How Did We Get Here- Mathematics (and a little physics) by Frances Hao

Math in Mesopotamia

fhao3 — 2024-01-19 16:13:19 UTC

The Sumerians used the sexagesimal system, which was a system in base 60. This was achieved by taking the unit, and alternating between multiplying 6 and 10, such that every other symbol is a power of 60. (see picture above). The symbol for "1" is a small cone-shaped object, and the symbol for 10 is a small circle. Six small circles formed a big cone, and ten big cones formed a cone with a circle in it. The reasons for these circles and cones was because they could easily be made with pressing a stylus into clay; a circle by pressing down vertically, and a cone by pressing down at a slightly slanted angle.

However, when the language of the Sumerians evolved from little pictures of everyday goods to actual writing (called cuneiform), their numeric system also underwent a dramatic change. A small vertical wedge represented 1, evolved from the cone; and a small vertical wedge represented 10.

Citations

fhao3 — 2024-01-19 16:22:04 UTC

See Google Doc:

Ancient Egypt Math

fhao3 — 2024-01-24 15:48:17 UTC

In Ancient Egypt, scribes were developed with their evolution of writing, and they were often used to keep track of taxes and even were dealing with the persecution of war. One problem states : "A ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits."

The Root of Geometry

fhao3 — 2024-01-24 15:48:48 UTC

The second-most studied book in the world (the first is the Bible, of course) is a book dating back to Ancient Greece, Euclid's Elements. This book, written by the Greek Mathematician Euclid, gives a detailed "tour" of geometry and each concept thoroughly.

Euclid's 5 axioms

fhao3 — 2024-01-24 15:48:59 UTC

Perhaps one of the most famous contributions by Euclid was his 5 Axioms and Postulates. While the axioms focused more on general mathematics, the postulates are centered more on geometry.

His axioms are as follows:

Things which are equal to the same thing are also equal to one another.
If equals be added to equals, the wholes are equal.
If equals be subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
The whole is greater than the part.

Although these may seem like they are common knowledge, without the automatic assumption that they are true, many mathematical theories cannot be formed today.

These are his geometric postulates:

A straight line segment can be drawn joining any two points.
A ny straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate

Galileo and Kepler

fhao3 — 2024-01-25 16:14:45 UTC

Galileo and Kepler were two astronomers from Europe (Galileo was from Italy and Kepler from Germany) and both claimed that the solar system was centered at the Sun, rather than the Christian belief that it was centered around Earth. Both received extreme backlash, and Galileo, also the inventor of the Telescope, was criticized as creating "an instrument of the devil". The Catholic Church was so against the idea of a heliocentric (sun-centered) solar system that it put Nicholas Copernicus’s “De Revolutionibus” on its list of banned books. Galileo, who was a supporter of this claim, was also put on trial. On the other hand, Kepler was an astronomer who supported Copernicus's theory and even pushed it a step further. He came of with three laws of planetary motion:
the planets move in elliptical orbits with the Sun at one focus
the time necessary to traverse any arc of a planetary orbit is proportional to the area of the sector between the central body and that arc
there is an exact relationship between the squares of the planets’ periodic times and the cubes of their mean distances from the Sun

Archimedes and his Law of the Lever

fhao3 — 2024-02-08 21:48:15 UTC

Archimedes was a Greek mathematician and physicist. One of his famous accomplishments was discovering the relationship between the surface area and volume of a sphere. He was also an inventor who invented the Archimedes Screw, a device used for raising water. However, he is most remembered for devising the Law of the Lever, which states that if two objects with (not necessarily equal) masses are balanced on a weightless stick (lever) with a support on the ground (this support is called the fulcrum), the mass of one object divided by its distance to the fulcrum is equal to the mass of the other object divided by its distance to the fulcrum.

Euler

fhao3 — 2024-02-08 22:22:26 UTC

Leonhard Euler was a Swiss mathematician. He worked on abstract aspects of math, such as imaginary and complex numbers. He devised two famous formulas: his Formula, which states that (eⁱ^θ = cos θ + i sin θ) where θ is any angle, and his Identity, e^iπ+1=0.

Newton

fhao3 — 2024-02-08 22:28:46 UTC

Although the famous story of Newton getting hit by an apple falling from a tree and therefore discovering gravity is unfortunately false, there is no doubt that he contributed to much of what is physics today. His 3 famous laws of motion are what many of the world's most famous physicists refer to even today. They state:

Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it
Force equals mass times acceleration: F=MA
For every action there is an equal and opposite reaction.
These laws are fundamental to what the universe is made up of, no matter where it is.

Katherine Johnson

fhao3 — 2024-02-08 22:33:51 UTC

Katherine Johnson was an American mathematician. She was one of three Black students selected to be offered spots at the state flagship school. She worked for NACA starting in 1953 and retired in 1986 from NASA.