Why Is a Negative Times a Negative a Positive?

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)

and

[(-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…)

22 thoughts on “Why Is a Negative Times a Negative a Positive?

  1. Hey, your blog, your rules. You can prove Fermi’s last theorem for all I care. 🙂 Or did someone already do that? (Better yet, find a way to put e, i, phi and pi all in one equation without doing the cheap slight-of-hand that the equation I sent you does.)

    And if you would like to try an interesting test that requires negative numbers in order to work (even though that only means that the results are displayed on a Cartesian graph), go here:

    http://www.politicalcompass.org/

    I rated very close to center on the Compass: just slightly above and to the left of the origin. That’s when I put in the balanced ideals I strive for as a Christian. When I redid the test and just put in my carnal knee-jerk reactions, I ended up being economically centrist and socially highly authoritarian. Just one more reason to make sure the old man stays dead…

  2. Doug Young

    Hmmm, I can see you are a math expert…. 6 + 6 equals 14! LOL! Seriously, thanks for the math insight and the great work you do for the church.

    Doug Young
    San Diego

  3. Good proof, Mr. Smith! Here’s another one, slightly unrelated:

    x^2 – x^2 = x^2 – x^2

    x * (x – x) = (x + x) * (x – x)

    x * (x – x) = (x + x) * (x – x)

    x = x + x

    1 = 2

    Have a great day! 🙂

  4. Mr, Young —

    Thanks for the comment, and the correction! Nice to know I have folks looking out for me! 🙂

    I have edited the entry to reflect your observation (see above), lest I lead anyone astray… (Does Mark 9:42 apply to math stuff, too? I *really* hope not…)

    Thanks, again!
    Wallace Smith

  5. Mike —

    You are a bad, bad boy. Dividing by zero (x – x = 0) is a capital mathematical offense! Once you start throwing the rules out, you can prove anything. For example, in a story attributed to various mathematicians (such as Hardy or Russell), a professor says (something to the effect of), “If you assume 1 + 1 = 1, you can prove anything.”

    “Then prove you’re the pope!” one student shouts out.

    “Alright,” the mathematician says after some thought. “Assume 1 + 1 = 1. Since 1 + 1 also equals 2, then 2 = 1. The pope is one, and I am one. The pope and I are two. Therefore, the pope and I are one.”

    There are lots of versions of that story — maybe in the second resurrection, we can track the original teller down! (I like to think it really was G. H. Hardy.)

    Thanks for the comment —
    Wallace Smith

  6. Hey, wait a minute…on the Divine level, 1 + 1 = 1, and 1 + 1 = 2. God the Father is one, and Jesus Christ is one. God the Father and Jesus Christ are two. Therefore, God the Father and Jesus Christ are one. (No wonder Rabbinic Judaism rejected any professing Christian theology that denied God’s absolute unity.)

    In fact, in our theology some day (1 + 1 + 1 + 1 + … to the limit n) = 1. God the Father will be one, Jesus Christ will be one, the resurrected saints each will be one, they all will be n, and they all will be one. (No wonder people think we’re crazy for saying that.)

    OK, Mr. Mathematician, resolve THAT. 🙂 (I hope I haven’t misstated the math in the first place!)

    Of course you realize what I’m leaving out: the difference between unity of Person and unity of Being, the latter of which is made possible by the Holy Spirit. In theological language, God is one Being, but more than one Person. By the same Holy Spirit, many more Persons will be added to the one Being. (Incredible, isn’t it?)

  7. Howdy, Mr. Wheeler —

    Hopefully you will give me the leeway to suggest the following without considering me theologically committed any particular mathematical ideas…

    Perhaps with the Father and Son (and the future family expansion, as well), we are looking not at 1 + 1 = 1, which is false given the assumption domain used in this blog entry, but rather something more akin to

    \aleph_{0}+\aleph_{0}=\aleph_{0}

    (or, if that is not as visible as I’d hoped, aleph null + aleph null = aleph null)

    which is true (under at least some additional reasonable assumptions). Just as 1 is the first finite cardinal number, aleph null is the first infinite or transfinite cardinal and, being infinite, it doesn’t obey the same rules as mere “mortal” cardinals. (You, especially, might also appreciate the fact that it is mathematical convention to denote this “infinte unit” with the Hebrew letter aleph.)

    If you have never read of the transfinte cardinals or of the concept of “different sizes of infinity,” you might consider perusing the very brief Wikipedia article:

    http://en.wikipedia.org/wiki/Transfinite

    and the related article on Absolute Infinity:

    http://en.wikipedia.org/wiki/Absolute_infinite

    Playfully submitted,
    Wallace Smith

  8. Ed Ewert

    I must admit I’m one of those who are put off by what appears to be a non-symmetric relationship between the positive and negative numbers, and I can appreciate why historically, there was a reluctance to publish a negative number even if the arithmetic involved used negative numbers. I tried searching on the internet for a radically alternate approach involving negative number arithmetic (and extentions thereof), or explanations that are of a much deeper nature, but I gave up after looking at a hundred or so sites which just had standard explanations.

  9. Howdy yerself, Mr. Smith! And happy Sabbath!

    Yes indeed, I can read the Hebrew letters just fine. And I can read the words in the Wiki articles, but doing so is like my trying to parse Greek: I understand the syntax, but what do the words actually mean? 😉

    I’ve heard of the term “transfinite” (thanks to C.S. Lewis on the one hand and Star Trek on the other) and “different sizes of infinity” thanks to a magazine article on Cantor (allegedly, that idea drove him insane), but it was interesting to see the concepts spelled out (what little I could understand of them).

    At least what you say here makes sense to me. I found it interesting that Cantor considered the Absolute Infinite as equivalent to God. (Other links from that page started leading me to Godel’s ontological proof and other things, but I realized I’d better not get in over my heard and stopped.)

    שלום
    יוחנן רכב

  10. Mr. Ewert —

    I can appreciate your comments. I was told by a professor many years ago (lifetimes ago?) about a mathematician who was so utterly despondent about negative numbers that he walked into the ocean and drowned himself. (I have not been able to find a reference to that story online, and I might have about it in William Dunham’s excellent book, Journey through Genius: The Great Theorems of Mathematics — GREAT book!)

    A taste for how the negative numbers have been regarded throughout history from the dawning of their discovery can be read at the Wikipedia entry on negative numbers — specifically in the “First usage of negative numbers” section on the page.

    I’m rather fond of them, myself, and am glad to have them around, though I do sympathize with those for whom they cause discomfort — especially for reasons such as you mention.

    Thanks for writing!
    Wallace Smith

  11. From the Wikipedia article:

    > In Hellenistic Egypt, Diophantus in the 3rd century CE referred to the equation equivalent to 4x + 20 = 0 (the solution would be negative) in *Arithmetica, saying that the equation was absurd, indicating that no concept of negative numbers existed in the ancient Mediterranean.

    It’s surprising that this much earlier reference isn’t cited (and perverted, as I suspect it often is):

    (Ecclesiastes 1:15 ESV) What is crooked cannot be made straight, and *what is lacking* [*chesron (חסרון), deficiency] cannot *be counted* [verb root *manah (מנה), here “to count, number”].

    But this is not a denial of the possibility of negative numbers. Solomon is saying, in effect, that what is zero in quantity (and thus is “lacking”) cannot be assigned a positive real value (and thus be “numbered”). People back then certainly had a concept of and a way of reckoning (say) debts, just not a direct symbolic means of representing them by negative numbers. It seems to me that the lack of such means was more a matter of logical fallacy than of asymmetry in physical reality. When the fallacy was uncovered, the relevance of negative numbers to physical reality became clear. On our level of daily existence, the Universe couldn’t function if there could be negative real values for the number of objects (try to imagine -5 apples in a bowl, for example). But I’m sure a physicist could point out that on other levels, the Universe couldn’t function without real negative values (and even some imaginary values) for many physical quantities.

    I confess I don’t understand what the fuss is about. But then, I think like an analytical geometer — positive and negative numbers make perfect sense on a Cartesian graph, and that’s enough for me. If I walk north from my house, I could consider myself going, say, +1 miles on a y axis. If I walk south, I could consider myself going -1 miles on the same y axis. Either way I am going a real, positive distance, but then the negative number represents the direction in which I’m going, and that is no less relevant to the physical order of things.

    Shabbat shalom (שבת שלום),
    John Wheeler (יוחנן רכב)

  12. P.S.: Sorry for this short insert, but if I’ve misused the mathematical term “real” (or any other) anywhere above, then you are free to edit it out or correct the concept in whatever reply you give.

  13. Austin House

    I might just be being stubborn, but I fail to see how any of this proves ANYTHING. First off, where the heck did all those extra
    bits come from in your example equation? Further, correct me if I’m wrong, but -a does NOT equal a. Certainly, if I put that on
    my final exam, I’d get the whole question wrong. Simplifying it down may give us two different versions, but doesn’t that present
    a contradiction? In Algebra, if I boil down an equation and get “a number equals another number”, I’m required to call it so. So,
    in the end, how exactly does -a x -b equal a x b? Why are some rules held in place, while others are ignored? Please let me know?

  14. Greetings, Mr. House! Let me help where I can. First, some basics:

    (1) What you would get on your exam would depend on what you were asked. If you were asked, “What is the capital of Texas?” you are correct: you wouldn’t get much. 🙂 But if you were asked to demonstrate that a negative times a negative is a positive, then you should get full credit, because the content of this post does exactly that.

    (2) You are correct: –a does not equal a (unless a = 0). However, I did not say anywhere that it did, as best I can tell. If I did, please help me find it so that I can correct it. Thanks for the help!

    (3) The “extra bits” you refer to in the starting expression are designed exactly so as to produce the final result. It is, admittedly, contrived, but that’s exactly what scientists do (mathematicians are a kind of scientist) — create situations and see what happens (a little overgeneralized there, to be sure). The situation in this expression is set up so that it can be evaluated in different ways by starting in different directions.

    This shouldn’t seem so strange. If I said to simply 12/8 to a mixed number, you could go in two different directions: (A) you could reduce it to 3/2 and then change that to 1½; or (B) you could first change it to 1 4/8 (one & four-eighths) and then reduce that to 1½. Either direction you go, you will, of course, get the same result.

    That’s the key: No matter what order we apply the rules in, as long as the rules are good rules and we apply them properly, the result always has to be the same number, even if it looks different. (That numbers can look different but still be the same number shouldn’t be shocking. For instance, ½ = 0.5.) And in the expression above, since simplifying the expression one way leads to (-a) × (-b) and simplifying it another way leads to a × b, it has to be true that (-a) × (-b) = a × b.

    This is the same as saying that a “negative times a negative is a positive.” For instance, it tells us that (-2) × (-3) = 6, since (-2) × (-3) must give the same result as 2 × 3, which is 6.

    (4) Getting back to your questions, you asked if getting two different “versions” means that there is a contradiction. No it doesn’t; rather, it means that those two different “versions” are actually the same number. That is, they may look different, but they must represent the same number.

    (5) You are right, if “‘a number equals another number’, I’m required to call it so.” That’s correct! And since, in this case, the number we started with equals (-a) × (-b) AND equals a × b, then it must be true that (-a) × (-b) and a × b are the same number.

    This comes from another property in math: the transitive property of equality. Formatted for our situation, it says that if A = B and A = C, then B = C. For instance, if x = 2 and x = y, then it must be true that y = 2.

    Put another way, if one expression equals one number AND another number, then those two numbers must be the same number. Otherwise, as you have suggested, there would be a contradiction.

    To try and summarize all of this… You are correct in your intuition: the expression that was created for simplifying is contrived. It is purposefully designed to be able to be simplified in two different ways. But this implies that the final results of these two simplifying efforts must be equal to each other, since simplifying a number shouldn’t change the number at all. Therefore, the result of one, (-a) × (-b), must be the same as the other result, a × b.

    This implies that the result of multiplying two negative numbers together is the same result you would get if you multiplied the two positive numbers together: a positive number.

    I know it’s hard to get it all just by reading it in a blog without any verbal interaction or explanation. If you are really desperate and this is not doing the trick, let me know in a further comment, and I’d be glad to explain in another format. Regardless, thanks for writing!

  15. Hello Mr. Smith!

    This is Larry Bruce. I met you at the 2012 F.O.T. at Lake of the Ozarks.

    Here is my version of a rather simple proof about the product of two negative numbers. It is presented like a geometric proof or a proof in Set Algebra, similar to what I learned in high school. (Please don’t laugh; I’m an old guy!)

    On the other topic, there is a website that shows how the Fibonacci numbers appear in the table for Pascal’s Triangle. I will find the link in a bit and print it here.

    I hope this proof has some value.

    Enjoy!

    Proof that the product of two negative numbers is a positive number:

    1. (-1) x c = -c definition; by convention (this is the customary way to write it)

    2. a x b = ab by convention

    3. let c = ab
    (-1) x ab = -ab by #1

    4. (-1) x (a x b) = -ab by #2

    5. [(-1) x a] x b = -ab associative law

    6. (-a x b) = -ab by #1: [(-1) x a] = -a

    7. ab + (-a x b) = -ab + ab = 0 add (ab) to both sides

    8. (-1) x ab + (-1) x (-a x b) = 0 multiply each side by (-1)

    9. -ab + [-a x (-1) x b] = 0 by #3 and the associative law

    10. -ab + [-a x -b] = 0 by #1

    11. ab + -ab + (-a x -b) = 0 + ab add (ab) to each side

    12. (-a x -b) = ab That’s All Folks!

    Larry Bruce

    ifosz@sbcglobal.net

    P.S. You sure are right about the texting in this program. My “That’s All Folks!” was in script similar to the Bugs Bunny Cartoons.

  16. Comment allez-vous, cette apres-midi?

    How’s that for a greeting?

    I realize it is the Sabbath where you are, but here in central California, it is still late afternoon.

    In my proof about the product of two negative numbers equaling a positive, the format went greatly awry. The original was nice and neat, but it lost some of its beauty when I copied and pasted it on this webpage. If you want, I can email it to you to someplace??? in its original form so it is easier to read and looks more elegant. It is my own proof, of course, and I did like it, but not because it is my own.

    It is fun to read about the inner workings of math. The number e is especially astounding. I will have to read that book on “The Story of e.”

    Thank you very much.

    Larry Bruce

What are you thinking?

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s