Base 16

Date: 07/07/97 at 13:14:36
From: Cristy White
Subject: Addition and subtraction in base 16.

Dr.Math,

I've been playing a game called RAMA. They have these puzzles where I 
have to convert symbols to numbers. There are only 16 symbols, 0 
through 15. Knowing which symbol represents what number, they give you 
several symbols telling you to add or subtract. The confusing part is 
that you have to add and subtract in base 16. How do you add and 
subtract in base 16?  

Please help, 
Cristy White

Date: 07/10/97 at 03:30:19
From: Doctor Marko
Subject: Re: Addition and subtraction in base 16.

Hi Cristy,

In order to talk about different bases, we first need to agree on the 
digits that we shall use in that bases. So for example, in the 
traditional base 10 system we use 10 digits: 0,1,2,3,4,5,6,7,8, and 9.  

In base 16 then we must have 16 digits, and to try to simplify things 
as much as possible, we use lots of base 10 nomenclature. This means, 
for example, that 0 in base 16 represents the same thing as 0 in base 
10; we put this in equation form: 0(16) = 0(10), where the 16 and the 
10 in parentheses refer to bases.  

Similarly, we can say 1(16) = 1(10)... 9(16) = 9(10). But what now? We 
ran out of our "regular" digits once we reached 9, so we must come up 
with the other six digits. Traditionally, people have used capital 
letters A through F, but you can really adopt any system you wish.  
For consistency, though, I will use the traditional system, which 
means: A(16) = 10(10), B(16) = 11(10), ..., F(16) = 15(10).

The first thing one should know about any numbers in any base is how 
to convert them to something we know about - base 10. In base 10, the 
digit left of the decimal point is often called the singles digit, the 
one left of that is the tens digit, yet more to the left the hundreds 
digit, and so on. Why these names?  Well, I will give you an example: 
143(10) has 1 as its hundreds digit, 4 as its tens digit and 3 as its 
singles digit, which means that 143(10) = 1*100+4*10+3*1, which people 
write as 1*10^2+4*10^1+3*10^0 for educational purposes. 

The thing to notice and remember from this little example is that the 
digits on the left side of the number carry more weight (i.e. are 
worth more) than those on the right side - you'd rather have $100 then 
$10, right? :)  

Also, you should learn that the relative strength goes as the 
appropriate power of the base that you are dealing with. In general, 
if the base is m, the digit to the left of the decimal place has
m^0(=1) weight, the next one over to the left has m^1(=m) weight, and 
so on. It is important to notice that this whole scheme works together 
with our original definition of the digits, for example, just as we 
defined it: A(16) = 10*16^0 = 10(10).

Now you are ready to do the conversion from base 16 to base 10. As an
example BF(16) = 11*16^1+15*16^0 = 176+15 = 191(10)

The conversion in the other direction can be a little tricky but works 
pretty much the same way. The trick here is to figure out what is the 
highest weight nonzero digit and then take it from there. What I am 
trying to say is that 1(10) and 0000001(10) are the same number - the 
zeros in front are irrelevant.

So let us try to convert 71(10) to base 16. First we figure out how 
many digits this number has in base 16. Well, since 16^0 = 1 < 71, it 
is clear that the digit to the left of the decimal point is 
significant, though we do not know what it is yet. Similarly for 16^1. 
However, 16^2 = 256 > 71, which implies that this number must be 
0??(16) where the question marks are yet to be determined. But the 0 
is for sure, because if it were nonzero, this number in base 16 would 
have bigger value than 71(10) does. The same holds for all the digits 
to the left - they are all zeros. So now the job is to figure out the 
two remaining digits, again remembering that we want to find the ones 
with most weight first.

Okay, in order to find the leftmost digit (which has 16^1 weight) we 
need to find the largest digit which, when multiplied by its weight 
factor, yields a number not greater than 71(10). Again, if the digit 
is too large there is no hope of this number of base 16 ever 
representing 71(10). So this digit happens to be 4, thus making the 
number 4?(16) = 71(10). 

Now you since you know how to convert from base 16 to base 10, you 
can check where we are: 4*16^1+?*16^0 = 64+? = 71(10). Thus the digit 
? must be 7, and 47(16) = 71(10). The procedure is similar for 3 or 
more digit numbers except that you have to repeat the step of finding 
a digit that doesn't make your number greater than what is given in 
base 10.

Now that you know how to convert from base 10 to base 16 and back, the
question of addition and subtraction is simple. Convert the numbers to 
base 10, add/subtract, and then convert back to base 16.

But there is also a slightly quicker but trickier way to do this, and 
it is exactly the same as when you add/subtract numbers of base 10. 
So, let's review:

      35
     +17
     ---
      52

Easy, but what is important to note here is the mechanism. First you 
add 5 and 7 to get 12, but the first spot is for digits of weight 
1(=10^0) so you put a 2 there, and "carry the 1" since the 1 
represents digit 1 of weight 10(=10^1). Now, you add 3 and 1 and the 
carryover 1 to get 5.

Adding in base 16 is very similar:

     1A(16)
    +28(16)
    ---
     ??(16)

Much as in the case of base 10, you add A to 8 and get 18(10), but you 
need to put only one digit down. Since 18 is bigger than any of the 
digits of base 16 (remember there are A..F, too) you realize that this 
becomes a two digit number, 12(16). Can you verify this? So the one 
digit you put down is of course 2, and you carry the 1, since it 
represents a digit of weight 16^1. Now, 1 and 2 and the carryover 1 
add up to 4(16), which makes your addition work out to 42(16).

And the subtraction works in an exactly the same manner. Hope this was 
of some use!

-Doctor Marko,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/