Examples of Mathematical Proof

EXAMPLE 1: Pythagorean Theorem

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.

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. https://www.cut-the-knot.org/pythagoras/

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. https://www.cut-the-knot.org/pythagoras/

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

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

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

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

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

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

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

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.

Pythagorean theorem c c2=a2+b2 b a

EXAMPLE 2: 𝟐 is irrational

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.

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.

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

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.

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.

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.

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

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).

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.

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.

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