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Thread: Prove that -X * -Y = +Z

  1. #1 Prove that -X * -Y = +Z 
    Forum Freshman IAlexN's Avatar
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    Is there anyway to mathematically prove that any number; say X times Y equals a positive number Z.

    Where X < 0, Y < 0 and Z > 0.

    This is not homework. I'm just curious, since all my teachers have told me that any negative number times a negative number equals a positive number. Though, they never explained why.

    Thank You!


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  3. #2 Re: Prove that -X * -Y = +Z 
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    Quote Originally Posted by IAlexN
    Is there anyway to mathematically prove that any number; say X times Y equals a positive number Z.

    Where X < 0, Y < 0 and Z > 0.

    This is not homework. I'm just curious, since all my teachers have told me that any negative number times a negative number equals a positive number. Though, they never explained why.

    Thank You!
    Let x be a real number such that x < 0 and let y be a real number such that y < 0, where the following relationship holds true:

    , for all z such that z > 0.







    for all real numbers such that x < 0, y < 0, and z > 0.

    QED

    Now, in the above proof, we don't originally use the expression , but , look at what this does to the set that x and y belong:









    As you can see from the above, using the negative coefficient of -1 for both numbers makes both of them positive, and we know that two positive numbers multiplied are positive, and we proved that an expression where both numbers had negative coefficients was equal to an expression where both numbers had the positive coefficient of one.


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  4. #3 Re: Prove that -X * -Y = +Z 
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by IAlexN
    Is there anyway to mathematically prove that any number; say X times Y equals a positive number Z.

    Where X < 0, Y < 0 and Z > 0.

    This is not homework. I'm just curious, since all my teachers have told me that any negative number times a negative number equals a positive number. Though, they never explained why.

    Thank You!
    Let x be a real number such that x < 0 and let y be a real number such that y < 0, where the following relationship holds true:

    , for all z such that z > 0.







    for all real numbers such that x < 0, y < 0, and z > 0.

    QED

    Now, in the above proof, we don't originally use the expression , but , look at what this does to the set that x and y belong:









    As you can see from the above, using the negative coefficient of -1 for both numbers makes both of them positive, and we know that two positive numbers multiplied are positive, and we proved that an expression where both numbers had negative coefficients was equal to an expression where both numbers had the positive coefficient of one.
    actually, in this example, you have to define x and y as positive if you are going to initially invert them and make them negative. because -x for x<0 is positive.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  5. #4 Re: Prove that -X * -Y = +Z 
    . DrRocket's Avatar
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    Quote Originally Posted by IAlexN
    Is there anyway to mathematically prove that any number; say X times Y equals a positive number Z.

    Where X < 0, Y < 0 and Z > 0.

    This is not homework. I'm just curious, since all my teachers have told me that any negative number times a negative number equals a positive number. Though, they never explained why.

    Thank You!
    It is a matter of how one constructs the integers starting from the natural numbers and what is meant, precisely, by negative integers and by multiplication.

    So the answer is not so much that you prove it as that you must build the integers and arithmetic from scratch and this fact comes out of the construction.

    If you would like to see this done, find a copy of Landau's Foundations of Analysis and read it. It is a very thin book, but also rather dry and with a telegraphic style.
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  6. #5 Re: Prove that -X * -Y = +Z 
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    Quote Originally Posted by Arcane_Mathematician
    Quote Originally Posted by Ellatha
    Quote Originally Posted by IAlexN
    Is there anyway to mathematically prove that any number; say X times Y equals a positive number Z.

    Where X < 0, Y < 0 and Z > 0.

    This is not homework. I'm just curious, since all my teachers have told me that any negative number times a negative number equals a positive number. Though, they never explained why.

    Thank You!
    Let x be a real number such that x < 0 and let y be a real number such that y < 0, where the following relationship holds true:

    , for all z such that z > 0.







    for all real numbers such that x < 0, y < 0, and z > 0.

    QED

    Now, in the above proof, we don't originally use the expression , but , look at what this does to the set that x and y belong:









    As you can see from the above, using the negative coefficient of -1 for both numbers makes both of them positive, and we know that two positive numbers multiplied are positive, and we proved that an expression where both numbers had negative coefficients was equal to an expression where both numbers had the positive coefficient of one.
    actually, in this example, you have to define x and y as positive if you are going to initially invert them and make them negative. because -x for x<0 is positive.
    No. The point of the proof is to prove that (-x)(-y) = xy.
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