Section One

Question 1

What was MATH in the EGYPTIAN ERA?

Answer 1

Hash-mark system
Numbers and measure, empirical uses
Reciprocals

The odd formula for pyramids, area, volume.

Question 2

What was MATH in the BABYLONIAN ERA?

Answer 2

More advanced hash-mark system!
Numbers and measure still, empirical uses
Chunks of base 60, no fractions
Astronomy! Use in timekeeping and angle measurements.

Question 3

PYTHAGORAS vs THALES. What what??

Answer 3

Discrete vs continuous! Pythagoras believed the universe was discrete, Thales believed the universe was continuous.

One argument against discrete was the study of commensurable numbers. Could you express the side and the diagonal of a square as the ratio between two whole numbers?

Question 4

How many PRIME NUMBERS are there?

Answer 4

INFINITE. PROOF: Let Ep be all the prime factors of p multiplied, then + 1. Even though the result isn’t prime, it decomposes into smaller primes that were LARGER than the largest prime factor in Ep

Question 5

Can you FIND PRIME NUMBERS?

Answer 5

There are several attempts. Mersenne primes, Fermat’s attempt,

Question 6

What’s the difference between POTENTIALLY INFINITE and COMPLETED INFINITE?

Answer 6

“Potentially” means it can theoretically go on forever, but it hasn’t yet. Like time! Or primes.

“Completed” means it’s sitting there right now, staring back at you. Like all the rational numbers between 0 and 1!

Question 7

ONTOLOGY vs EPISTEMOLOGY. What what??

Answer 7

Ontology: What is math?
Epistemology: How do we know math is math is works?

Question 8

BABYLONIAN CLAY TABLET??

Answer 8

I dun understand this but it’s early on in the notes

Question 9

MORE about ONTOLOGY.

Answer 9

PLATO and ARISTOTLE.

PLATO believed that mathematical objects existed in a higher dimension, casting their shadows onto our cave-world.

ARISTOTLE believed mathematical objects are only abstractions (A for Aristot-al Abstractionista!). That we take the idea of a straight stick, and subtract wood and width till we get the line.

Question 10

MORE about EPISTEMOLOGY.

Answer 10

Math is governed by logical deduction, but everyone agreed that we all had to start with some unprovable truths.
EUCLID built the Elements of Mathematics with five axioms (he needed a sixth though. the cheater.) and went on to prove everything mathy from the ground up.

Question 11

ELEMENTS OF MATHEMATICS. What were the five axioms?

Answer 11

Segments, circles, angles.

1) A line can be constructed from two points.
2) This segment can be extended indefinitely.
3) A circle can be constructed from a radius r
4) All right angles are the same.
5) Parallel Postulate/ Playfair’s axiom

(6): Lines don’t have holes, and can intersect each other.

Question 12

CONIC SECTIONS! What do you know?

Answer 12

Ellipses: PF1 + PF2 = 2r
Parabola: PF = PD
Hyperbola: PF1 - PF2 = constant

Eccentricity: PF/PD = e, where e is the eccentricity

Question 13

GREEK CONTEMPORARIES: PAPPUS

Answer 13

PAPPUS: Conic sections work! Found the directrix of an ellipse.

Question 14

GREEK CONTEMPORARIES: ARCHIMEDES

Answer 14

290-212 BC: Did geometric calculus before the invention of calculus! Also did volume and surface area work. Also wrote the Sand Reckoner, what a tragedy.

Question 15

GREEK CONTEMPORARIES: EUDOXUS

Answer 15

400BC: Similar triangles, proportions, LIMIT IDEA!! (method of exhaustion). Approximating a circle by inscribing polygons of increasing size.

Question 16

GREEK CONTEMPORARIES: PTOLEMY

Answer 16

200 AD? Astronomers, trigonometry, Almagest, the silly Earth-Is-Still model that persisted till 1600

Question 17

GREEK CONTEMPORARIES: DIOPHANTUS

Answer 17

200 AD? Diophantine equations, whole numbers such that 3x + 5y = 17 for example.

Question 18

GREEK CONTEMPORARIES: HYPATIA

Answer 18

Last head of the Museum in Alexandria.

THE END OF GREEK MATHEMATICS