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### Operations in Nondecimal Bases

```
Date: 10/16/1999 at 20:02:32
From: Tommy Paley
Subject: Operations in non-decimal bases

Hi doctor,

I understand how to add numbers in other bases, but I was wondering if
it was possible to subtract, multiply, and divide numbers in other
bases?

I tried to multiply the base 9 numbers:

35
x 28
-----

When I get 5 x 8 = 40, where do I go from there? I was thinking that
40 = 4(9^1) + 4(9^0). So I thought I'd place a four in my answer, and
carry the other 4:

4
35
x 28
-----
4

and then I went 3 x 8 + 4 = 28 = 3(9^1) + 1(9^0). Is this right so
far? If it is, I think the answer should be 1124.

Can you show me how to divide and even square numbers in other bases?

Thanks in advance. I am looking forward to teaching my math 9 classes

Tommy
```

```
Date: 10/16/1999 at 22:58:12
From: Doctor Peterson
Subject: Re: Operations in non-decimal bases

Hi, Tommy.

You'll want to look through our FAQ on bases,

http://mathforum.org/dr.math/faq/faq.bases.html

It includes some help on operations in various bases, especially
binary, as well as a long discussion of "adding in hexadecimal." Some
of the other information on the meaning of bases and how to convert
may be of use to you as well.

The operations all work the same in any base; the only things that are
different are the tables. Your multiplication is exactly right; it
might go faster if you actually wrote up a multiplication table in
base 9, so you could see instantly that 5 * 8 = 44 (base 9). That will
especially help if you try dividing in base 9 - a good table is a
must, as you may recall from your early experiences with base 10
division.

Actually, the operations are almost trivial in binary, because there's
practically no table to learn; that allows you to focus on the
algorithm alone, so it can be a great teaching tool for kids who
follow binary well in the first place, but need to see the
multiplication and division algorithms more clearly. I sort of taught
my son to multiply in binary before I taught him to do it in decimal.

Since we already cover operations in binary pretty well in our
archives, let's try a different small base. I'll use 3. First we make
the tables:

+ | 0 | 1 | 2 |     * | 0 | 1 | 2 |
--+---+---+---+     --+---+---+---+
0 | 0 | 1 | 2 |     0 | 0 | 0 | 0 |
--+---+---+---+     --+---+---+---+
1 | 1 | 2 |10 |     1 | 0 | 1 | 2 |
--+---+---+---+     --+---+---+---+
2 | 2 |10 |11 |     2 | 0 | 2 |11 |
--+---+---+---+     --+---+---+---+

Now for subtraction, the only trick is borrowing. In decimal, you
borrow by subtracting one from the column to the left, and adding 10
to the column you are working in. Here, you do the same thing,
remembering that 10 now means 3. Or if you want, you can add 3 in
decimal to the digit, since within one digit the base doesn't matter.
Let's subtract 10 from 15:

13
//
15 -->   120
- 10 --> - 101
----     -----
012 --> 1*3 + 2 = 5

You've shown me how to multiply, but I'll write one out in base 3 so I
can use it for division:

20 -->   202
* 15 --> * 120
----     -----
000
1111
202
------
102010 --> ((((1*3 +0)*3 +2)*3 +0)*3 +1)*3 + 0 = 300

(Here I've used one of the easy conversion methods in the FAQ.)

Now let's divide:

_____120_
202 ) 102010
202
---
1111
1111
----
00

Since there's such a small table, this was pretty easy - all I can
multiply a number by is 0, 1, or 2, so it's easy to guess a digit for
the quotient.

To square a number, of course, you just multiply it by itself.

Hope this helps. Let me know if you need any more ideas for teaching
this subject - it's one of my favorites. (I work with binary and
hexadecimal all the time in my programming job.)

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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