then continue placing numbers below it in a
triangular pattern. Each number is the numbers directly above it added together.
Euclid (c. 365 – c. 275 BC)
Nationality: Greek
Famous For: Father of geometry
The earliest known “math books” is one written by Greek mathematician Euclid, Elements is its title. It
serves as a textbook to teach geometry and mathematics. His mathematical system is known as
“Euclidean geometry.”
Ptolemy (c. 90 – c. 168 AD)
Nationality: Greco-Roman
Famous For: Almagest
Ptolemy was a mathematician of the highest order. In his book Almagest, or The Mathematical
Compilation, Ptolemy provides mathematical theories related to the solar system.
Ada Lovelace (1815-1852)
Nationality: English
Famous For: Work on the Analytical Engine
English mathematician Ada Lovelace is recognized as the worlds first computer programmer. Her
mathematical skills were evident at an early age. As part of her work, she produced a mathematical
algorithm that would be later used in computers.
Alan Turing (1912-1954)
Nationality: British
Famous For: Father of computer science
Turing’s fame as a mathematician can be attributed to his formulating of algorithms and computations
for a computer, the Turing Machine. His mathematical background helped device techniques in code
breaking, specifically in world war 2. In 1948 Turing became interested in mathematical biology.
Eratosthenes (276 – 194 BC)
Nationality: Greek
Famous For: Sieve of Eratosthenes
Eratosthenes provided the concept of a simple algorithm as a way to locate prime numbers. The Sieve
of Eratosthenes that has been used to find prime numbers.
John von Neumann (1903-1957)
Nationality: Hungarian
Famous For: Operator theory and quantum mechanics
The mathematical evaluation of self-replication by John von Neumann came before the DNA model
was introduced. Other mathematical subjects he tackled include the “mathematical formulation of
quantum mechanics”, “game theory,” mathematical statistics and mathematical economics. His
contribution to the study of the “operator theory” is equally important.
Pierre de Fermat (1601-1665)
Nationality: French
Famous For: Fermat’s Last Theorem
As an amateur mathematician, de Fermat is given recognition for his work that has led to infinitesimal
calculus. He applied the use of “adequality” in explaining his mathematical constructs. De Fermat’s
also contributed to the math fields of analytic geometry, differential calculus, and number theory.
FERMAT'S LAST THEOREM - states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.
John Napier (1550-1617)
Nationality: Scottish
Famous For: Inventing “logarithms”
John Napier is responsible for manufacturing logarithms. It was also he who applied the everyday use
of the decimal point in mathematics and arithmetic. Napier’s bones was an abacus created by John. The
device was used mainly for multiplication problems.
Gottfried Wilhelm Leibniz (1646-1716)
Nationality: German
Famous For: Infinitesimal calculus
The work of Leibniz on infinitesimal calculus was one completely separate from Isaac Newton. His
mathematical notation continues to be in use. He also proposed the mathematical principle known as
the Transcendental Law of Homogeneity. His refining of the binary system has become foundational in
mathematics.
Andrew Wiles (1953)
Nationality: Proving “Fermat’s Last Theorem”
Famous For: British
Andrew Wiles was successful in proving “Fermat’s Last Theorem”. He also used the “Iwasawa theory”
to identify elliptic curves using its complex multiplication system. Wiles, with a colleague, worked on
rational numbers under the “Iwasawa theory”.
Daniel Bernoulli (1700-1782)
Nationality: Swiss
Famous For: Bernoulli principle
Hydrodynamica by Daniel Bernoulli was a book that touched on mathematical principles applied in
other sciences.
BERNOULLI PRINCIPLE - states that an increase in the speed of a fluid occurs simultaneously with a decrease
in pressure or a decrease in the fluid's potential energy.
Luca Pacioli (1445-1517)
Nationality: Italian
Famous For: Father of accounting
Fifteenth century friar and mathematician Luca Pacioli developed an accounting or bookkeeping
methods that are still in use today. Because of this, Pacioli is viewed by many as the “father of
accounting.”
Georg Cantor (1845-1918)
Nationality: German
Famous For: Inventor of set theory
One of the basic theories in mathematics is the set theory, thanks to the work of Georg Cantor. He
helped define the importance of the “one-to-one correspondence” principle as well as introduce
cardinal and ordinal numbers.
SET THEORY - a branch of mathematical logic that studies sets, which informally are collections of objects.
George Boole (1815-1864)
Nationality: English
Famous For: Boolean algebra
George Boole and his ideas on mathematics were in the field of algebraic logic and differential
equations. He is the source of what is known as “Boolean logic” in algebra. This and other
mathematical concepts are part of his book The Laws of Thought.
BOOLEAN ALGEBRA - is the branch of algebra in which the values of the variables are the truth values true and false,
usually denoted 1 and 0 respectively.
Évariste Galois (1811-1832)
Nationality: French
Famous For: Theory of Equations
Galois worked on abstract algebra and the theory of equations. He also set forth a solution to the
polynomial equation which is know as the “Galois theory.”
Edward Witten (1951)
Nationality: American
Famous For: String theory
Edward Witten specialized in the field of mathematical physics. He brought together math concepts
and basic physics.
STRING THEORY - is a complex theory that describes our reality with superstrings as the most basic and fundamental
piece of all matter
FIBONACCI SEQUENCE
- is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1,
the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the expression is Fn = Fn-1 + Fn-2
also known as the Recurrence Relation.
FIBONACCI'S RABBITS
Fibonacci first noted the sequence when pondering a mathematical problem about rabbit breeding. Beginning with a male
and female rabbit, how many pairs of rabbits could be born in a year? The problem assumes the following conditions:
 Begin with one male rabbit and female rabbit that have just been born.
 Rabbits reach sexual maturity after one month.
 The gestation period of a rabbit is one month.
 After reaching sexual maturity, female rabbits give birth every month.
 A female rabbit gives birth to one male rabbit and one female rabbit.
 Rabbits do not die.
FIBONACCI SPIRAL
- is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for
dimensions. The squares fit perfectly together because of the nature of the sequence, where the next number is
equal to the sum of the two before it. Any two successive Fibonacci numbers have a ratio very close to the
Golden Ratio, which is roughly 1.618034. The larger the pair of Fibonacci numbers, the closer the
approximation. The spiral and resulting rectangle are known as the Golden Rectangle.
SOME INTERESTING THINGS
Here is the Fibonacci sequence again:
n=
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 ...
xn=
0
1
1
2
3
5
8
13 21 34 55 89 144 233 377 610 ...
There is an interesting pattern:
 Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)
 Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)
 Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)
And so on (every nth number is a multiple of xn).
 Sum of Fn and Fn+2 is equal to the square of Fn+1
GOLDEN RATIO
- When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to
the Golden Ratio "φ" which is approximately 1.618034...
- We find the golden ratio when we divide a line into two parts so that: the whole length divided by the long
part is also equal to the long part divided by the short part
USING THE GOLDEN RATIO TO CALCULATE FIBONACCI NUMBERS
And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:
𝜑 − (1 − 𝜑)
𝑥 =
√5
The answer always comes out as a whole number,
number, exactly equal to the addition of the previous two terms.
terms
HONEYCOMB
- is a space filling or close packing of polyhedral or higher-dimensional cells,, so that there are no gaps.
The Roman scholar Varro, in his 1st century BC book-long poem De Agri Cultura (“On Agriculture”), briefly states
“Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that
this hexagon inscribed in a circular figure encloses the greatest amount of space1.”
This quote is the earliest known source suggesting a link between the hexagonal shape of the
th honeycomb and a
mathematical property of the hexagon, made more explicit a few centuries later by Pappus of Alexandria (sometimes
considered to be the last Ancient Greek mathematician). Writing after the Roman Empire’s glory days, Pappus points out
bees, in their
that there are three regular polygons that tile the plane without gaps—triangles,
gaps triangles, squares and hexagons—and
hexagons
wisdom, choose the design that holds the most honey given a set amount of building material