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Base 26

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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

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

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/

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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

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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