1. ## 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

2. ## 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 $\frac{x- a}{1}$:

$\frac{1}{\frac{1}{x- a}}= 1\times \frac{x- a}{1}= x- a$

3. ## 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?

4. ## Re: Confused with fractions

Originally Posted by maxpancho
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

I can't write with my mouse. So sorry for that horrible mouse-writing(i.e. hand-writing)

*deleted*

6. ## 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.

7. ## Re: Confused with fractions

Thanks guys. I've cleared everything up for me.