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Homework Help: -a^2 != (-a)^2, Clarification needed

  1. Jul 30, 2010 #1
    This isn't really a homework problem but more of a clarification request. I'm studying prealgebra stuff right now and this rule kind of confused me.

    [tex]-a^2 \neq (-a)^2 [/tex] Ermmm....I typed out (-a)^2 but it's not showing.

    I understand that the former produces a negative answer i.e. -2a while the latter produces a positive number i.e. +2a. But for this to be true then -(a) = (-1) * (a).

    This might be silly but where does the extra 1 come from? I was given a list of some of the basic rules of algebra at the beginning of the book but that never popped up.

    Some relevant rules that I know of are the same sign rule for multiplication, negative plus a negative is equal to a positive.

    My guess is that the extra 1 comes because - = (-1). But this doesn't make sense because if you apply this to a simple problem then the neighboring rule squashes it. i.e.
    -5 -3
    = -(-1)5 -(-1)3
    = -(-5) - (-3)
    = +5+3
    = 8
    D'oh...
     
  2. jcsd
  3. Jul 30, 2010 #2
    [itex]-x[/itex] is a notation for the additive inverse of an element [itex]x[/itex] belonging to any field:

    [tex]
    x + (-x) = (-x) + x = 0
    [/tex]
    where [itex]0[/itex] is a notation for the additive neutral element.

    In a field, the distributive laws hold. Therefore, one has:

    Using the distributive Law:
    [tex]
    (x + 0) \cdot y = x \cdot y + 0 \cdot y
    [/tex]
    for all [itex]x, y[/itex].

    Using the definition of the additive neutral element:
    [tex]
    (x + 0) \cdot y = x \cdot y
    [/tex]

    Since the left sides of above equalities are the same, the right sides must also be the same. If we denote [itex]z = x \cdot y[/itex], then:

    [tex]
    z + 0 \cdot y = z
    [/tex]

    This equality holds for all z, so we must have:

    [tex]
    0 \cdot y = 0
    [/tex]

    since [itex]0[/itex] is a unique neutral element for addition. This holds for all [itex]y[/tex]. Now, let us represent the [itex]0[/itex] as:

    [tex]
    0 = 1 + (-1)
    [/tex]

    where [itex]1[/tex] is the notation for the multiplicative neutral element and [itex]-1[/itex] is its additive inverse. Then we have:

    [tex]
    (1 + (-1)) \cdot y = 0
    [/tex]

    Using the distributive law, the left hand side is:

    [tex]
    1 \cdot y + (-1) \cdot y = 0
    [/tex]

    Using that [itex]1[/itex] is the multiplicative inverse, we have:

    [tex]
    y + (-1) \cdot y = 0
    [/tex]

    But, this means that [itex](-1) \cdot y[/itex] is the additive inverse of [itex]y[/itex], so, we may write:

    [tex]
    (-1) \cdot y = -y
    [/tex]
     
  4. Jul 30, 2010 #3

    Mark44

    Staff: Mentor

    Yes, it's showing.
    ??? The expressions above don't have anything to do with 2a or -2a. -a2 is the negative of the square of a, while (-a)2 is the square of the negative of a. The same operations are involved, but in the reverse order.
    You have a bunch of extra (and erroneous) signs in what you have just above.
    -(something) is the same as (-1) * (something). You are also confusing negation (the negative of something) with subtraction. Both use the same symbol, but negation is an operation on a single number, while subtraction is an operation on two numbers.

    you can always rewrite an expression involving subtraction as the sum of the first expression plus the negative of the second expression. IOW, it's always true that a - b = a + (-b).

    So -5 - 3 = -5 + (-3)
    = (-1)*5 + (-1)* 3
    = (-1)*(5 + 3)
    = (-1) * 8 = -8

    Compare my answer, -8, with what you got, which was +8, which is incorrect.
     
  5. Jul 30, 2010 #4

    vela

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    I think you meant -a2 and a2, not -2a and 2a.
    When you write -5-3, you're using the symbol - in two different ways. The first one acts on just the 5, denoting the negation of 5. The second one acts on two numbers, the -5 and 3, and gives an answer that is their difference, and what that really means is to take -5 and add to it the additive inverse of 3, i.e.

    -5 - 3 = -5 + (-3)

    Now, if you want, you can say that -5=(-1)(5) and -3=(-1)(3), so you'd get

    -5 - 3 = (-1)(5) + (-1)(3)

    Note that the minus signs in front of the 5 and 3 get replaced by (-1); you can't just insert (-1) in there without getting rid of the signs.
     
  6. Jul 30, 2010 #5
    Hey mark44 & vela just ignore that last part I wrote. It contains a whole bunch of extra and erroneous symbols because I was trying to figure out why – (a) = (-1) * (a)

    I assumed that it was because -=(-1) which is where all the extra stuff came from. But it didn’t make sense to me when I applied it to that simple problem so that’s gotta be wrong.

    Now I’m not even sure if –(a) = (-1) * (a) is true but you see in my algebra book the author mentioned this rule in passing:

    [tex]-a^2 \neq (-a)^2[/tex]

    Afterwards he gave just one contrasting example and moved on.
    [tex] -3^2
    \\= - (3)^2
    \\=- (9)
    \\=-9[/tex]

    Versus
    [tex](-3)^2
    \\=(-3)*(-3)
    \\=+9 [/tex]

    Now how can the first one be true if the following isn’t true:
    –(3) = (-1) * (3) or –(a) = (-1) * (a)

    If I’m not mistaken, then where does this hidden 1 come from?
    I imagine DickForce goes into explaining this in detail but it’s waaay over my head. I don’t even know what an inequality is yet. I haven’t even gotten to exponents yet. However, this thing is like a splinter in my mind. I can’t figure it out. Maybe I should just not question it for now and come back to it later.
     
  7. Jul 30, 2010 #6

    statdad

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    Homework Helper

    Think this way - for the mechanics of operations, just as [itex] a [/itex] is short for
    [itex] 1a [/itex], it is true that [itex] -a [/itex] is short for [itex] (-1) a[/itex]
     
  8. Jul 30, 2010 #7
    D'oh... How did I miss that -__-;

    Well, now that I know where 1 comes from lol, I still don't understand why the 1 is separated from the a if 1 IS a.

    -a = (-1)a

    -(a^2) = (-1) * (a^2)
     
  9. Jul 30, 2010 #8

    Mark44

    Staff: Mentor

    1 ISN'T a; what statdad was saying that a is the same as (= to) 1 times a.
    So a = 1a, and -a = (-1)a.
    We're not saying anything about the value of a, which could be anything.
     
  10. Jul 30, 2010 #9

    vela

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    It is true. Dickfore's post shows why it's true, but as you noted, it's not particularly helpful for your understanding. It really just comes down to the last few lines of his post. If you accept the following to be true:

    1. a x 1 = a
    2. a x 0 = 0
    3. 1 + (-1) = 0
    4. a(b+c) = ab + ac

    Then it follows that (-1)a = -a. It's probably not terribly important at this point that you understand this train of thought (though it's not like it's out of your reach either), and you could just assume that (-1)a = -a is just another property of real numbers to learn.
     
  11. Jul 30, 2010 #10

    vela

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    When you write -a = (-1)a, the two sides of the equation mean different things. When you write, for instance, -2, you're referring to the additive inverse of 2. In other words, it's the number you add to 2 to get 0. When you write (-1)2, it means you take the number -1 and multiply it by 2. To a mathematician, it's not obvious that the quantities -2 and (-1)2 are, in fact, equal to each other, but because the real numbers possess certain properties, it turns out they are. So anywhere you see something like -a, you can replace it by (-1)(a), and vice versa.
     
  12. Jul 30, 2010 #11
    I have a feeling that I'll be hanging around this forum a lot for the next six months.

    I recognize the first 3 properties but not the last. And I don't understand how each premise follows from one another but it's pretty cool that you can prove it with those very basic properties.

    I'll try to wrap my head around this in the next few days.

    Anyways, thanks a lot for the clarification Vela, Mark and statsdad.
     
  13. Jul 30, 2010 #12

    vela

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    The last property is the distributive law. You know, the one that lets you say 2(3+4) = 2x3 + 2x4. You should be familiar with that, I hope.

    I didn't mean that the properties follow from one another. They're four independent assumptions. If you accept that they're true for the real numbers, then you can show that -a = (-1)a.
     
  14. Jul 30, 2010 #13
    Whoops... I need to read more carefully. I didn't realize that that was the distributive law because I always write it down with the multiplication signs in between a and b. But actually.... That makes things a TON clearer. That and particularly number 1 makes it clear.
     
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