Proof in Mathematics

Synthetic Geometry ,[object Object],[object Object]

Checklist: ,[object Object],[object Object],[object Object],[object Object]

Constructions ,[object Object],[object Object],[object Object],[object Object],[object Object]

Proof in Maths ,[object Object],[object Object]

Deductive Reasoning ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object]

So how does it work? ,[object Object]

Here is an example.. ,[object Object],[object Object],[object Object],All students in the class have brown hair. Brian is a student in the class. Brian has brown hair.

Another.. ,[object Object],[object Object]

[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],Consider the function:

What is meant by mathematical proof? ,[object Object]

Yes? ,[object Object],[object Object],[object Object],[object Object]

Satisfying feature. ,[object Object],[object Object]

[object Object],[object Object],Proof by Contradiction

[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Euclid’s Proof of the Converse of Pythagoras’ Theorem (I.48) To prove that : If the square on the hypotenuse is equal to the sum of the squares on the other two sides then the triangle contains a right angle . α To prove that angle α is a right angle Given c 2 = a 2 + b 2 The Proof b c a C A B E D

Incommensurable Magnitudes (Irrational Numbers) 1 1 √ 2 The whole of Pythagorean mathematics and philosophy was based on the fact that any quantity or magnitude could always be expressed as a whole number or the ratio of whole numbers. Unit Square The discovery that the diagonal of a unit square could not be expressed in this way is reputed to have thrown the school into crisis, since it undermined some of their earlier theorems. Story has it that the member of the school who made the discovery was taken out to sea and drowned in an attempt to keep the bad news from other members of the school. He had discovered the first example of what we know today as irrational numbers .

1 1 √ 2 1 It is possible to draw a whole series of lengths that are irrational by following the pattern in the diagram below and using Pythagoras’ Theorem. Continue the diagram to produce lengths of  √ 3, √ 5, √ 6, √ 7 , etc. See how many you can draw. You should get an interesting shape.

1 1 √ 2 1 √ 3 1 √ 4 1 √ 5 1 √ 6 1 √ 7 1 √ 8 1 √ 9 1 √ 10 1 √ 11 1 √ 12 1 √ 13 1 √ 14 1 √ 15 1 √ 16 1 √ 17 1 √ 18

Theorem 12 ,[object Object],[object Object]

Theorem 13 ,[object Object],|AB| = |BC| = |AC| |DE| = |EF| = |DF|

[object Object],[object Object],[object Object],Activity A right angle each time! This proves the converse of Pythagoras is true!

Using Proof by Contradiction ,[object Object],[object Object]

How to write a formal proof? ,[object Object]

Proof in Mathematics

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Andere mochten auch

Andere mochten auch (20)

Ähnlich wie Proof in Mathematics

Ähnlich wie Proof in Mathematics (20)

Mehr von mscartersmaths

Mehr von mscartersmaths (19)

Proof in Mathematics