1. Examples of Mathematical Proof

2. EXAMPLE 1: Pythagorean Theorem

3. c2=a2+b2. Pythagorean theorem
For a right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse:
c2=a2+b2.

4. These are rotated 90°, 180°, and 270°, respectively.
Pythagorean theorem
PROOF by Bhaskara: Suppose we have four copies of congruent right triangles where the length of the sides are equal to a, b and c, as shown in the figure.
These are rotated 90°, 180°, and 270°, respectively.

5. These are rotated 90°, 180°, and 270°, respectively.
Pythagorean theorem
The region enclosed by a right triangle has area ab/2. Without loss of generality, assume a > b.
These are rotated 90°, 180°, and 270°, respectively.

6. Pythagorean theorem
Let us put them together (without additional rotations) so that they form a square with length of the side equal to c.

7. Pythagorean theorem
This forms a smaller square with length of a side equal to a – b.
right angles

8. b a b Pythagorean theorem a – b
This forms a smaller square with length of a side equal to a – b. b a
a – b b

9. a b Pythagorean theorem
The sum of the area of the region enclosed by each right triangle is 4(ab/2) = 2ab. a b

10. Pythagorean theorem a – b
The area of the region enclosed by the smaller square is (a – b)2.
a – b

11. a b Pythagorean theorem a – b
Hence, the area of the bigger square is c2 = (a – b)2 + 2ab.
a – b a b

12. Pythagorean theorem c2 = (a – b)2 + 2ab c2 = (a 2 – 2ab + b2) + 2ab
End of proof

13. Pythagorean theorem
TRIVIA: The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the two legs.

14. Pythagorean theorem c
c2=a2+b2 b a

15. EXAMPLE 2: 𝟐 is irrational

16. 𝟐 is irrational What is a rational number?
It is a number that can be expressed as a fraction of the form 𝒂 𝒃 , where a and b are integers and 𝒃≠𝟎.

17. 𝟐 is irrational
If 𝟐 is irrational then this means that it CANNOT be expressed as a fraction of the form 𝒂 𝒃 , where a and b are integers and 𝒃≠𝟎.

18. 𝟐 is irrational We will prove this using
Proof by Contradiction (an indirect proof).

19. 𝟐 is irrational
Remember: an axiomatic system cannot have contradictions!
So what we are going to do is to assume that 𝟐 is rational then we will arrive in a contradiction. Hence, we can conclude that 𝟐 must be irrational.

20. 𝟐 is irrational
PROOF:
Suppose 𝟐 is rational, which means it can be expressed as a fraction of the form 𝒂 𝒃 , where a and b are integers and 𝒃≠𝟎.

21. 𝟐 is irrational
PROOF:
Also, let us assume 𝒂 𝒃 as a fraction in lowest terms (Note: we can always reduce a fraction to lowest terms).
This means that a and b are relatively prime to each other.

22. 𝟐 is irrational PROOF: Since 𝟐 = 𝒂 𝒃 or 𝟐= 𝒂 𝟐 𝒃 𝟐 then we have
𝟐 𝒃 𝟐 = 𝒂 𝟐 .

23. 𝟐 is irrational
PROOF:
𝟐 𝒃 𝟐 = 𝒂 𝟐 means that the perfect square 𝒂 𝟐 is even, so 𝒂 must also be even (Note: a square of an even integer is always an even perfect square, and the converse is also true).

24. 𝟐 is irrational PROOF: So we can write 𝒂=𝟐𝒌 where 𝒌 is an integer.
We now have 𝟐 𝒃 𝟐 =𝟒 𝒌 𝟐 , resulting in 𝒃 𝟐 =𝟐 𝒌 𝟐 .
This also means that 𝒃 𝟐 is even, and so is 𝒃.

25. 𝟐 is irrational PROOF: We have shown that 𝒂 and 𝒃 are both even.
However, this contradicts our assumption that 𝒂 and 𝒃 are relatively prime. Two even numbers cannot be relatively prime.

26. 𝟐 is irrational
PROOF:
Therefore, 𝟐 cannot be expressed as a rational fraction; hence 𝟐 is irrational.
End of proof