TOK Math Unit Review

Which postulate of Euclid is questionable?

"Given on a plane, a line L and a point P not on L. Then through P exist one and only one line parallel to L."

Be able to explain how General Relativity uses non-Euclidean geometry.

(It uses non-euclidean geometry because it isn't on a single plane?)

Who formulated these alternate views?

(Riemann and Lobachevsky?)

What implications does this have for the structure of the universe?

(Universe isn't flat?)

What is the Copenhagen Interpretation?

(in my notes)

What is the golden ratio (PHI)?

1: 1.6180

What did the worksheet on inductive reasoning show?

A rule that is observed to work many times can suddenly stop working.

What is the Lorenz attractor; be able to draw it.

A strange attractor in the form of a two-lobed figure formed by a trajectory that spirals around the two lobes, passing randomly between them.

What is the God Particle (Higgs Boson)?

A subatomic particle whose existence is predicted by the theory that unified the weak and electromagnetic interactions. It explains why all other particles have mass.

What is the quantum, Schrodinger's cat?

A thought experiment that says that a cat that has a 50/50 chance of dying is in superposition (both dead and alive) until the box is opened and the wave function collapses.

How many cases must be tested in order to "prove" the validity of the theorem?

By testing it infinite times.

What are Dark Matter and Dark Energy?

Dark matter is an unknown thing that has been identified by gravitational effect. Dark energy shows that the universe is expanding faster and faster.

What is chaos theory?

Deals with complex systems that highly depend on slight changes in conditions, so that small changes can cause great consequences.

Explain how mathematics produces a unique kind of knowledge using axioms and deductive reasoning. Why is it unique?

Deductive reasoning uses axioms and substitution to create theories. It is unique because it's absolute truth.

What is quantum non-locality?

Entanglement

Be able to give examples of fractals

Flowers, clouds, architecture, coastlines and trees.

Why we are in the golden age of astronomy?

Gravitational lensing and the composition of the universe (Visible matter (4), Dark matter (26), Dark energy (70))

Explain how gravitational lensing is used and what it tells us.

Gravity causes light to be warped in space-time. This means that light sources, like stars, are actually not where we see them. Gravitational lensing tells us that light is warped.

Explain what Godel's theorem is and its implications for mathematics.

His incompleteness theorem suggested that any system of axioms relies on something that is not proven and therefore, math will always be incomplete.

What's the difference between inductive reasoning used by science and deductive reasoning used by mathematics?

Inductive reasoning (science) comes from observation and is unreliable. Deductive reasoning is absolute truth and is based on axioms and substitution.

What is the double slit experiment?

It proved that light is a wave because of wave addition and cancelling. With 2 slits, particles will make 2 lines while waves will make an interference pattern. Normally, light acts like a wave but when observed, it acts like a particle.

How does string theory try to unify quantum mechanics and relativity? (The elegant universe)

It says that particles are not little balls but rather little strings. That allows the quantum world to be slowed down to be compatible​ with relativity.

What is the twin paradox?

It's a thought experiment in relativity involving identical twins. One goes to space returns to find that the twin who stayed on Earth has aged more.

What is the game of life and why is it interesting?

It's an animation that depends on little dots to be filled in. The

What is meant by calling mathematics a language?

Math is a universal language because anyone can understand despite language barriers because it's true anywhere in the world.

What is the Heisenberg Uncertainty Principle?

Math tells us that information is limited. If the position is known, momentum can't be found.

What does math describe?

Nature?

Why does it have problems being believed?

No observations or experiments can be done to disprove it and it suggests multiple universes.

Pollock and Chaos (found in the group presentation links)

Pollock's splatters create fractal patterns like the ones that have been seen in nature.

Give some examples of Fibonacci numbers from nature.

Snail shells, seed patterns in flowers, plant leaves, pine cones, tree branches and cauliflower.

What is the entanglement?

Sub-atomic particles can be entangled (connected) even if they're far from each other. They have properties that oppose each other but haven't been set. When one is observed, the function collapses.

Relate the golden ratio of Fibonacci to beauty.

The golden ration plays roles in nature, art and even in the human body. The length from the bellybutton down to the height of a person is approximately 1:1.618.

What is particle-wave duality?

The idea that light can be both a particle and a wave at the same time.

What is the polarized light?

The intensity of the light equals the cosine of the angle that the filter is at

What are Fibonacci numbers?

The previous 2 numbers give the next one. (1,1,2,3,5,8,13,21,34)

What are Quantum Computers?

They are computers that use entanglement and will be able to perform millions of task in seconds. Uses qubits instead of the traditions 1s and 0s.

What did Fibonacci numbers suggest about mathematics and nature?

They suggested that math and nature are interconnected and decribe each other.

What is non-Euclidean geometry?

They're 3-dimensional and curved. Based on Euclid's first 4 postulates with their own versions of the 5th.

What is spooky action at a distance?

This is what Einstein called the predicted duality of waves and particles. When observed, the particle's wave function collapses and it only appears in one place, not in more than one.

Is there order in chaos? Explain

Yes, in May's equation sets, the bifurcations eventually led to chaos and one could not predict future results but there were visibly little strips of order in white.

Illustrate how mathematics can be superior to common sense in certain cases.

gamblers fallacy, some math is too complex but still axiomatic, common sense can vary