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| practical problems of counting and recording numbers |
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| earliest technique to visibly express numbers |
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| What moved math beyond tallying and why? |
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| Agriculture-count harvest, measure land, calender for harvest, taxing |
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| first symbolic representation of numbers |
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decimal system (base ten)
new symbol for every multiple of ten
not place value |
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| almost base 60 place value system |
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| why wasn't the babylonian system a true place value system? |
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Definition
| didnt have a symbol for zero |
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Definition
-Egytian document containing 85 math problems
-translated by the rosetta stone
-most of what is know about egyptian math came from Rhind Papyrus |
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Term
| what limited the egyptians from progressing beyond solving linear equations? |
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Definition
| answers had to be written in distinct unit fractions |
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Term
| What quirk did the babylonians have when dealing with numbers? |
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Definition
| they didnt like nonterminating fractions and avoided them whenever possible |
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| How did babylonians approach math differently than the egyptians? |
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Definition
More theorectical but no proofs
--ex: knew how to solve quadratics but quadratic formula was never stated. instead they gave several examples and said "such is the way" |
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| had an approximation of pi: 3.1506 |
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has a list of pythagorean triplets proving that the pythagorean thm was known to the babylonians more than 1,000 yrs before pythagoras
-values in table are so large that trial and error is not possible
-left side of table is broken off so how it was constructed is unknown |
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| What is the defining difference b/t Greek Mathematics from all previous math |
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Definition
| wanted knowledge for its own sake rather than just for practical reasons |
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Term
| What contribution to math did the greeks offer? |
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Definition
abstraction and structure
"math began with the Greeks"
made math into one disciplin
proofs and theoretical structure |
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| first to introduce logical proof |
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Definition
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| what characterizes a logical proof |
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Definition
| based on deductive reasoning rather than experiment and intuition to support a conjecture |
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| What was important to thales when studying math |
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Definition
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Term
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Definition
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| What specific conributions did thales make |
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Definition
| several geometric propositions |
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| 1st of the 7 sages of greece |
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Definition
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Term
| Who was the first mathematician to have a "school" |
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Definition
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| What was pythagoras' school? |
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Definition
| -300 member secret society |
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Term
| what were the aims of pythagoras's school |
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Definition
| political, philisophical, and religious |
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Term
| What was studied at pythagoras's school? |
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Definition
| arithmetic (number theory), music, geometry, and astronomy |
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Term
| What did the pythagoreans believe about knowledge? |
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Definition
| knowledge is the greatest form and purification and to them knowledge meant math |
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| According to the pythagoreans what was the goal of math |
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Definition
understanding of man's place according to a rational scheme
-find order in chaos
-purification
-essential in life and religion |
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Term
| How was one accepted into the pythagorean's? |
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Definition
| 3yr period of listening behind a curtain and approval from pythagoras himself to be allowed in to even see him speak |
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| What is one exceptional characteristic of the pythagoreans considering the time period? |
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Definition
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| Who had a proof of the pythagorean thm that dates earlier than phythagoras's proof |
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Definition
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| which US prespublished his own proof of the pythagorean thm? |
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| The square root of two is not rational wad discovered by who? |
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Definition
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| what did the Greeks think of the irrationals? |
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Definition
| referred to them as the unutterables and ignored the concept |
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Term
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Definition
infinite processes
-zeno's arguments worred greek mathematians more than irrational
-made them feel math wasnt exact
-methods of infinity were banned |
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| originated idea of presenting geometry as a chain of propositions, each derived on the basis of earlier one |
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| First to use letters to designate points and lines in geomatric figures |
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| 1st known proffesional math teacher |
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| Three problems of antiquity |
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Definition
1. duplication of the cube
2. Trisection of an angle
3. Squaring a circle |
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Term
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Definition
| -constructing the edges of a cube having twice the volume of a given cube |
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Definition
a square having the same area to that of a given circle
-squaring a circle |
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| what kind of number is pi |
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Definition
| transcendental => not constructable |
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| showed that a lune is squarable |
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Definition
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| a curve that solves boht the quadrature and trisection problem |
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Term
| what was the problem with the quadratix? |
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Definition
a mechanical device other than a straight edge and ruler is needed to draw the curve
-hippias was basically seen as a cheater |
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Definition
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| there are infinitely many primes |
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| came within 225 miles of computing the circumfrence of the earth |
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Definition
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Term
| made a mean finder, what was it used for, what did he do with it |
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Definition
-Eratstothenes
-helped solve the doubling the cube
-had it cast in bronze and gave it to ptolemy |
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Term
| Where did Eratstothenes study? |
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Definition
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| "The wise man of Alexandria" |
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Definition
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| appointed as the chief librarian at the university of Alexandria |
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Definition
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| What areas did Eratstothenes study? |
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Definition
| Geography, philosophy, history, astronomy, mathematics, literary criticism and poetry |
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| demeaning nickname given to eratstothenes because he was so well rounded that he was not first in any one field |
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| what friends called Eratstothenes |
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Definition
| technique to find all primes less than or equal to a given value n |
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Term
| Summarized everything known about astronomy into 8 book known as...which was... |
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Definition
Alexander Claudius Ptolemy
"Geography"
accepted blueprint of solar system for 14 centuries |
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| What discoverer used geography |
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Definition
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| considered the greatest creative genius of the ancient world |
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| Where did archimedes likely study |
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Definition
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| What was Archimedes' profession and what advantage did this give him |
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Definition
| personal mathematician for the king so he was able to completely devote himself to the study of mathematics |
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Term
| what tool did archimedes devise that involved moving water and why |
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Definition
Archimedian Screw
water pump for raising canal waters over levels into irrigated fields |
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| What kind of mathematics was archimedes interested in |
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Definition
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Term
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| What did he do in Sicily that was significant |
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Definition
invented war machinnes and defended Sicily from roman invasion for two years at the age of 75
-catapults and crossbows that could fire in a variety of specified ranges |
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| Aimed at producing small tracts of limited scope addressed to the eminent math scholars of his day unlike euclid who wanted to educate the general students |
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Archimedes
knew it wasnt exact but very near |
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Archimedes invented a new system for representing very large numbers
used this notation to count the grains of sand on earth |
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Definition
Letter written to eratstothenes by archimedes.
contains the beginnings of integral calculus
monk prayer book was written over it
is being restored at a museaum in Baltimore |
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Term
| "Give me a place to stand and i can move the earth" |
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