1. TOK: Mathematics Unit 1 Day 1

2. Introduction

3. Opening Question Is math discovered or is it invented? Think about it. Think real hard. Then discuss.

4. "Mathematics is the language in which God has written the universe." -Galileo Galilei

5. Activity! Go with your groups and find an example of math in the real world and brainstorm as many real world examples as you can that involve math.

6. Math Math might be characterized as the search for abstract patterns Patterns are everywhere For example: 2+2=4 It doesn't matter what it is, if you have two of something then add two more, you have four of the same item Circles: If you divide the circumference by the diameter, you always get pi.

7. Mathematical Paradigm Defined: "The science of rigorous proof" Dates back to the Greeks Euclid first person to consider Formal system of reasoning. Three key elements Axioms Deductive reasoning Theorems

8. Axioms Starting points or basic assumptions. Premises. Can’t prove axioms (Infinite regress) Four traditional requirements for a set of axioms. Consistency Once proven inconsistent, you can prove almost anything Independence You should not be able to deduce any more axioms from an axiom Simplicity Clear and simple as possible Fruitfulness Should be able to prove as many theorems as possible using the fewest number of axioms

9. Euclid's Axioms 1.It shall be possible to draw a straight line joining any two points. 2.A finite straight line may be extended without limit in either direction. 3.It shall be possible to draw a circle with a given centre and through a given point. 4.All right angles are equal to one another 5.There is just one straight line through a given point which is parallel to a given line. Euclid later used these five axioms to form theorems

10. Deductive Reasoning Ex. Syllogism (1) All humans beings are mortal (2) Socrates is a human being (3) Therefore Socrates is mortal If we say that (1) and (2) are true, then (3) must be true.

11. Theorems Theorems are like Conclusions Can be used to construct complex proofs Derived by Euclid based on his five axioms and deductive reasoning Lines perpendicular to the same line are parallel Two straight lines do not enclose an area The sum of the angels of a triangle is 180 degrees The angles on a straight line sum to 180 degrees

12. Example Theorem Pythagorean Theorem a² + b² = c² The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares equals (the area of) the big one.

13. Proofs and Conjectures Conjectures A hypothesis that seems to work Not necessarily true Proofs A theorem is shown to follow logically from the relevant axioms Necessarily true Inductive Reasoning Particular to general Not completely certain Don't jump to conclusions (Ex. on pg 193)

14. Goldbach's Conjecture Famous mathematical conjecture Every even number is the sum of two primes Proven up to an 18 digit number Still can't be considered a theorem

15. Beauty, Elegance, and Intuition An elegant proof might even be described as beautiful Paul Erdos spoke of “the BOOK” of math Exercise 1 (Pg. 195) There are 1,024 people in a knock-out tennis tournament. What is the total number of games that must be played before a champion can be declared? Exercise 2 (Pg. 195) What is the sum of the integers from 1-100?