How to position negative sign of fraction

Question

For example we have:

$$ \frac{-1}{2} $$

Does this mean that only the numerator of the fraction is negative?

Can we put it like this?

$$ -\frac{1}{2} $$ Does this means that the whole fraction is negative?

And then can we put it like this?

$$ \frac{1}{-2} $$

Does this mean that only the denominator of the fraction is negative?

Do the above suggestions [look] right? Or how should I understand when in some examples the minus sign is moving from numerator to the whole fraction?

Answer 1

The value is the same for all three versions, although the value might be seen as the result of a different operation: \begin{align} \frac{-a}{b} &= \frac{(-a)}{b} = (-a) / b = -(a / b) \\ -\frac{a}{b} &= -\left(\frac{a}{b}\right) = -(a / b) \\ \frac{a}{-b} &= \frac{a}{(-b)} = a /(-b) = - (a / b) \\ \end{align} The middle version is the most symmetric and IMHO most often used, the other version might show up during calculations or in explanations how one calculates with fractions.

Answer 2

As vadium123 remarks in the comments, these are all the same number.

To say that it has a "negative denominator" or a "negative numerator" or that the "whole fraction" is negative is not mathematically meaningful. A number is either negative or it isn't, and the number in question is $- \frac{1}{2}$.

Since dividing $-1$ by $2$ and dividing $1$ by $-2$ both give $-\frac{1}{2}$, it is only natural in our notation for $\frac{-1}{2}$, $\frac{1}{-2}$, and $-\frac{1}{2}$ to all mean exactly the same thing, but to say that "denominator" is negative, the "numerator is negative", or the "whole fraction" is negative doesn't really mean anything, because it's the number itself ($-\frac{1}{2}$) that is either positive or negative, not pieces of its written appearance.

Answer 3

You might want to ask yourself what is meant when you put a minus sign in front of something. The answer could be, for example, that $-\frac ab$ is the solution to the equation $$x+\frac ab = 0$$ where the unknown is $x$. Now, you want to know whether $\frac {-a}b$ is the same as $-\frac ab$. Just check if it solves the equation! $$\frac {-a}b + \frac ab = \frac{-ab+ab}{b^2} = \frac 0{b^2} = 0$$ But lo and behold, the same goes for $\frac a{-b}$ $$\frac a{-b} + \frac{a}{b} = \frac{ab + a(-b)}{(-b)(b)} = \frac{ab-ab}{-b^2} = \frac 0{-b^2} = 0$$

We conclude that both $\frac {-a}b$ and $\frac a{-b}$ are opposites of the number $\frac ab$, i.e. they are both $-\frac ab$. This works because of the way that fractions and their operations are defined. Note that I have used a primitive way of adding fractions, by taking as the common denominator the product of denominators (to avoid using known tricks).

Answer 4

When working with fractions, it's important to understand two separate but interrelated issues: expression in lowest terms, and canonical expression. If both the numerator and denominator are positive, the fraction represents a positive number, but depending on your calculations to get that fraction, you might or might not have it in lowest terms. To give a simple example: $$\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}.$$

The fraction with 4 as numerator and 8 as denominator is a valid result of the addition, but it is not in lowest terms. The fraction with 1 as numerator and 2 as denominator is in lowest terms. Now consider $$\frac{3}{8} - \frac{7}{8} = \frac{-4}{8} = \frac{-1}{2}.$$

Since $3 - 7 = -4$, the fraction with $-4$ as numerator and 8 as denominator is a valid result of the subtraction. But it's not in lowest terms. Now, what algorithm do you use to get a fraction in lowest terms? The way I do it, I see if I can divide the numerator by factors in common with the denominator, I perform the divisions and write down the new numerator, which in this case is $-1$. Then I perform the same divisions on the denominator, in this case obtaining 2, and then I write the fraction bar and the new denominator.

But this in some cases involves a purely stylistic choice: I could decide I don't like the way that $\frac{-1}{2}$ looks so I could erase it and change it to $-\frac{1}{2}$. My personal preference is for the negative numerator, as sometimes the minus sign in front of the fraction looks a bit too thin for my taste, but either way, we're talking about the same number, namely $-0.5$. There are situations in which I prefer to position the minus sign in front of the fraction, as in for example $$\left(\frac{\sqrt{65}}{2} - \frac{1}{2}\right) \left(\frac{\sqrt{65}}{2} + \frac{1}{2}\right).$$

Though perhaps I'd prefer to express that as $$\left(-\frac{1}{2} + \frac{\sqrt{65}}{2} \right) \left(\frac{1}{2} + \frac{\sqrt{65}}{2}\right).$$

Here we are getting even more into stylistic issues. In both instances we're talking about the same numbers, but your preferences for how to write these numbers may differ from mine. A publication might set rules dictating that certain notations are canonical or preferred, but there may also be circumstances in which the editorial staff determines these rules are not applicable to a particular case.

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Lastly, as plenty of other people have already told you, a positive numerator with a negative denominator may occur in intermediate calculations but is somewhat uncommon in properly edited, published documents. Maybe there is an innate dislike for dividing by negative numbers. But sometimes it just can't be helped, at least in intermediate calculations.