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Base 26


Date: 01/27/97 at 21:41:44
From: ac
Subject: Base numbers

Question 1   Consider a base-twenty-six number system wherein the
letters of the alphabet are the digits.  That is, A=0, B=1, C=2, ..., 
Z=25 in base ten.  Calculate TWO+TWO in this system.  Express your 
answer in base ten.

Question 2   Express 1997 in base twenty-six using the number
system described in Question 1.


Date: 01/28/97 at 15:20:22
From: Doctor Wallace
Subject: Re: base numbers

Dear ac,

First, you need to know what a "base" number system is and how it 
works. You can use our number system, which is base 10, as a model to 
use for any other base you want.  How does base 10 work?

Well, the base tells you how many digits you have.  Base 10, ten 
digits.  Each digit will have a symbol.  Base ten uses 0, 1, 2, 3, 4, 
5, 6, 7, 8, and 9.  Each digit in a number you write out has a value 
dependent on where it is in the number.

For example, look at 45.  The rightmost digit is the "ones" column, 
and the one next to it is the tens column.  Each digit we place to the 
left gets a value 10 times as great as the one to the right.  In 
another base, the value with be n times as great, where n is the base.  
In other words, what we're looking at is a sequence of powers.

With base 10, it's powers of 10.  With base 26, it'll be powers of 26.  
So, we have 10^0 = 1 for the ones place.  10^1 = 10 for the tens 
place.  10^2 = 100 for the hundreds.  Then thousands, and so on.

Now, to find the value of a number, we multiply the digit in that 
spot by the power of ten which it corresponds to, and add up them up.  
So, with 45, we have

        45 = (4 x 10^1) + (5 x 10^0)

Another example:

      1425 = (1 x 10^3) + (4 x 10^2) + (2 x 10^1) + (5 x 10^0) 


Okay?  Now let's look at another base.

Let's look at base 2 or binary as a sample.  Since the base is 2, we 
have 2 digits.  We use 0 and 1 for these.  The 0 corresponds to 0 in 
base 10, and the 1 to 1.  We only have two digits, so that's all there 
are.  Each place will be a power of 2.  So we'll have 2^0 = 1 for the 
ones place.  2^1 = 2 for the next place.  2^2 = 4 for the next place.

So we might have the number 111 in base two.  What is this in base 10?

Well, we have three places, so we use the first three powers of 2, 
which will be 2^0, 2^1, and 2^2.  Then we multiply out the places, 
and add:

111 (base 2) = (1 x 2^2) + (1 x 2^1) + (1 x 2^0)       

and we get

             = (1 x 4) + (1 x 2) + (1 x 1)
            
             = 4 + 2 + 1

             = 7 (base 10)

Now, to go the other way, and convert from base 10 to base 2, we have 
to express our number as powers of the base.  So we need to factor it.

Say we want to express 8 in base 2.  We know that 8 is 7 + 1.  It 
would be nice if we could just add 1 to our 2^0 place, but we can't, 
because we only have 2 digits.  Since 7 = 111 (base 2) we need to use 
another digit.  We find the highest power of our base (2) which 
doesn't go over our number.  We know that 2^3 = 8, our power is 3.  
So there will be a one in the place corresponding to 2^3 (the fourth 
digit over).  That would give us:

              1000 = (1 x 2^3) + (0 x 2^2) + (0 x 2^1) + (0 x 2^0)
           
                   =  8        +    0      +    0      +    0
 
                   =  8 (base 10)

What about 9?  To get that, we can add 1 to 8, using the 2^0
column, because when 0 is the exponent, the value is always 1.  So 
9 in base 2 is written as 

              1001 = (1 x 2^3) + (0 x 2^2) + (0 x 2^1) + (1 x 2^0)
           
                   =  8        +    0      +    0      +    1
 
                   =  9 (base 10)

I hope you can apply this little lesson to your base 26 problems.  You 
already know that you have 26 digits, and you know what they equal in 
base 10.  To find TWO + TWO, just write out the places, multiply, and 
add the way I showed you to find what this is in base 10.

To convert 1997, begin by asking yourself what is the highest power of 
26 which doesn't equal 1997?  You can use a calculator and figure this 
out quickly.

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



Date: 01/29/97 at 10:51:59
From: Doctor Lorenzo
Subject: Re: base numbers

This is really three questions:

1) Add TWO + TWO in base 26.
2) Convert a number in base 26 (namely your answer to Question 1) 
   to base 10.
3) Convert a number (1997) from base 10 to base 26.

I'll show you how to get started on (1), and work two DIFFERENT 
problems for (2) and (3).  You should then have enough info to do the 
rest yourself.

Part 1) 

O is the 15th letter of the alphabet, so O=14.  O+O = 28, which is
26 + 2.  That's one 26 and 2 1's.  So you keep the 2 (C) and carry the 
26 (B, one digit over).  In other words,

                        B
                       TWO
                       TWO
                       ----
                         C

Now keep going.  What is B+W+W (1+22+22)?  The answer is 45, which is 
BT, so you have to carry the B, and have T left over.  Then what is B+
T+T?  Keep going.

Part 2) 

Let's convert the number BAEG to base 10.  B means 1, and it's in
the 26^3 place.  A means 0, and is in the 26^2 place.  E means 4, and 
is in the 26's place, and G means 6, and is in the 1's place.  So the 
answer is

1 x (26)^3 + 0 x (26)^2 + 4 x 26 + 6 = 17,686

Part 3) 

Let's convert 997 to base 26.  26^2 = 676, and 676 divided into 997
is 1 with a remainder of 321, so 

997 = 1 x 26^2 + 321

321 divided by 26 is 12 with a remainder of 9, so

997 = 1 x (26)^2 + 12 * 26 + 9 = BMJ
      B             M        J

See if you can use my examples in Parts 2 and 3 above to complete your 
problem.

-Doctor Lorenzo,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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