On even and odd numbers, and kid's questions

Kids ask the best questions. Unburdened by prior knowledge or assumptions, their questions are driven by pure curiosity and a deep desire to learn something new. As adults we can learn a lot from how kids ask questions and seek knowledge. And we can also learn or strengthen our understanding of a particular topic by answering kids questions.

For example, my son asked me a question about even and odd numbers, and why we get an even number when we add two even numbers or two odd numbers, and an odd number when we add an even and an odd number. Because this knowledge is drilled into us since our younger years, we stop wondering why.

Intuitively we know the proposition is true, and we can cite a couple of examples to prove it to ourselves. For example, adding 2 and 4--both even numbers, gives us 6, also an even number. When we add 3 and 5--both odd numbers, we get 8, which is an even number. If we choose a mix of an even number and an odd number and add them, we got an odd number. For example, adding 2 and 3, we get 5, which is an odd number. It seems that our intuition is onto something. Sometimes intuition leads us astray, and reality is reflected by something non-intuitive. The theory of relativity—both special and general, and quantum mechanics for example offer non-intuitive description of reality and how things work.

The addition examples above are not enough to gain confidence that we answered the original question. Since there is an infinite number of natural numbers (1,2,3, …etc), we have to find a different way of establishing or refuting the statement without listing an infinite number of examples. The clue is in how we write even and odd numbers.

An even number is a number divisible by $2$. We can write an even number as the product of two numbers, $2$, and another natural number $k>0$ in the form of $n_1=2k$. This number is even, because it is divisible by $2$. An odd number is one that is not divisible by $2$, or one where if divided by $2$ would have a remainder of $1$. Thus, we can express an odd number as $n_2=2m+1$, where $m>0$ is a natural number greater than zero.

Now we can see what happens when we add two even numbers, two odd numbers, and a mixture of numbers. Let’s add two even numbers first $n_1=2k$ and $n_2=2m$. The sum is $n_1+n_2=2k+2m=2(k+m)$ which is an even number.

How about adding two odd numbers $n_1=2k+1$ and $n_2=2m+1$? The sum is $n_1+n_2=2k+1+2m+1=2(k+m)+2=2(k+m+1)$ which is an even number.

Finally, how about adding mixed numbers one even $n_1=2k$, and one odd $n_2=2m+1$. Their sum is $n_1+n_2=2k+2m+1=2(k+m)+1$ which is an odd number.

By working through this simple math, we have gained insight into something drilled into us long time ago. Indeed, kids ask the smartest questions; don’t dismiss them, but use them to learn something new, or understand something you already know better.