Even & Odd Numbers [Version 2 of 2 / More Technical]
Hi there.
The previous post looked at even and odd numbers in a way that is easy and simple for young students. As students get older and learn more mathematics, the mathematics starts to get more structured and tougher with rules, math symbols, math notation, letters and abstract concepts. With that being said, here is a more technical look at even and odd numbers.
A More Technical/Mathematical Version Of Even & Odd Numbers
Given a natural number k where k = 1, 2, 3, 4, 5, 6, ..., an even number can take on the form as:
If k = 1, I have the first even number as 2, if k = 2 I then get 2 x 2 = 4 and so on.
For the odd number case with k = 1, 2, 3, 4, 5, 6, ..., the form would be:
An alternate form for odd numbers would be:
for k = 0, 1, 2, 3, ... (The variable k now starts at 0 versus 1.) An odd number can be thought of as an even number plus (or minus) 1.
Some Properties
Proof Of 1)
Let k and m be natural numbers from 1, 2, 3, 4 and so on. Define the first even number as 
and the second even number as 
. Since two is a common factor the sum can be shown as follows.
Two times any whole number is an even number.
Proof Of 2)
Let a and b be natural numbers from 1, 2, 3, 4 and so on. Define the first number as a even number in the form of 
and the second number as an odd number as 
. We show that the sum of an even and odd number is results in an odd sum.
Notice how the first two terms are factored and the minus one is unchanged.
Proof Of 4)
Let p and q be natural numbers from 1, 2, 3, 4 and so on. Define the first even number as 
and the second even number as 
. The product of 
and 
would result in an even number.
It does not matter what (natural) numbers p and q are as the product is multiplied by an even number of 4.
Notes
- The rest of the proofs can be done as exercises.
- This type of topic was one of the first topics I was exposed to while learning math proofs.
- As you can see, the math you learn early on can be generalized. It becomes more theoretical and abstract.
- The math text in the form of images was done in LaTeX with QuickLaTeX.com
- I consider the number zero as neither even nor odd. A zero ending digit in a two digit number or higher is even though.
