General math Intuition-Proof-and-Certainty.pptx

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1. Intuition, Proof, and Certainty

2. topics - What is intuition? - Examples of intuition - What is Proof? - Examples of Proof - What is certainty? - Polya’s Problem Solving Strategy 2

3. What is Intuition? Intuition is an immediate understanding or knowing something without reasoning Intuition is that feeling in your gut when you instinctively know that something you are doing is right or wrong An immediate unconscious perception Direct understanding of truths - Independence of analytical process A non-linear process of knowing through physical awareness, emotional awareness, and making connection between them An irrational unconscious type of knowing

4. What is Intuition? There are a lot of definition of an intuition and one of these is that it is an immediate understanding or knowing something without reasoning. It does not require a big picture or full understanding of the problem, as it uses a lot of small pieces of abstract information that you have in your memory to create a reasoning leading to your decision just from the limited information you have about the problem in hand. Intuition comes from noticing, thinking and questioning. As a student, you can build and improve your intuition by doing the following: a. Be observant and see things visually towards with your critical thinking. B. Make your own manipulation on the things that you have noticed and observed. C. Do the right thinking and make a connections with it before doing the solution.

5. Examples of Intuition The most symmetric 2-d shape possible The shape that gets the most area for the least perimeter (see the isoperimeter property) All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string) The points ( x,y ) in the equation x2 + y2 = r2 (analytic version of the geometric definition above) The points in the equation r * cos(t), r * sin(t), for all t (really analytic version) - The shape whose tangent line is always perpendicular to the position vector (physical interpretation)

6. 1. Based on the given picture below, which among of the two yellow lines is longer? Is it the upper one or the lower one? 6 The figure above is called Ponzo illusion (1911). There are two identical yellow lines drawn horizontally in a railway track. If you will be observing these two yellow lines, your mind tells you that upper yellow line looks longer that the below yellow line. But in reality, the two lines has equal length. For sure, you will be using a ruler to be able to determine which of the two is longer than the other one. The exact reasoning could goes like this. The upper yellow line looks longer because of the converging sides of a railway. The farther the line, it seems look line longer that the other yellow line below.

7. What is Proof and Certainty? By definition, a proof is an inferential argument for a mathematical statement while proofs are an example of mathematical logical certainty. A proof in mathematics is a convincing argument that some mathematical statement is true. A proof should contain enough mathematical detail to be convincing to the person(s) to whom the proof is addressed Proof Techniques used in Mathematics Direct Proof Proof by contradiction Proof by induction Proof by contrapositive

8. Examples of direct proof ► proof :the sum of any two consecutive numbers is odd • 2+3 = odd • 4+5 =odd first, we define some definition: definition 1: integer is even if n = 2k definition 2: integer is odd if n = 2k +1 definition 3: two consecutive integers a and b are consecutive if b = a+1 let a = integer let b = consecutive integer = a+1 sum : a + b a + a+1 2a+1 based on our definition 2: integer is odd if n = 2k +1 therefore, we have a direct proof that the sum of two consecutive integer is odd. 8

9. Example of Proof by Contradiction • Proof – Assume that a and b are consecutive integers Assume that (a + b) is not odd, then no integer k such that ( a+b ) = 2 k + 1 - Sum : a + b a + a +1 Statement 1: 2a+1 Statement 2: (a + b) = 2k+1 If k is not an integer, then why ( a+b ) = 2a+1, where a is an integer?? These statements are contradicting!

10. Example of Proof by Induction • Proof - The sum of any two consecutive numbers is odd. - 1+2 = 3 -> odd - 5+6 = 11 -> odd - 7+ 8 = 15 -> odd Based on observations, (inductive reasoning) we have proven that the sum of any two consecutive numbers is odd

11. Example of Proof by Contrapositive • Proof - If the sum of (a + b) is NOT odd, then a and b are NOT consecutive numbers Make a statement that is consecutive and check the sum - We know that (a + b) = K + (K+1) = 2k+1 K does not hold for any integer K. K +1 is the successor of K. In order for - • Therefore, we have proven that The sum of any two consecutive numbers should be odd

12. Certainty • the quality of being reliably true. Certainty is the state of being definite or of having no doubts at all about something •examples: probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty

13. Polya’s Four Steps in Problem Solving

14. George Polya is one of the foremost recent mathematicians to make a study of problem solving. He was born in Hungary and moved to the United States in 1940. He is also known as “The Father of Problem Solving”. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. Heuristic , a Greek word means that "find" or "discover" refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal

15. 15 The George Polya’s Problem-Solving Method are as follows: Step 1. Understand the Problem. Step 2. Devise a Plan Step 3. Carry out the plan Step 4. Look back or Review the Solution

16. 16 The George Polya’s Problem-Solving Method are as follows: Step 1. Understand the Problem. Step 2. Devise a Plan Step 3. Carry out the plan Step 4. Look back or Review the Solution

17. THANK YOU 17