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26 Cards in this Set

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Incommensurable
--The side and Diagonal of a Square are non measureable

--Pythagoreans fell apart because of this discovery. It proves that everything they discovered was wrong.
Squaring the Circle
Greek
Pythagoras
-went to Egypt to study
-came back to Greece but left becuase it was being run by a tyrant
-went to Italy
-sfter his death there were colonied of pythagoreans
---liked music (chords)
--harmony as an arithmatical phenomenon
---whole #'s are the underlying principle of the whole universe
Greek Social Structure
--High Class Aristocracy
--Did math just to do it
--Indepently wealthy mathemat.
-Thales & Pythagoras-famous
Thales & Pythagoras lived during the...
Golden Era of Greece
Positionsal Notation
Babylonian
Egyptian System for Numerals
-non positional
-symbols for 1, 10, etc..
--somewhat less efficient than Bab.
--Treatment of fractions arose
--there were symbols for the recipricals of whole numbers but not general ones
--general fractions were writtne as sums of recipricals
Descibe the Pythagorean Method of Geometrical Proof.

Why was the discovery of incommensurabel magnitudes such a blow to this method?
Given: 2 triangles with the same height and equal bases have the same area

Prove: 2 triangles with the same height have areas proportional to their bases

--dissect both triangles into triangles of the same h & b U
--U divides both evenly





--You couldn't always find U.
Method of Mediation & Duplation
.
Interpolation Methods
Babylonians
Incommensurable
Proves that square root of 2 is irratioansl

Shook mathematics foundations

Babylonian Social Structure
--Figure Ruled
--Scribes change and become less subscribed to preist began to show off
--1st of their kind
--used more algerbraic mathe
--dissection
Pythagorean Theorem
Babylonians invented
Geometric Algebra
Greek
Thales
--1st person to have theore,s attributed to him
--Congruence thms (ship rods)
--Similarity thms-Tri. Shadows
The Mathematics of Astronomy
Babylonian
Babylonians
--organizers
--efficiency-wanting to make things fit together
--loved making tables
-tables of pythagorean triples, square roots, powers, conversion, & astronomical
The mathematics of Music
Greek
Eudoxus
-ratio between two lengths as whole #'s even if they weren't
-tried to save math
-method of exhaustion
-almost came up with concept of a limit
Duplation and Mediation
Egyptian
Doubling the Cube
Given a cube construct another cube whose volume is twice that of the original
The method of False position
Babylonian
Surveying
Egyptian
Figurate Numbers
Greek
Egyptian social structure
--Religious
--Centralized State
--Scribes --Utilitarian
--Temples and Pyramids
--Math dealt with Practical Matters
--Started when they figured out hiw ti survey land due to the flooding of the Nile.
Impact of Geograhy of Babylonia on the development of Math.
X-Roads central to all civilazations of the time
--Irrigation Canals (Geometry)
--Plots of land irregular shape learned disection
--Taxation and Commerce-1st
--Interest calculations
--Astronomy with Agri. Cycle
--Clay influenced written form restrictions.
--Number base 60