new simplified math

Hey all,

Here are some thoughts.

If you were to add the numbers 24 and 18, how would you
do it?

Or better yet, if you were teaching an beginning math
student how to add two numbers -- what approach would
you use?

Perhaps you would start by noting that 24 = 6 * 4
and that 18 = 6 * 3. That's interesting! Both contain
the number 6. Let's go further. 6 * 4 = 3*2 * 2*2 and
6 * 3 = 3*2 * 3. Now both numbers have been broken into
prime factorizations. That's a useful thing to be able
to do. But it's getting kind of complicated looking.
Let's use the caret notation to mean 'raised to the
power of', so we have 3 * 2 * 2 * 2 = 3^1 * 2^3 and
3 * 2 * 3 = 3 * 3 * 2 = 3^2 * 2^1.

So what we are REALLY doing is trying to add:

3^1 * 2^3
+ 3^2 * 2^1
-----------
?????????

How do we do the next step? Well, the calculator tells
us that 24 + 18 = 42 and 42 is *really* 21 * 2 which is
7 * 3 * 2 which is 7^1 * 3^1 * 2^1. But don't forget
that 5 is between 7 and 3 and is also a prime number,
so we should properly represent 42 as 7^1 * 5^0 * 3^1
* 2^1.

So what we really are getting to is:

7^0 * 5^0 * 3^1 * 2^3
+ 7^0 * 5^0 * 3^2 * 2^1
-----------------------
7^1 * 5^0 * 3^1 * 2^1

Hm, it's still not entirely clear how to get the sum
here, perhaps we should simplify it a bit:

Now we know what the prime numbers are, so we
don't really actually need to write them, we
could just write the exponents and that would tell us
all the information we need to know. Let's write the
exponents as a vector why not:

<0 0 1 3>
+ <0 0 2 1>
------------
<1 0 1 1>

Hm, certainly there must be some sort of intuitive
vector operation we could apply here to get the sum
from the result. Something easy so we can teach our
student to do it in the most efficient way possible.

Any ideas? My student is getting frustrated it seems
and is looking out the window at the goats.

- Jeff