1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Exponent on Negative Number

  1. Jun 3, 2014 #1
    How come when you do something like -3^2=-9. Does it mean that the dash is a subtraction symbol and not a negative symbol?
     
  2. jcsd
  3. Jun 3, 2014 #2

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    This is basically convention. When you see something like -3^2 without parenthesis, you take the exponent of the positive number 3 and then leave the negative symbol. Essentially -3^2 is shorthand for -(3^2). We put the parenthesis between the number and the exponent by convention. If you want to get the positive number 9, you would write (-3)^2=9.

    Really, this is just convention, like reading an equation from left to right, or PEMDAS.
     
  4. Jun 3, 2014 #3
    Sure, but what I was wondering is that the dash makes the number a negative. Why would you have to put it in parenthesis? The dash is simply showing the number is negative. Why would you put 3^2 in parenthesis then multiply by negative 1? Just as if 3 was positive it would be +3. Isn't the dash just saying the number is negative?
     
  5. Jun 3, 2014 #4

    Mentallic

    User Avatar
    Homework Helper

    And keep in mind that the reason (-3)2=9 is because

    [tex](-3)^2=(-1\times 3)^2=(-1)^2\times (3)^2 = (-1)\times(-1)\times 9= 1\times 9[/tex]
     
  6. Jun 3, 2014 #5
    So a negative number is always -1 multiplied by that number? -3 = -1 * 3, but can't it as well just be 1 * -3? Negative 3 is just another number like positive 3, I thought.
     
  7. Jun 3, 2014 #6

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    It's just the way people decided to do things. There's nothing really wrong if you define -3^2 to equal (-3)^2, but we defined it the other way as -(3^2).

    People could be completely pedantic and just demand the usage of parenthesis everywhere. So that 1+2*3 must be written 1+(2*3). But this is a hassle, so we use PEMDAS. This eliminates the need for a lot of parentheses. Exponentiation comes before all other operations (other than parenthesis) so you should read -3^2 as -(3^2).
     
  8. Jun 3, 2014 #7
    If it's convention wouldn't it be better to just write (-1)3^2 instead of saying just -3^2 = -9 which is a bit confusing? (Or maybe it's just me :D)
     
  9. Jun 3, 2014 #8
    OHHHH I see now! :D thanks man. Now it is clear!
     
  10. Jun 3, 2014 #9

    Mentallic

    User Avatar
    Homework Helper

    -3 IS just another number, but each number can be expressed in varying ways, and it turns out that -3 = -1*3 helps us, while -3 and 1*-3 give us no extra insight into why (-3)2=9 and not -9. If you know the rule [itex](ab)^n=a^n\times b^n[/itex] then you can use the fact that [itex]-3=-1\times 3[/itex] to prove to yourself that it is indeed the case.
     
  11. Jun 3, 2014 #10
    Yes yes I see that :D but just as you can say 3 = -1 * -3. I forgot about pemdas, but now I see. I just think it makes more sense (to me) to say that -3^2 = 9 instead of -9 (but of course mathematicians follow pemdas, so yea :D)
     
  12. Jun 4, 2014 #11

    symbolipoint

    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    You are going backward from what you began to understand.

    3^2 is 9.
    +3^2 is 9. The plus sign was not needed.
    -3^2 is the OPPOSITE of 3^2; therefore -3^2 is the opposite of 9.
     
  13. Jun 4, 2014 #12

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    Your suggestion would get really confusing in expressions like 1-3^2. We should be consistent in using PEMDAS, otherwise, one would have to always write 1-(3^2).
     
  14. Jun 4, 2014 #13

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    That is a result of the "precedence" conventions for arithmetic: PEMDAS (or "Please Excuse My Dear Aunt Sally") Any calculations in Parentheses are done first, then exponents, then multiplication and division, then addition and subtraction. Here the exponentiation is done first, 3^2= 9, then the "multiplying" by -1. (It really doesn't matter whether you think of this as "negation" or "subtraction" from 0.)
     
  15. Jun 4, 2014 #14
    But (-3)^2 =9. I'm wondering why you have to write it in parenthesis to get (-3) * (-3) = 9
     
  16. Jun 4, 2014 #15
    But isn't negatuve 3 just another number like positive ? So they negative sign is just telling that 3 is negatuve. Why would you have to put that in parenthesis?

    -3^2 to me should still me -3 * -3
     
  17. Jun 4, 2014 #16

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If we used your convention, such that negation has higher priority than exponentiation, then our notation for polynomials starts to fall apart. For example, under your convention,
    $$x^2 - y^2 = x^2 + (-y)^2 = x^2 + y^2$$
    or maybe, if we don't generously insert the implied ##+##, we interpret it as follows:
    $$x^2 - y^2 = (x^2)(-y)^2 = x^2 y^2$$
    If we wanted to subtract the square of ##y## from the square of ##x## we would have to write
    $$x^2 - (y^2)$$
    Also, we would have the curious situation that
    $$-x^2 = (-x)^2 = x^2$$
    but
    $$-1x^2 = (-1)(x^2) = -(x^2)$$
    so now we can no longer even multiply by ##1## without potentially changing the value.
     
  18. Jun 4, 2014 #17

    Mentallic

    User Avatar
    Homework Helper

    Like Matterwave brought up in post #12, expressions such as 1-32 would get confusing if it were calculated the way you've proposed.
     
  19. Jun 4, 2014 #18
  20. Jun 4, 2014 #19

    adjacent

    User Avatar
    Gold Member

  21. Jun 4, 2014 #20
    I think maybe where you're getting confused is that you think that the grouping of sybmbols "##-3^2##" has inherent meaning. It does not. It's just a bunch of squiggles on the page until we've given it meaning.

    If you believe in alternate universes, then in one of them "##-3^2##" is interpreted as meaning ##(-3)\times(-3)=9##, and this is perfectly reasonable. In this universe, however, we have chosen to interpret "##-3^2##" as meaning ##-(3\times3)=-9##.

    Now others here have tried to persuade you by giving you reasons why the notation means what it means. I'm going to tell you that there are no reasons. It's (basically) completely arbitrary. There are two completely reasonable interpretations, and as a group we decided a long time ago to use the second. It's fine if you disagree with that decision; I have a ton of issues with mathematical notation that I think is "wrong". But disagreeing with the chosen notation does not change the fact that, to everyone that "matters", "##-3^2##" means ##-(3\times 3)##.

    Now to those here that argue that the notation would get confusing if we chose the alternate interpretation or a different order of operations, understand two things; (1) it only seems confusing to you because you are so used to the commonly accepted conventions and (2) the conventional notation is as confusing to your students (if not more so) than unconventional notation is to you. In a different universe, the distributive law looks like ##c\times a+b=(c\times a)+(c\times b)##, and in another it looks like ##c+(a\times b)=(c+a)\times(c+b)##. This isn't because the math is different in that universe; it's that the symbols used to express the mathematics are interpret differently.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Exponent on Negative Number
  1. Negative exponents (Replies: 4)

  2. Negative Exponent (Replies: 8)

Loading...