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Math Help - Square of real numbers.

  1. #1
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    Square of real numbers.

    Ok our lecturer asked the question if we knew WHY the square of 2 real numbers is always positive.
    I know the simple rules we learned at school like the product 2 different signs always make a positive and the product of 2 matched signs always make a positive and the idea that the two negative statements make a positive (eg I don't know nothing = I know something) but I have never learned why this is so.
    Ive looked around on the net a lot but can't find anything that satisfies me (its all aimed at high school students).
    Thanks for the help and I hope this is the right sub-forum haha.
    Ryan

    One thing I can think of is that the square root of a negative number is a complex number but that doesn't really help me in seeing the question asked. (Its not an assignment question by the way...I'd never cheat like that)
    Last edited by rushton; February 25th 2011 at 09:45 PM. Reason: Merged posts.
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  2. #2
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    You must mean non-negative. This is a consequence of the distributive property. First note that
    b(-a)+ba=b(-a+a)=b(0)=0,
    so by definition b(-a)=-(ba), and similarly we can get (-b)a=-(ba).

    This relation is true for any b, and in particular it's true when b is replaced with -b. Making this replacement gives
    (-b)(-a)=-[(-b)(a)]=-(-(ba))=ba, so the negatives cancel.

    If you're really perceptive, you'll notice that I used b(0)=0 without justification, but this can also be proved using the distributive law and the definition of zero. Try it.

    I'm not sure what you mean by the responses you found being aimed at high school students. This is high school material, although at least in the US, it's almost never explained, which probably contributes to the feeling of so many students that math is a bunch of arbitrary rules.
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  3. #3
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    I understand that but the part that keeps me wondering is -(-(ba)=ba. Is there a justification of why the negatives cancel?
    Maybe I'm missing something when it comes to understanding the distributive property but is seems that knowing that the 2 negatives cancel is assumed.
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  4. #4
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    By definition, a+(-a)=0. This means that the additive inverse (negative) of -a is a, or in other words, -(-a)=a. This isn't assuming that the negatives cancel. It just follows right from the definition of -a as the element that gives zero when added to a.

    There are even some things in this post that need to be justified, for example I said "the" additive inverse instead of "an" additive inverse, which would only make sense if inverses are unique, which can be proved. I suggest you look at the axioms for a ring, possibly checking out the axioms for a group first.
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  5. #5
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    Thanks its clicked now!
    Yeah I have looked into the axioms for a group but I will go over it again and look into the axioms for a ring.
    It makes sense to me now that I am thinking in terms of the inverse of a number rather then negatives being their own number (even though they are haha).
    Thanks again!
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  6. #6
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    No problem! Glad I could help.
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  7. #7
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    This is an excellent answer. I would only like to add that there is a chapter on negative numbers in the book "Penrose tiles to trapdoor ciphers" by Martin Gardner. Part of the chapter is available in Google Books. Basically, his point is, "Minus times minus equals plus. The reason for this we need not discuss." Just kidding: he really has this quote, but that's not his point. He says:
    [W]hen mathematicians found it desirable to enlarge the concept of number to take in zero and negative numbers, they wanted the new numbers to behave as much like the old ones as possible.
    Then he goes into the argument about distributivity. The next page is missing in Google Books, so here is a double translation to Russian and back:
    Therefore, it is wrong to assert that mathematicians can "prove" that the product of two negative numbers is positive. We are talking not about a proof, but about a choice of rules that would allow negative numbers to satisfy the same old rules that natural numbers satisfy.
    The chapter also has many examples appealing to intuition to show that the rule that the product of two negative numbers is positive is a reasonable one.
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