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TOK

Area of Knowledge

Diana Denisova & Victoria Starkova

Invented: Golden Ratio

History of Mathematics

"There is not a shred of evidence to back up the claims, and every reason to assume they are both false. Simply a myth"

Invented: Zero

"The symbol "0" is an extremely great invention . It helped in writing smallest to biggest numbers with ease. The place value notation is great. The entire progress of science and technology was due to invention of zero."

2000 BC Egyptians

Moscow papyrus:

  • word problems
  • volume

Rhind papyrus (instruction manual):

  • ×, ÷, +, -
  • fractions, area formulas
  • composite and prime numbers
  • arithmetic and geometric means and series
  • first linear equations
  • second order algebraic equations

3000 BC Babylonians

  • multiplication tables on clay tablets
  • no decimal point
  • no positional notation (O!)
  • true place value system
  • (digits on the left -> larger value)
  • base 60 numerical system
  • (60 seconds, 60 minutes, 360°)

"Man has fingers equal to 9+1. They wanted to make the "( 9+1) th " number repetetive . They deliberately wrote "10" instead of "9+1"."

"Amusingly, in 1994, a relation was 'discovered' even between the 'number of the Beast' and the Golden Ratio"- Just a coincidence resulting from residual fascination in numerology

“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”

–Galilei, Galileo (1564 – 1642), Opere Il Saggiatore p. 171.

Discovered: Golden ratio

"Another spurious claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense."

"Once upon a time there was no zero. Of course people knew if they had nothing, but there was no mathematical notation for it. Zero was independently invented only three times."

1.61803398875

600 BC Greeks

Geometry

Use of deductive reasoning

  • (before only inductive -> repeat observations to establish rules)
  • Logic!!

Thales' Theorem

Pythagoras

  • "All is number" school
  • Pythagorean theorem

Plato

  • Platonian Academy
  • analytical method

Eudoxus & Archimedes

  • "method of exhaustion" -precurser to integration

Euclid

  • "Elements"
  • axiomatic method: definition, axiom, theorem, proof
  • logical framework of mathematical knowledge

Ptolemy

  • some use of O
  • "How can nothing be something?"

Discovered: Zero

Renaissance

Algebra and accounting intertwined

Luca Pacioli

  • "Summa Arithmetica" for merchants' bookkeeping
  • 1st published book on algebra
  • "plus and minus" signs

Solutions to cubis and quartic eguations

Trigonometry emerges for navigation

  • chart of sines and cosines published

"There is no doubt that anybody who grew up in a western or mideastern civilization is a pupil of the ancient Greeks, when it comes to mathematics, science, philosophy, art, and literature.

Medieval

  • God had ordered "all things in measure, and number, and weight".
  • Quadrium: arithmetic, geometry, astronomy, and music
  • Boethius, Fibonacci, Oxford scholars
  • physical problems: velocity, force, resistance, uniform acceleration

Ancient joke: "He found out about zero; we will have to kill him."

"Nothing comes from nothing"

(ex nihilo nihil fit)

MODERN DAY

"KING LEAR: ..what can you say to draw

A third more opulent than your sisters? Speak.

CORDELIA: Nothing, my lord.

KING LEAR: Nothing?!

CORDELIA: Nothing.

KING LEAR: Nothing will come of nothing, speak again.

“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

–Johannes Kepler

During Scientific Revolution

logarithms first used

Galileo, Brahe, Kepler-astronomic discoveries

Rene Descartes- analytical geometry -"Discourse on the Method

Leibnitz & Newton - calculus

Pascal & de Fermat - probability

Euler:

  • graph theory- "Seven Bridges of Königsberg" Problem
  • standardized math terms and notations "π" " i"

Gauss- geometry, series, fundamental theorems

Is mathematics discovered or invented?

What is mathematics?

Pure mathematics is, in its way, the poetry of logical ideas.

~Albert Einstein

Methodology

n. "the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations"

-Merriam Webster

First known use:

1573

Black holes result from God dividing the universe by zero. ~Author Unknown

Deductive reasoning

Logic

rationalistic thinking

axiomatic method: definition, axiom, theorem, proof

language for science

precise & accurate

"always the right answer"

little room for emotion

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