Direct and Indirect reasoning

Good evening everyonethis is m@rk scribing for yesterday's class. I've been banged up by high fever that is why i was unable to go to class today.

To start yesterday's class we proved using direct reasoning that the product of 2 even numbers is even too.First of all an even number is something that is divisible by 2 and has a factor of 2.

Let 2x=an even number
2a=another even number

2x * 2a =4ax
2x * 2a= 2 (2ax)

∴ Any product by both even numbers is also even as seen from above.

Then, we proved using direct reasoning that the product of 2 odd numbers is odd too.

Let (2x+1) =an odd number
(2a+1) = another odd number

(2x+1) (2a+1) =4ax+2x+2a+1
(2x+1) (2a+1)=2 (2ax+x+a) 1

∴ Any product by both odd numbers is also odd as seen from above because the product does not factor perfectly with 2.

Now, from the previous class we proved that √2 is irrational using indirect reasoning.

First, assume that √2 is rational.

√2=a/b

Note: a/b is in reduced form meaning they have no common factors.

2= a²/b² Multiply both sides by b²
2b² =a²

a² must be an even number and a must be even too so, let a=2c.

2b² = (2c)²
2b²=4c²
b²=2c²

b² and b are both even numbers so, let b=2d. This is a contradiction! Both a and b cannot have any common factors because of our assumption that they are in reduced forms in the beginning.

∴ √2 is irrational

Then, we prooved that √3 is irrational using indirect reasoning.

First, assume that √3 is rational. THis problem is similar to the the previous one.

√3= a/b

Assume that a/b is in reduced form.

3= a²/b²

3b² = a²

On the other hand lets look at square numbers first.
4= 2*2
9=3*3
16=2*2*2*2
25=5*5
36=2*3*2*3
49=7*7
64=2*2*2*2*2*2
81=3*3*3*3
100=5*2*5*2

Can you see the pattern that there is always a pair of prime numbers in square numbers?

So let b²=3c

3(3c)² =a ²

3* 9c² =a²

BUt it cannot happen because a and b will share the same factor which is 3. Therefore contradicting with our first statement that a and b is in reduced form.

∴√3 is irrational

Lastly, we talked about pardoxes. A paradox is a statement that cannot be true but also cannot be false.

Example:
This sentence is false

In this case you cannot decide whether it is true or false.

Thats it for my scribe for tonight. I dont know if im going to be able to go to class tomorrow. It all depends if im feeling better . LIke i told Mr. K the scribe will be Natnael. The homework is the Unusual school play