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26 Cards in this Set
- Front
- Back
Incommensurable
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--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. |
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Squaring the Circle
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Greek
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Pythagoras
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-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 |
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Greek Social Structure
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--High Class Aristocracy
--Did math just to do it --Indepently wealthy mathemat. -Thales & Pythagoras-famous |
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Thales & Pythagoras lived during the...
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Golden Era of Greece
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Positionsal Notation
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Babylonian
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Egyptian System for Numerals
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-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 |
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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. |
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Method of Mediation & Duplation
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Interpolation Methods
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Babylonians
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Incommensurable
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Proves that square root of 2 is irratioansl
Shook mathematics foundations |
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Babylonian Social Structure
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--Figure Ruled
--Scribes change and become less subscribed to preist began to show off --1st of their kind --used more algerbraic mathe --dissection |
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Pythagorean Theorem
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Babylonians invented
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Geometric Algebra
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Greek
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Thales
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--1st person to have theore,s attributed to him
--Congruence thms (ship rods) --Similarity thms-Tri. Shadows |
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The Mathematics of Astronomy
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Babylonian
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Babylonians
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--organizers
--efficiency-wanting to make things fit together --loved making tables -tables of pythagorean triples, square roots, powers, conversion, & astronomical |
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The mathematics of Music
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Greek
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Eudoxus
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-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 |
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Duplation and Mediation
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Egyptian
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Doubling the Cube
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Given a cube construct another cube whose volume is twice that of the original
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The method of False position
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Babylonian
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Surveying
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Egyptian
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Figurate Numbers
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Greek
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Egyptian social structure
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--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. |
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Impact of Geograhy of Babylonia on the development of Math.
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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 |