The rules of subtraction involving different bases are not that dissimilar to the rules of subtraction in base 10.  Just as we “carry” when we add in base 10, and certainly when we add in other bases (as we have recently seen) then also we “borrow” when we subtract using different bases just as we do in base 10, however we must be alert at all times to the fact that we have to make certain base 10 conversions when we do this.


Let us take a look at a simple example, this does not involve any letters as it is in base 9:



Hopefully by now you’re not still in the habit of saying things like “six hundred and forty-seven minus thirty-eight” because other than in base 10 this sort of expression means nothing.


Step 1 - “7 takeaway 8 =?”,  While straight away we have a problem there is a fairly straightforward way to get round it, we “borrow” from the 4 reducing it to 3, so let us restate the problem (and also the diagram):



now we can say “17-8” but bearing in mind that “1-7” is a base 9 number and its decimal equivalent is 16, we must say that 16 – 8 = 8 and this is what we put into the answer box underneath the “units”:



The rest of this particular example is quite straightforward, 3 - 3 = 0 and we put this to the left of the 8 in the answer box, likewise 6 minus, well…. there’s nothing to take away so we simply drop the 6 down into the answer box.


Our final answer is therefore this:



In the next example, which will be considerably harder you will see the value of writing at the small “A to F” number table because you will be referring to it quite a bit:



This one looks like a real monster, but follow the rules and you’ll find that it’s not as difficult as it appears.


Step 1 - “7 takeaway 8 equals?” - Here we go, time to borrow already.  We have to borrow from the next column, in fact we have to borrow from the A, reducing it to 9.  This effectively makes our 7 into 17 but remember that we are talking in base 16 so in fact “17” has a decimal representation of 23.  23 takeaway 8 is 15 which is hexadecimal “F”:



So far so good? Okay let’s carry on:


Step 2 - this time a simple step 9 takeaway 9 equal 0, so we place 0 into the answer box to the right of the F. 


Step 3 - “B” – “A” is 1 so we enter one into the answer box to the left of the 0 that we have just entered.  At this stage our answer looks like this:


Step 4 - “6 take away 7”, we have to borrow again and this time we borrow from the E in the top line reducing it to a D and effectively making 6 into “16” but remember that our “16” is in fact a hexadecimal number, the decimal representation of which is 22 so in fact what we have to say is 22 take away 7 = 15, which in hexadecimal is F.  We enter F to the left of the one in the answer box.


Step 5 - in the final step using our table, D (which used to be E) takeaway A comes to 3 which we enter into the answer box to the left of the left most F.


The final answer is shown below in the diagram, along with all of the annotations that got us there.  


What I strongly suggest you do now is study this particular example very carefully to make sure that you fully understand it.


I hope by now it’s becoming a bit clearer to you that while we are going all through these examples we are constantly passing through interim base 10 stages.