Just as in base 10, basic arithmetic operations such as add subtract multiply or divide can also be carried out on numbers in other bases, but you have to be constantly on guard that when you’re performing any part of the calculation you don’t allow the answer, or any part of it to exceed the base that you are working with.


For example, if you’re working in base 7 and you somehow end up with a 7 or above in the answer or part of it then you’ve gone wrong and you will need to recheck your working.


Just as a basic recap on a previous section, I’m going to carry out a very basic addition in base 10, then I’m going to carry out the same addition in another base.


Consider the simple addition below:


Well, because you’ve been reading this wonderful publication you know all about addition in base 10 and you’re saying to yourself “easy peasy lemon squeezy” (or at least something similar!)  and you’ve already worked out that the answer is correct.


Okay, let’s do the same addition, but this time in a different base:



Anybody flicking open the book at this page would look at that addition and would say “that can’t possibly be right!”  But it is if you’re thinking in base 5!  Let’s go through the steps of the addition and work out why:


Step 1 - 2+ 1= 3, and because the answer does not exceed the base value we write it as it is, so we put a 3 in the answer box.


Step 2 - 4+3= 7, but we can’t exceed 5 because that’s our working base so we ask ourselves this question “what is 7 in base 5?”,  Well it is “12” because the 1 represents a 5 and 2 represents the units so this time we enter 2 into the answer box and we carry the 1.


Step 3 - 3+ the carried 1 equals 4 and because this doesn’t exceed the base value 5, we write it in the answer box as it is.


Our answer is therefore “4 2 3”, not “four hundred and twenty-three” because such an expression is only valid in base 10.


I suggest that you spend a few minutes reading over these last few pages again just to make sure that you got the principles of working in different bases properly, then read on where you will find one or 2 further examples in different bases to either base 10 or base 5.


In the further examples that I do, I will in fact append the base as a suffix as seen previously because this way it takes away any ambiguity as regards which base you are actually working with.


Let’s take a look at an addition in base 7:


This time I have included base suffixes, or subscript but please note that the subscript does not just apply to the 5 on the top line and the 4 the bottom line, the subscript applies to the whole of the top line “665” and the bottom line “54” because the whole number is base 7 not just the (what would probably be regarded as) the units bit.


Okay, let’s get on with the addition then:


Step 1 - remembering all the time that this is a base 7 problem, “5+4”= 9, but we can’t have a 9 in base 7 so convert 9 (because this is a base 10 number) into its base 7 equivalent which is “12”.


We place the 2 in what would be the units column and we carry the 1:

Step 2 - we now add up the central column and we say that “6+5+1 = 12”, but once again this isn’t 12 because it’s not denary, it is in fact a base 7 number and we can establish that the base 7 equivalent is “15”, we put the 5 into the answer box to the left of the 2 and once again we carry the 1:



Step 3 - we now total the third column 6+ the carried 1= 7, but of course in base 7 we can’t have 7, the base 7 equivalent of the decimal 7 is in fact “10”, and this is what we put to the left of the 52 in the answer box:



When we start to look at bases higher than 10, we start to introduce letters as you seen previously because we need representations for the decimal numbers 10, 11, 12 and so on.  Earlier on in the section I showed you that the letters, at least for the first dozen or so bases above 10, are the capital letters A, B, C and so on, where the values are as follows:



It would appear that the letter “I” is used to represent “18” but I’m sure that have seen somewhere on the Internet that it is discouraged, in any event as long as you are careful and consistent your calculations should not suffer.

In calculations involving bases higher than 10, my advice when you write the question, or if the question is already written for you, at the side of it in a small table written by yourself would be to make a note of the letter values up to and including the base value less 1 so that when you come to use letter manipulation you have a little guide to work from.


Let’s take a look at an example:




This is probably something you’ve never seen before, and ordinarily you probably wouldn’t have a clue where to start but it is in fact a base 16 (hexadecimal) addition problem where instead of hundreds tens and units, we have 256’s, 16’s and units.


We will just approach this problem in the usual way:


Step 1 - starting from what would be the “units” column (i.e. the right-hand column) we say “F + E = ?”.  Straight away the table comes in useful, because we can see that the answer here is in fact 29, however, this is a decimal, or denary 29 which we need to convert to base 16.  The hexadecimal equivalent of the denary number 29 is “1D” so we enter the D into the answer box underneath the “units”, and we carry the one:

Stop for a minute and satisfy yourself that 1D in base 16 is equivalent to 29 decimal (think about it as(1 x 16)+(13 x 1) = 29).


Step 2 - we now move across to the next column then we say to ourselves “C+E+1” and again we refer to the table to come up with an answer of 27, but once again remember that this is a decimal 27 which we now need to convert to hexadecimal, the hexadecimal equivalent of the denary 27 being “1B”.  We enter the B into the answer box to the left of the D and once again we carry the 1:




Step 3 - 7+ the carried 1 equals 8, there is no conversion to do on the 8 because it is way less than the base of 16 and so it is entered into the answer box, to the left of the B as it stands.  The answer to question is therefore:



Have a go at this next one yourself, satisfy yourself that the answer given is in fact correct:


9FE16 + 7A616 = 11A416