Skip to main content

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.


Comments

Popular posts from this blog

Virtual machine could not be started because the hypervisor is not running

I wanted to experiment with TensorFlow, and decided to do that in a Linux VM, despite the fact that Windows Subsystem for Linux exists. In the past I used Sun’s, and then Oracle’s VirtualBox to manage virtual machines, but since my Windows install had Hyper-V, I decided to use that instead. The virtual machine configuration was easy, with disk, networking, and memory configurations non-eventful. However when I tried to start the virtual machine to setup Ubuntu from an ISO, I was greeted with the following error: “Virtual machine could not be started because the hypervisor is not running” A quick Internet search revealed that a lot of people have faced that problem, and most of the community board solutions did not make any sense. The hidden gem is this technet article , which included detailed steps to find if the Windows Hypervisor was running or not, and the error message if it failed to launch. In my case, the error was: “Hyper-V launch failed; Either VMX not present or

Why good customer service matters?

I am not an Apple fan, but I do like their computers, and recommend them to colleagues and friends for a variety of reasons. They are well designed, and in addition to an excellent user interface, they run a flavor of Unix--which makes the life of computer programmers a lot easier. But most importantly, Apple's customer support is impeccable, that despite all the hardware issues I experienced in the past, I still recommend Apple computers. Let me explain why. A year and a half ago, I bought a Mac Book Pro for work. At the time it was the first generation unibody laptop, that had an i7 processor, lots of memory, and lots of disk space. Alas, like first generation models everywhere, it also had a lot of hardware problems. The most annoying of which was the screen randomly turning dark, with the hard drive spinning out of control. The only way to get out of this state was by forcing a reboot by holding down the power button, and losing everything I have been working on. At first

MacOS Catalina, OneDrive, and case sensitive file systems

Over the weekend, I dusted off my old Macbook Air to search for some old family photos. I have not used the laptop for a long time, and it was completely out of charge. I plugged it in, and it quickly booted. Shortly after, I got bombarded with notifications that many of the applications needed updating, and that a new version of the OS was available.   I waited till I found the photos I was looking for, before attempting to upgrade anything. I also wanted to install OneDrive to get my old files to the cloud, so that I can access them from any of my devices, instead of dusting off old computers to get to them. The MacOS upgrade experience has always been fantastic, and this one was no different. The OS upgrade files downloaded quickly and after a restart and a quick install, the Macbook Air was ready to go.   Upgrading the installed applications was also a breeze, however in the process I discovered that a large majority of the applications installed were not compatible with Cata