Who Decided that the Product of Two Negatives Should be a Positive?

By Deane Barker on May 16, 2007

Why is the product of two negatives a positive? I understand why this —

-5 * 6

— should be -30. It’s because if you add -5 to itself 6 times, you come up with -30.

But, one must wonder, why is the inverse so odd? How do you “add” something to itself a negative number of times? These leads us to the odd situation of -5 * 6 making sense, but 6 * -5 seeming weird.

These leads me to wonder who decided that two negatives reverse polarity and become a positive. Multiplication is a shortcut for addition, right? “Two times three” is the same as “two plus two plus two,” just shorter.

But the “shortcut for addition” theory breaks down with negatives because you can’t add something to itself a negative number of times. So it feels like this one was arbitrary. Someone just decided, apparently, that multipltying two negatives makes a positive. I can’t find a logical truth to it.

Perhaps it just felt right.

Gadgetopia

Comments

  1. But the ?shortcut for addition? theory breaks down with negatives

    it does.

    you can use this, however, to discover what happens by following patterns (and this is something your kids math teacher should be teaching them – at least this is how i teach my students at the same age this)

    you understand the positive times a negative via repeated addition. look at the sequence:

    5 * -4 = -20 4 * -4 = -16 3 * -4 = -12 2 * -4 = -8

    there is an immediately recognizable pattern, both on the product side and the answer side. you may want to use a number line to make sure you follow the pattern on the answer side. continuing with this pattern for a couple more steps should illustrate that when the multiplicand of -4 becomes negative, the answer will become positive.

  2. btw – this seems to be awfully late in the year to be learning some of this stuff. is the teacher cramming for a state test right now?

  3. there is an immediately recognizable pattern, both on the product side and the answer side

    Yes, it’s “balanced.” But I still don’t think it’s logical. It seems that a lot of arbitrary rules in math are that way just for the sake of balance. Would you agree?

    this seems to be awfully late in the year to be learning some of this stuff. is the teacher cramming for a state test right now?

    Oh, I’m way off the curriculum at this point. Order of Operations was about a month ago, and that just sparked the whole “arbitrary rules” thing in my head, which is what has led to all these subsequent questions. It has no correlation to what my son is learning right now (they’re on probabilities now, I think).

  4. It’s not that arbitrary, certainly not once you accept treating multiplying positive numbers as adding a number to itself a certain number of times.

    Just look at the multiplication not as “shortcut for addition”, but as “shortcut for repetition”.

    So 35 is 3 * (+5), and you repeat the “+” operation (adding) on 3 for 5 times. And -3-5 is then repeat subtracting (the “-” operation) -3 for 5 times. If you accept that subtracting a negative is adding (i.e. that -(-3) is +3 ) , then you’re done.

    Though, if it would make you happier, I do think that the whole “shortcut for whatever” analogy would drop dead quite quickly if you’d mix imaginary numbers into it. ;-)

  5. Do you understand that multiplication is commutative operation for Integers? a * b = b * a. That’s why you can add a number b times, or b number a times. That should explain how you can add 5 number -4 “times”. 5 * -4 = -4 * 5.

    Also negative number you can get by multiplying some number by -1 -4 = (-1) * 4.

    So if x<0 and y<0, then x * y = (-1) * |x| * y = -(1) * (add y number |x| times).

    For me all this number stuff seems very logical.

  6. Things get even more fun when you introduce “i” (square root of negative one).

    Then you get to prove fun things like:

    -2i ln (i) = 3.14159… (pi)

    or, rephrased:

    i ^ i = e ^ (-pi/2)

    I think a lot of this stuff is cool, but I’m often reminded why I chose an applied field rather than a theoretical one.

  7. The formatting swallowed some of my multiplication signs. Oh, well, still should be clear enough…

  8. Things get even more fun when you introduce “i” (square root of negative one).

    Then you get to prove fun things like:

    -2i ln (i) = 3.14159… (pi)

    or, rephrased:

    i ^ i = e ^ (-pi/2)

    I think a lot of this stuff is cool, but I’m often reminded why I chose an applied field rather than a theoretical one.

  9. If it makes it easier, think of it as repeated subtraction.

    5 * 6 means 5 added to zero six times.
    -5 * 6 means -5 added to zero six times.
    5 * -6 means 5 subtracted from zero six times.
    -5 * -6 means -5 subtracted from zero six times.

    5 ^ 6 means 1 multiplied by 5 six times.
    5 ^ -6 means 1 divided by 5 six times.
    -5 ^ 6 means 1 multiplied by -5 six times.
    -5 ^ -6 means 1 divided by -5 six times.

    Zero is unity for addition. (x plus 0 = x)
    One is unity for multiplication. (x * 1 = x)

    One is unity for multiplication because x added to zero once will equal itself.

    All perfectly logical.

    (“plus” referred to above instead of the literal plus because Markdown isn’t letting me backslash-escape it. Which does not make me happy.)

  10. Please stop with the embarrassing math posts already. Or at least start another blog specifically for remedial math questions, so all 2 people who are interested can subscribe to it.

  11. Please stop with the embarrassing math posts already.

    You should see what I have ready for tomorrow…

    Additionally, your browser has a Home button. It probably leads to a page that is not on this site. Press it and find out.

  12. Multiplication is not a shortcut for addition… Multiplication and addition are just two binary operations that happen to meet certain criteria and they happen to please our eye (sometimes). There are other sets and pairs of binary operations that meet the same criteria and they behave the same way as multiplication, addition and whole numbers, but there is no sense in applying the repetition shortcut terminology to them.

  13. I thought the question was, Who, not Why. Whyis a plenty interesting question and deserves a nice little proof, along the lines of proving that 1 is not equal to 0 – also a fundamental mathematical point. But when it comes to Who, that’s also a fabulous question, as we spend lots of time on the why of math but not enough on the history.

    So: speaking to the question you asked but not apparently intended.. To Euclid we owe much of the early formalization of maths, but I sadly cannot state that he actually dealt with this topic himself, nor (if he did) if was the first to enunciate it.

  14. Previous Post with no name: There is no answer to “who”.

    The body of real numbers comes with 18 axioms which are quite abstract in nature. The list can be found anywhere online. Also note that in some axiomatic versions, 1 by axiom is distinct from 0. We were asked to prove for example that 1 > 0. But not that 1 was different from 0.

    Anyways, from those 18 axioms, you can indubitably prove that – times – makes +.

  15. PS. To answer original blog post: these 18 axioms are quite simple really. They don’t purvey any esotheric hidden agenda to numbers. Only codify the way you already expect them to work.

  16. If we think of a positive number a, then it’s negative -a is the the product -1a, let us call it A. Now let’s have another positive number b with its negative -b notated as B. Then, AB = (-1a)(-1b) which can be written as AB=-1a-1b, which is the same as -1(a-1b). We see that AB=-1(-ab). Which is the same as computing the opposite of (-ab). which is ab. Wee that the two negative sign do logically cancel themselves out. This does not seem arbitrary to me. Or maybe am I missing a point…

  17. I think you can get there from considering debts. If I add to my debt, it increases. (-1 + -1 = -2). That is, I added to my debt +1 times. If I add to my debt a negative number of times, it decreases.

    As to who – this was one of the great mathematically controversies of the Middle Ages. The physical significance of negative numbers was a hot topic for a long time.

  18. i really dont get this stuff im in year 7 and our thick teacher is trying to teach us this but she doesnt bother to explain

  19. There is a hidden base 0 in multiplication that most people dont talk about, but it’s there. You start at 0, and then add a given number X times. With -2 X -2 you cant add because you are subtracting a negative 2 from 0 two times. ^^^^^^^^^^ ^^^^^^^^
    There is no addition.

    When multiplying 2 x -2 one can actually use both additive or subtractive processes to achieve the logical outcome.

    Because

    a) -2 added to 0 TWICE = -4 b) 2 subtracted from 0 TWICE = -4

  20. using the axioms of a field can prove why a negative multiplyed by a negative equals a poistive, e.g -a-b= ab. As the axioms of a field use group structure along with left and right distributive.

  21. the explenation they give on the dr.math inconection with the product of 2 negative s is a positive is faulty.why?because you can’t you the distrubutive law if you did’nt prove the rule first.

  22. really easy actually, if you just think of the negative symbol as having a negative effect eg always reduces toward 0 eg a subtraction from the perspective of a negative number would be a positive number so 2+-2 = 0 it was addition but the – reversed it therefore -2 x -2 = 2 you start with -2 can just as easily be written -2 +2+2

    not easy to explain actually oh well

  23. I’ve seen similar threads regarding this interesting question (and I have understanding) and someone often pompously has to remark (as above) how stupid a question it is. Please do what I have done: ask the heads of neurosurgery departments of Ivy League universities and like people, “Why do two negatives multiplied together equal a positive?” These people at the top of the intellectual heap in various fields often don’t know. So maybe it’s not such a dumb question, after all.

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