Writing Proofs

Slideshow 43, MathematicsMr Richard Sasaki, Room 307

Objectives

• Try to develop reasoning when answering a question

• Learn to overly explain all parts of a problem

• Apply this to making proofs

ProofsThe content of this lesson will be on number (not shape).

Writing proofs can be very difficult, especially for those people who rush maths (lazy people) and like to calculate things as quickly as possible.

These people understand the process but not always the reasoning.

ProofsWhat is a proof?A proof is an explanation to prove something. It isn’t something that can be answered in sentences or something with only calculation. Both are necessary.

Let’s try an example.

ExampleProve that 17 is an odd number.Odd numbers are numbers that end in 1, 3, 5, 7 and 9.Plan: Calculate the nth term for the sequence of odd numbers and prove that 17 is within it.Sequence (ascending) of odd numbers: 1, 3, 5, 7, 9, …Let n be the position in the sequence above. The nth term is _______.2n - 1If 17 is an odd number, we can express 17 in the form 2n – 1 where nℤ.Let 17 = 2n – 1, nℤ.

18 = 2n = n

⇒𝑛=9As n = 9, nℤ so 17 appears in the sequence above. is an odd number.

Question 1 – Prove that 23 is not an even number.

Sequence (ascending) of even numbers: 2, 4, 6, 8, 10, …Let n be the position in the sequence above. The nth term is ____.2nIf 23 is an even number, we can express 23 in the form 2n where nℤ.Let 23 = 2n, nℤ.

11.5 = n = 11.5

As n = 11.5, nℤ so 23 doesn’t appear in the sequence above. is not an even number.

Question 2 – Prove that the sum of two even integers is even.Let .As and are even, we can say that and where .Let’s check if is even. can be written as where .H. must have a factor of 2, has a factor of 2.

The sum of two even integers is even.

Question 3 – Prove that the sum of two odd integers is even.Let .As and are even, we can say that and where .Let’s check if is even. can be written as where .H. must have a factor of 2, has a factor of 2.

The sum of two odd integers is even.

Question 1 – Prove that 0.00732 is a rational number.

If 0.00732 is rational, then it can be written in the form where .

The number 0.00732 written in a fractional form is which can be written as .

W and .

0.00732 is rational.

This implies that must be rational.

Question 2 – Prove that if is an odd integer, is also an odd integer.

Let .As is an odd integer, we can say that and therefore, where .Let’s rewrite so that it’s clear that must be an odd number.𝑛2=(2𝑎−1 )2=(2𝑎−1 ) (2𝑎−1 )

Simplifying this, we get . This can be written in the form . We know that as .

is in the form where , must be an odd integer.

If we expand the brackets, we get .

Question 3 – Prove that if are odd integers, is also an odd integer.

Let .As are odd, we can say that and where .

Let’s calculate .𝑚𝑛=(2𝑎−1 ) (2𝑏−1 )

Factorising this, we get .

is in the form where , must be an odd integer.

If we expand the brackets, we get.