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Math Help - Confused with fractions

  1. #1
    Member maxpancho's Avatar
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    Question Confused with fractions

    If we have $\dfrac{1}{\dfrac{1}{x-a}}$, is it:

    a)
    $\dfrac{1}{1}$ and then divided by $\dfrac{x-a}{1}$, or is it
    b)
    $1$ divided by $\dfrac{1}{x-a}$
    ?

    Because that would be two different results:
    a) (1/1)/((x-a)/1) - Wolfram|Alpha
    b) 1/(1/(x-a) - Wolfram|Alpha
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  2. #2
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    Re: Confused with fractions

    Yes, they give two different results because they are two different things.

    \frac{a}{b} means "a divided by b" so \frac{1}{\frac{1}{x- a}} means 1 divided by \frac{1}{x- a}, your second choice.

    But you should have learned that dividing by a fraction is the same as multiplying by its reciprocal:
    \frac{1}{\frac{1}{x- a}} is the same as 1 multiplied by \frac{x- a}{1}:

    \frac{1}{\frac{1}{x- a}}= 1\times \frac{x- a}{1}= x- a
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  3. #3
    Member maxpancho's Avatar
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    Re: Confused with fractions

    Well, I know which result each of them should give and why, but how do I know if it's variant a or b that I should solve?
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    Re: Confused with fractions

    Quote Originally Posted by maxpancho View Post
    Well, I know which result each of them should give and why, but how do I know if it's variant a or b that I should solve?
    You find that by the length of the lines separating numerator from denominator or by the parentheses present if any when writing.
    If it is in the form of latex, you can observe the difference by font size.

    For example $\frac{1}{\frac{1}{x-a}}\text{ or }\frac{\frac{1}{1}}{x-a}$.

    I think you can see the difference very well even when writing

    Confused with fractions-capture.gif

    I can't write with my mouse. So sorry for that horrible mouse-writing(i.e. hand-writing)
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  5. #5
    Member maxpancho's Avatar
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    Re: Confused with fractions

    *deleted*
    Last edited by maxpancho; June 29th 2014 at 08:29 AM.
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  6. #6
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    Re: Confused with fractions

    I can see why you think that $\dfrac{1}{\dfrac{1}{x - a}}$ is ambiguous and could be interpreted as $\dfrac{1 \div 1}{x - a}$ or $\dfrac{1}{1 \div (x - a)}.$

    Post 4 mentions a convention that the length of the bars should prevent that ambiguity. I was not aware of that convention, and it does appear to me that in LaTeX the bars do have a slightly different length, but I would never have noticed such a slight difference.

    Nevertheless, I think anyone familiar with mathematical notation would interpret $\dfrac{a}{\dfrac{b}{c}}\ as\ \dfrac{a}{b \div c}.$

    The reason is this: if I want to divide (a/b) by c I can just write $\dfrac{a}{bc}$ without further ado.
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  7. #7
    Member maxpancho's Avatar
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    Re: Confused with fractions

    Thanks guys. I've cleared everything up for me.
    Last edited by maxpancho; June 29th 2014 at 08:29 AM.
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