Counting in Hexadecimal

by kirupa   |   28 November 2009

In college, one of the classes I took was called Computation Structures. In that class, you learn how applications work, how memory is handled, what instructions a processor needs to do something, what pipelining does behind the scenes, etc. You even write code (and lots of it!) in languages that resemble assembly and machine code. I took this class years ago, so most of it is fuzzy at this point (most of it was fuzzy when I took the class also!), but the one thing I still remember is how to count in hexadecimal.

Knowing how to count in hex (what the cool kids call hexadecimal) is not as important for many types of code you may end up writing today. Chances are, you may never consciously have to write it in any of your projects. At most, you will have to copy and paste some hex values that define a color. Despite its seeming rarity in your world, there is some value to learning how all of this works.

Let's Look at Normal Numbers

So, the best way I can explain how to count in hex is by looking at how we count normal numbers - numbers which are categorized as base-10. In base-10, we have our 10 numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. To represent a number like 314, you do the following:

Notice that you subdivide your number by digits. The 1’s digit is actually 10^0, the 10’s digit is actually 10^1, and so on. So a number like 314 is actually the sum of:

It is the above process that you use to represent all of the numbers you see every day in the real world. Much of what goes on inside your computer, though, is represented in powers of 2. The most common ones are binary and hexadecimal which are base-2 and base-16 respectively.

The above table approach is great for taking an existing number and breaking it down by 10’s. For non-base 10 pairs, unless you are really good at subdividing by whatever base you are in, it is very difficult to use the above method and go from a decimal number to something like a hexadecimal/base-16 number. After all, where do you draw the cut-off line for something like 314 when dealing with powers of 16? Let’s revisit our base-10 numbers and try a different approach.

Let’s do a division based method instead where you keep dividing your number by your base, in this case 10, until you get an answer of zero. It is the remainders that are helpful as you will see:


Now, you just put your numbers back together in backwards order: 3.1.4 or 314. This approach scales well across all bases (which are belong to us), so I will use this method for explaining how to convert normal numbers into hexadecimal values.

When it comes to hex, your numbering is a little odd. You start off as usual by counting up from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Once you get to 10, though, you switch to letters. 10 would be A, 11 would be B, 12 is C, 13 is D, 14, is E, and 15 is F. So your numbering goes from 0-9 and A-F to represent 0 through 15 in our decimal world.

It does get easier from there though! Let’s look at how to convert the decimal number 314 to hex using the division method:

The main thing to notice is that I am dividing by my base number, which in this case is 16. So, like before, reading our numbers backwards, we get 1.3.10 or 13A because decimal 10 is A in hex numbering. If you want to go from hex to decimal, just remember that you are in base-16 and use the earlier table approach:

You can now sum up the individual values:

That's all there is to counting in hex.


Hopefully this gives you a good understanding of how to think about representing decimal and hexadecimal numbers. I tried to make the explanation abstract enough where you can figure out what the values would be for a number of any base! Word on the street is that famous programmer Charles Simonyi prefers base -2 when interviewing candidates.

If you have a question about this or any other topic, the easiest thing is to drop by our forums where a bunch of the friendliest people you'll ever run into will be happy to help you out!


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