Discrete Math

A HISTORICAL OVERVIEW

Mathematics – What’s that?

• From Greek: μάθημα, máthēma, ‘knowledge, study, learning’.
• Discovering and proving properties of abstract objects.
• Abstractions from nature (such as natural numbers or lines),
• Development of abstract concepts defined by their basic properties, called axioms.
• Mathematics is the study of patterns, order, structure, relation, quantity, space, and rate of change through logical reasoning and abstract thinking.

Ancient Math

• Prehistoric Peoples used counting �(over 40,000 years ago).

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Lebombo Bone

• About 44 000 years old
• From South of Africa
• Assumed to be Lunar Phase Counter

Mesopotamia & Egypt

• Complex Math around 3 000 BC by Babylonians (now Syria/Iraq) and Ancient Egyptians.
• Used mathematics for counting, construction, astronomy, taxation, financial calculations.

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The Babylonian Mathematical Tablet, 1800 BC

Hieroglyphic Numbers �

Ancient Greeks

• Famous Theorem of Pythagoras 6^th century BC.

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• Pythagoreans: First systematic study of mathematics as such, Greece.

Ancient Greeks

• Euclid of Alexandria (300 BC): Introduced axiomatic method (definition, axiom, theorem, proof) still valid today
• Father of Geometry.
• His book Elements most important mathematical publication, used until 20^th century.

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Ancient Greeks

The Maya civilization

• Mayan Civilisation (1800 BC - 900 AD)
• Mexico and Central America
• Concept of zero as a number
• Vigesimal (base 20) numeral system

Temple of Kukulcán, Mexico

India

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• Brought Hindu-Arabic numeral system in use today (100-400 AD)
•  Trigonometry was further advanced in India
• Developed modern definition of sine and cosine
• Brahmagupta was the foremost Indian mathematician of his time
• He was the first to give rules to compute with zero 

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Brahmagupta

(598 - 670)

China

• The Nine Chapters on the Mathematical Art
• One of the most influential of all Chinese mathematical books
• Composed by several generations of scholars, 200 BC
• It is composed of 246 problems on agriculture, engineering, taxation, calculation, the solution of equations, and right triangles
• Solves systems of equations using methods similar to the modern Gaussian elimination and back substitution.

• The Chinese mathematician Zu Chongzhi (429-500) was most notable for calculating  π as between 3.1415926 and 3.1415927
• A record in accuracy which would not be surpassed for over 800 years.

Golden Age of Islam

• 9^th – 10^th century AD.
• Plethora of advances.
• Development of Algebra.
• Addition of Decimal Point.
• Advances in Spherical Geometry.
• Representatives: Muhammad ibn Musa Al-Khwarizmi and Omar Khayyam.
• The Compendious Book on Calculations by Completion and Balancing
• ٱلْكِتَاب ٱلْمُخْتَصَر فِي حِسَاب ٱلْجَبْر وَٱلْمُقَابَلَة

Early Modern Period

• Math developed at increasing pace in Western Europe.
• Development of Infinitesimal Calculus in 17^th century by Leibniz and Newton.
• Most important figure in 18^th century Leonhard Euler (founded topology and graph theory, ground-breaking contributions to many other areas).
• Karl Friedrich Gauß in 19^th century (Fundamental Theorem of Algebra, Number Theory).

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Foundational Crisis

• Appearance of Paradoxes in formal treatment.
• Example: Russel’s Paradox (1901).
• Split into Formalists (lead by Hilbert) and Intuitionists (lead by Brouwer).
• Hilbert’s Program: Build a complete finite set of axioms as base for all theories, prove it is consistent.
• Kurt Gödel (1931): Not possible! �(Gödel’s Incompleteness Theorem)

Contemporary Math

• Divided into two areas:
• Pure Mathematics and �Applied Mathematics
• Applied Math �Developed towards applications.
• Pure Math: Study of mathematical concepts for their own sake.

Applied Math Wiki

Vehicle Routing Problem

Pure Math Wiki

Group E8

Pure Mathematics

• Studies abstract mathematical ideas for its own sake.
• Provides intellectual stimulation.
• Appreciation of beauty in math.
• Listed along art, music, philosophy, poetry.
• Not application oriented, although applications often appear (possibly hundreds of years after theory).
• Example: Integer Factorization of Euclid (300 BC)
• Now used heavily in cryptography.

Pure Mathematics

Applied Mathematics

• Mathematical methods are developed regarding a real-world problem.
• Used in physics, engineering, medicine, biology, finance, business, computer science, industry.
• Based on tools from pure math and motivates pure math research.

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Applied Mathematics

Discrete Mathematics

Discrete Math cont.

• Meets many fields of mathematical research, e.g.
• logic
• set theory
• number theory
• abstract algebra
• combinatorics
• graph theory
• proof theory
• probability theory

Main Aim of the Course

• Present an overview about several areas of Pure Mathematics�
• Teach you critical thinking and arguing�
• Learn the skill of writing rigorous mathematical proofs�
• Have Fun!

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Logic Puzzle

On a day, a girl meets a lion and a zebra in the forest.

The lion lies every Monday, Tuesday and Wednesday and the other days he speaks the truth.

The zebra lies on Thursdays, Fridays and Saturdays, and the other days of the week he speaks the truth.

The lion told the girl: “Yesterday I was lying”.

“So was I”, said the zebra.

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What day is it?