Another quick entry, today, while we are “vacationing” here in sunny Texas. In this one I want to follow up on a threat I made an entry or two ago: I will demonstrate why a negative times a negative is a positive. That is, why is (-2) × (-3) equal to positive 6 and not -6?
[I know that some of you are groaning that I am doing another math-related entry. (Yes, I can hear you.) I do apologize, but I really wanted to follow up on that. And besides, I am (sort of) on vacation and this stuff is my way of relaxing, so I’m in the mood for it. If you are a faithful blog visitor and don’t think you can take another math entry without pulling your hair out like Ezra (or someone else’s, like Nehemiah) – feel free and skip this one. I won’t hold it against you! 🙂 ]
Some people often think that “a negative times a negative is a positive” is just some weird “math rule” and take it as a matter of faith. Others have been given odd little analogies as to why it makes sense (I have certainly come up with my share). But those analogies often seem artificially contrived – and for good reason, because they are artificially contrived. They don’t satisfy, because they don’t prove WHY the “rule” is true. They are just meant to make us feel comfortable about what is, for many, a “rule” that goes against their intuition (somewhat like the “just so” stories that are told about evolution).
It seems illogical, yet math is all about the logic. So let me demonstrate for you that the things we like to believe about arithmetic require a negative times a negative to yield a positive. That is, let me show you WHY…
First, there are five key math facts that we have to recognize first. To state them generally, we will use “a”, “b”, and “c” to indicate numbers in general, so that when you see “a” think “that could be a 0 or a 1 or a 2 or a 3… or even a 2,718,281. Plug in your favorite numbers and try them out. Here are those preliminary facts:
1. For each number a there is a number called the additive inverse of a which we write as “–a” such that if you add them you get zero. That is,
a + (-a) = 0.
Example: 3 + (-3) = 0.
(Note: We could have written that as “a + –a = 0”, but we tend to put parentheses around negative numbers when we are concerned that there may be some confusion caused by seeing the minus sign. We want to make sure that we are being clear that the “minus sign” is, indeed, attached to the number.)
2. For every number a, it is true that
a × 0 = 0.
Example: 5 × 0 = 0.
3. The operations of addition (+) and multiplication (×) are commutative, meaning that the two numbers can be reversed and the same result will be obtained. That is,
a + b = b + a, and
a × b = b × a.
Example: 3 × 2 = 2 × 3. (Both result in 6.)
4. Equality is transitive. That is, if a = b, and a = c, then b = c.
Example: if x = 2, and x = y, then it must be true that y = 2, also. I say “also” instead of “too” because this is not a ballet-oriented website. (If you don’t get that joke I just attempted, just think about it. Then again, don’t bother. Sadly, it is not that funny.)
Most of the above is probably familiar to most of us. The next one might be a bit fuzzier for us, but if you look back in your school books, it’s there!
5. Multiplication distributes over addition. That is,
a × (b + c) = (a × b) + (a × c).
It is also true that
(b + c) × a = (b × a) + (c × a),
since, as we said in point #3 above, multiplication is commutative (the order can be switched).
Example: 2 × (3 + 4) = (2 × 3) + (2 × 4). (Both expressions result in 14: 2 × 7 = 14 and 6 + 8 = 14.) (EDIT: A big thanks to DY for noticing I origially had “6 + 6 = 14” here (see comment, below). A second pair of eyes always helps! 😉 )
These things said, what we ultimately want to demonstrate in order to establish that a negative times a negative really is a positive is that
(-a) × (-b) = a × b.
That is, we want to establish that multiplying two negative numbers gives the same result as multiplying the two positive numbers. For example, we want to show that (-3) × (-4) equals the same result as 3 × 4. This would imply that (-3) × (-4) equals positive 12, since 3 × 4 equals positive 12.
So here we go. If you have always wanted to know WHY a negative times a negative is a positive, here is the reason. (I should also say that there might be better proofs, but this is one I really like. I saw it in Jerry P. King’s The Art of Mathematics, I think, and I like it for its simplicity.)
To Be Proved: (-a) × (-b) = a × b.
The Proof: Consider the following expression:
[(-a) × (-b)] + [(-a) × b] + [a × b].
Let’s call this “Expression 1.” What does it equal? I mean, it looks unnecessarily complicated. How can it be simplified?
Watch how we apply the rules discussed above to rewrite this expression in a more simple form (I will do this as a series of equations, keeping “Expression 1” on the left side, and writing the step-by-step simplified version of the expression on the right):
We can rewrite Expression 1 using the following steps:
[(-a) × (-b)] + [(-a) × b] + [a × b] = (-a) × [(-b) + b] + [a × b]
[(-a) × (-b)] + [(-a) × b] + [a × b] = (-a) × 0 + [a × b]
[(-a) × (-b)] + [(-a) × b] + [a × b] = 0 + [a × b]
[(-a) × (-b)] + [(-a) × b] + [a × b] = a × b
So, our original expression, “Expression 1,” is equal to a × b when simplified.
The steps we followed in those series of equations are these: We “undistributed” the “-a” to get the first equation, we noted that (-b) + b = 0 to move to the second equation, replaced (-a) × 0 with 0 to get the third equation, and then noted that adding 0 to anything just yields that very same anything. These were the rules we agreed to up above.
However, we could have simplified it differently. For instance, we could have “undistributed” the “b” from the latter two parts as a first step, instead. Let’s look at how that would progress (noting that we will use exactly the same rules, just in different places in the expression):
[(-a) × (-b)] + [(-a) × b] + [a × b] = [(-a) × (-b)] + [(-a) + a] × b
[(-a) × (-b)] + [(-a) × b] + [a × b] = [(-a) × (-b)] + 0 × b
[(-a) × (-b)] + [(-a) × b] + [a × b] = [(-a) × (-b)] + 0
[(-a) × (-b)] + [(-a) × b] + [a × b] = (-a) × (-b).
Therefore, “Expression 1” evaluates to (-a) × (-b), as well. So, if
[(-a) × (-b)] + [(-a) × b] + [a × b] = (-a) × (-b)
[(-a) × (-b)] + [(-a) × b] + [a × b] = a × b
it must be true that
(-a) × (-b) = a × b.
At this point, mathematicians like to say stuff like “Q.E.D.”, which is an abbreviation of some Latin words that mean, “Which was to be demonstrated.” Though in one of my college classes, we liked to end with “G.E.A.” for “Gig’em Aggies.” 🙂
So, we see that when you multiply two negative numbers together, you must get the same result as when you multiply the two “positive versions” of the same numbers together. That’s why (-2) × (-3) equals positive 6. It must be so, because logically the unusual looking (-2) × (-3) must give you the same result as more familiar 2 × 3 – which is still 6, no matter what lawyer or accountant you talk to.
There! That wasn’t too painful, was it?
BTW: If you’re a “math guy” or “math gal”, please don’t write me and mention that I could have been pickier and brought in the associative law, or that the transitive rule is normally written differently, and all sorts of other stuff like that. I suspect that I was pickier than I needed to be, anyway, but I did want to make sure I got across the correct impression that math follows laws and accepts the outcome, “odd” or not.
The point is that a negative times a negative is a positive BECAUSE if we agree that we like the other “rules” then this rule is a necessary consequence. If we don’t like having a negative times a negative be a positive, then we need to decide which of the other five facts I listed we are going to throw out (which would be a shame, as they are very polite facts, and they don’t normally cause much trouble when they are treated nicely). For instance, we could throw out fact or rule #1 and get rid of negative numbers entirely. But if we want to keep those rules around (which I recommend), then we must accept the logical consequences of that decision. We can’t keep the other rules and ignore their consequences. I know that ignoring the consequences of your actions is one of the great American pastimes, but it just doesn’t fly in math…
And that’s it! I think I have purged myself of the compulsion to write mathy stuff for a while, so the next entry will, hopefully, be more palatable to more normal folks. If you feel you are owed something by having read all of this, feel free to ask me to read an essay on one of your favorite-yet-boring topics as payment. 🙂
(And I apologize for saying this would be a “quick” entry. I think I underestimated that a bit…)