In Middle School and High School Mathematics, the discussions about mathematical proofs often focus on Geometry. In this post, I will discuss a very basic theorem in Algebra and its proof. The theorem states that the sum of two even numbers is even. Note that in this particular post, I mean integers when I say numbers.
As I have discussed before, I will assume that some readers of this blog are not mathematically inclined, so the proofs in the early part of this blog are going to be quite detailed.
The conjecture that the sum of two even numbers is even came from observations. For example, and are even, and is even. Other examples such as and also support the observation. Listing more example would convince us that this the conjecture is really true. However, no matter how many pairs of numbers we add, it is not yet enough to say that the sum of two numbers is even because we have not listed all pairs. Of course, it is impossible to list all pairs. We need a proof.
The sum of two even numbers is even.
Let and be even numbers.
Then and are both divisible by . That is, if we divide and by , we can find quotients and . Writing the two equations, we have
Clearly and are also integers (Why?). Multiplying both sides of the two equations by , we have
If we add and , we have
Now, is even since it is divisible by . But which is the sum of two even numbers. Therefore, the sum of two even numbers is even.
You would have realized by reading the proof above that all even numbers can be represented by for any integral . This is because in any even number, we can always factor out and determine the integer as the other factor. As a consequence, if is even, the or .