Adding in Base of 8 and in Base of 16 - CCNA

Just like I mentioned previously, I am going to teach you how to do additions in base of 8 and in base of 16 (Octals and Hexadecimals). So, to start off we are first going to look at what each base looks like:

Base of 8 {0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, etc.}

Base of 16 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, etc.}

If you notice, they act differently than the numbers we are used to in the base of 10, and the base of 16 even has letters. Don't worry though, I will show you that when adding, they are treated just like normal numbers.

To start off we are going to look at base of 8 to get the hang of it, and then you will see that the base of 16 is quite similar. Just like all addition, visually it is the easiest to do when they are stacked on top of each other like below, this will allow us to clearly work from right to left.

When doing additions in the base of 8 or base of 16, you have to be careful. Here you would naturally start off with 3+6 and if this were in the base of 10 you would think out of reflex that the answer is 9. But we aren't working in the base of 10, this is the base of 8 and 6+3=11. But why?

If you noticed above when I started to list out the base of 8, 8 and 9 seem to not exist. Since we are in the base of 8 we work with 8 numbers primarily 0 to 7, and after 7 we start again but now in the 10s and so on and so forth. This may be like a flashback to kindergarten when we all started to learn how to add, but I do recommend that you create a number line or a list of the numbers in base 8 when starting out. I created one below highlighting 3 and adding 6 to that to show we get 11.

Base of 8 {0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, etc.}

In all honesty, once that is understood you can continue the addition with zero issue. Since it is 11, that means the one is carried over. 2+5 =7 and 7+1 (that was carried over) =10. Again, the one is carried over. Which leave us with 1+7=10+1=11. That leaves us with the grand total of 1101 base 8.

Now for the base of 16. Just like the base of 8, I recommend looking at the list of numbers that makes of the base of 16 to get a clear understanding of it.

Base of 16 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, etc.}

You will quickly notice that there are numbers in between 9 and 10, but why is that. Well, since we are working in the base of 16 we have to reach 16 numbers before making it into the 10's, and we can't use 10, 11, 12 because that is used later on. At least that is how I understand it. So, I think you may be wondering how you add A, B, C, D etc. well I personally treat it like the number it is meant to be. For example, if it was 1+A, I would consider A as 10, and move 10 spaces on the number line to get find that the answer is B. Which is a correct way to do it, but I just don't want you all to start confusing A and 10 together when working with the base of 16 because they are not the same thing. Anyway, to continue let's look at the problem below:

Like we did for the base of 8, if you look at a number line for the base of 16. So, if we add B+5, we will get 10, thus we carry the one over. Now we have A+3=D+1=E. Which leaves us with the grand total of E0 in base of 16. Trust me if you are new to base of 16 it is common for that answer to look a little bizarre. It still does for me too.