Surprises at Infinity

What are Zeno’s two paradoxes and how do they work?

1. The Runner (Dichotomy): To reach the finish line, a runner must first go halfway. Then they must go half the remaining distance, then half of that, forever. Because there are an infinite number of "halfways," Zeno argued the runner can never finish.

2. Achilles & the Tortoise (Jesse Owens & the Professor): The fast runner (Jesse) gives the slow runner (the Prof) a head start. When Jesse reaches the Prof's starting spot, the Profhas moved forward a little. When Jesse reaches that new spot, the Prof has moved again. Zeno argued Jesse can never catch up because he is always running to where the Prof used to be.

What is an infinite sum? What is convergence and divergence?

Infinite sum is a series of numbers added together that goes on forever. The total of an infinite sum is what number it goes to. Convergence is when an infinite sum ends up being one number at the end, divergence is when it goes above a certain number (usually because it is being multiplied into much larger parts).

What are the categories of the infinite sums and how do we know what they are?

Geometric sum - this one is the same ratio always, so just divide the second one by the third one and then the first one by the second one to see.

Alternating sum - sum where the signs flip back and forth

Harmonic sum - one specific sum that goes 1, 1/2, 1/3, 1/4, 1/5 etc

What are the 7 types of numbers?

1. counting numbers (1, 2, 3)

2. Integers (same but negative)

3. Rational numbers (any a/b whether it be positive or negative)

4. Algebraic numbers (any number that can be generated from a polynomial equation)

5. Irrational numbers (numbers that cannot be written as fractions)

6. Transcendental numbers (those that cannot be solved algebraically)

7. Real numbers (all the rest)

What is a prime number?

An integer greater than 1 whose only divisors are 1 and itself.

How do you do induction?

1. First, prove the base case. Set n=1 and make sure the right side = the left side.

2. Then set n=k for the whole equation and add the same thing but n=k+1 but just on the left side.

3. Then add the right side answer to the n=k+1 version of the left side to get the final answer, ignore the n=k version on the left side.

What is an independent axiom?

When it cannot be proved by the other axioms. You prove this by making all the other axioms work together and one not.

What does it mean when axioms are consistent?

They don’t contradict one another and you can build a model with them.

What is Euclid’s parallel postulate?

Given a line and point not on that line, there is exactly one line that is parallel to the original line.

Hyperbolic Geometry

A place where there are infinite lines parallel to the main line on a separate point.

Elliptic Geometry

A place where there are 0 parallel lines because everything crosses (like on a globe).

What is the formula for a geometric sum?

Total = a/1-r (where a is the first term). You can find r by dividing the second term by the first term.

Why does a harmonic sum never converge?

Because although the numbers keep getting smaller, you can lump them into groups larger than 1/2, meaning it will keep climbing although slowly.

Why does the alternating harmonic sum converge?

Because the jumps keep getting smaller and smaller so you go back and forth onto a point eventually.

Why do rational decimals repeat?

Because there is only a limited number of remainder patterns for the number, so once it hits one that has already happened the whole sequence will repeat.

Describe a model of a Poincaré disk. What do lines look like? What do triangles look like?

The Poincaré model represents infinite space all squeezed inside a circle. The outside edges are curved infinitely more as you get closer to them. Straight lines are really curved lines that make 90 degree angles with the sides of the circle. Triangles are always less than 180 degrees.