-6 times -6

Date: Thu, 03 Nov 1994 12:26:28 -0600
From: Robert Trueblood
Subject: Question

I really need to know what  the answer to this problem is.

                     -6
                X   -6
             ____________

Any clue Doc?

Robert Trueblood

From: Dr. Ken
Date: Thu, 3 Nov 1994 12:55:02 -0500 (EST)

Robert-

We're not realy sure what you mean to ask.  Do you mean x - 6^(-6)?  
Or -6 times -6?  In the former case, it's x - 1/46656.  In the latter, it's 
36, since the negatives cancel out.


-Ken "Dr." Math

Date: Thu, 03 Nov 1994 13:16:17 -0600
From: Robert Trueblood

I Mean -6 times -6

Since multiplication is repetitive addition and multiplication is impossible
without repetitive addition or memorization.

-6 + -6 =-12
-12 + -6 = -18
-18 + -6 = -24
-24 + -6 = -30
-30 + -6 = -36

So the answer should be -36

BUT:

if you draw a bar graph and start at -6 at 6 6 times you get a positive 30

Hmm can you explain your answer of +36 

Thanks

Robert Trueblood

From: Dr. Ken
Date: Thu, 3 Nov 1994 14:28:56 -0500 (EST)

Here's how it would work.

If you want to use the technique for multiplication where x times y means
you add x to itself y times, then what you've found in your example is 
-6 x 6, because you added negative six to itself six times.  

To find -6 x -6, we'd have to add negative six to itself negative six
times.  Kind of a funny concept, but we can make some sense of it.  

If we add two negative sixes together (-6 x 2), we get -6 + -6 = -12.
If we add one negative six together (-6 x 1), we get -6.
If we add zero negative sixes together (-6 x 0), we get 0.
Notice that we are increasing by sixes each time.
If we add -1 negative sixes together (-6 x -1), we get -(-6) = 6.
If we add -2 negative sixes together (-6 x -2), we get -(-6) + -(-6) = 12.

You can see the pattern then, and how -6 x -6 = 36.

-Ken

Date: Thu, 3 Nov 1994 14:38:14 -0500
X-Sender: dbonaro1@cc.swarthmore.edu

    Hi, Robert, this is Demetri.

    I must admit that I, myself, was puzzled when I read your reasoning
and, although I knew that the result is not -36 but +36, I couldn't quite
see what was wrong with your approach. I think I have it now, though, so
here goes:

    What you have done above is taken +6 times the number -6 and not -6
times (in other words, you have found the result to 6 times -6, not to -6
times -6).  Adding -6 to itself is the same as taking -6 twice. This is
exactly what you have described above, all the way until we had -6 taken -6
times.

    So, one way to think about this is to take 6*(-6) (that is 6 times -6),
find the result to this (which is exactly what you have above, ie -36) and
then consider what -[6(-6)] is. This is nothing other than the negative of
6(-6) or, if you prefer, its opposite (opposite numbers are two numbers
whose sum is 0- you may know this, but I said it just in case you don't
know). Clearly, the opposite of -36 is +36, because -36 +36 = 0. Therefore,
-6*(-6) = +36

    I realise that it is rather hard to explain things like this via email,
but I hope that you've understood what I am talking about and that I
haven't confused you even more. If I have, though, do write back and we'll
have another go at it.

    Demetri- Dr. of some sort.