A Little Math Experiment

This post will be a little different, but that’s kind of the theme lately.  I’ve never been a fan of the many modern quirks of mathematics (negative and imaginary numbers, misrepresented concepts of zero and infinity, etc.). It’s not that I can’t do them – Doing what you’re told is easy and I did so quite well. It’s that I won’t do it the way most people seem to. When you are dealing with “can’t”, you respond with the “how”. But, when you’re dealing with “won’t”, you have to respond with the “why”. People seem to forget that. Anyhow, on with the experiment.


Multiplication using Positive Integers:

3 x 3 = 3 + 3 + 3 = 9

Good so far? 3 multiplied by 3 is simply 3 positive 3’s added together.

3 x 2 = 3 + 3 = 6

3 multiplied by 2 is simply 2 positive 3’s added together. No conflict yet and it is a beautiful thing.


Multiplication using Negative Integers:

-3 x -3 = -3 + (-3) + (-3) = -9
-3 x -3 = -3 – 3 – 3 = -9
(0 – 3) x (0 – 3) = 0 – 3 – 3 – 3 = (0 – 9)

Here’s where it starts to get tricky. -3 multiplied by -3 is simply(?) three negative 3’s added together. According to integer rules, negative integers added together will always end as a negative integer. Already we see a problem that we’ll address in the next example. To continue to address this example, the second iteration is expressed as negative 3 subtracted by positive 3 subtracted by positive 3. Again, according to integer rules, this should always end up as negative. However, we see that the addition of a negative integer is short-handed to subtraction of positives.  And finally, the third iteration – The same exact thing expressed purely in functions involving exclusively non-negatives! Negative numbers aren’t even necessary to express negative values, much less necessary in anything else in my opinion. Of course, there’s more to be said about this and elaboration would be warranted, but let’s keep this short.


-3 x -2 = -3 + (-3) = 6?

No need to break this one down into alternative iterations as you should be aware of them by now. There is a much more glaring problem here. -3 multiplied by -2 is negative 3 plus negative 3, negative 3 minus 3, whatever… or is it? According to “Da Rules”, a negative integer multiplied by an even negative integer equals a positive integer. That’s the how. But why? Why is this the case when we saw in our previous examples of both positive and negative integer multiplication that the multiplication operation is shorthand for addition. Even when the operation is explained in terms of scaling, again it falls short. For no reason whatsoever and in direct opposition to its own rule, -3 multiplied by -2 is positive 3 plus positive 3.



This is just a small and simple example of a principle of modern mathematics that stumbles over itself. For me to address all objections would mean going beyond the scope of this post and tackling reality-based computation and the nature of zero, nothing, and infinity. I’m not an expert on this stuff but there are those who are. Leave a comment below with what you think and your reasoning for where you stand on this. For further fun, talk to your school instructor or professor and get their take on it. Remember, the “How” has already been addressed – The main concern is “Why”. If that’s not something you care about, I understand, but to me it’s very important for humans in general to consider the question behind the action as time and situation allows.



~ by demonhide on May 12, 2013.

2 Responses to “A Little Math Experiment”

  1. -3 x -3 is not -3 + -3 + -3
    -3 x -3 is -(3 x -3) = -(-3 + -3 + -3) = -(-9) = +9
    (0-3) x (0-3) is not 0 – 3 – 3 – 3 = (0 – 9)
    (0-3) x (0-3) is (0x0) + (-3 x 0) + ( 0 x -3) + (-3 x -3) = 0 + 0 + 0 + 9 = +9
    -3 x -2 is -(3 x -2) = -(-2 + -2 + -2) = -(-6) = +6
    or -(2 x -3) = -(-3 + -3) = -(-6) = +6
    Negative numbers, zero and infinity, imaginary et al. are very old concepts in maths; albeit relative – perhaps you don’t think 300-1000 years very old.
    Maths can’t really be learned by just applying rules if you want to get further than the elementary stuff; learning maths is intrinsic – you have to come to understand each of the ideas for yourself to be able to move on.
    Questioning and asking why of every detail is good as long as you are actually thinking things out for yourself; not so good if you are just playing Devil’s advocate and being lazy about learning it for yourself.
    Interesting views.

    • Excellent! I had not considered putting it that way. There seems to be interest in this. I’ll continue the dissent in future posts in between regular content, perhaps with a little more time spent on my part getting the concepts down. But no promises :)


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