# Negative Fractions

Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?

Answer: As long as you write only one negative sign, it does not matter where you put it.

Two ideas are useful to keep in mind during the explanation that follows:
– Subtraction is the same thing as the addition of a negative.
– The negative of a number can be created by multiplying the number by negative one.

These principles apply to fractions as well, so:

$-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}$

Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to adding the same fraction, but with a negative numerator. But this is not the only option…

Recall how equivalent fractions are created: multiply the original fraction by a fraction that equals one, where numerator and denominator have the same value. If we multiply the above result by 1 in the form of a negative one divided by itself to create another equivalent fraction:

$\dfrac{-3}{~5}\\*~\\*~\\*=(\dfrac{-1}{-1})(\dfrac{-3}{~5})\\*~\\*~\\*=\dfrac{~3}{-5}$

we end up with the negative sign in the denominator. Summarizing these equivalent fraction results:

$-\dfrac{3}{5}~=~\dfrac{-3}{~5}~=~\dfrac{~3}{-5}$

As long as there is only one negative sign, either in front of the fraction, or in the numerator, or in the denominator, the fraction represents a negative quantity. As the examples above illustrate, you are welcome to move the negative sign around from where it is to either of the other two positions… whichever is most convenient for you.

However, note that the negative sign must be applied to the entire numerator, or the entire denominator. So, in cases with more than one term in the numerator or denominator, the negative sign will have to be distributed if it is moved to the numerator or denominator:

$-\dfrac{3-x}{x-5}~=~\dfrac{-(3-x)}{x-5}~=~\dfrac{-3+x}{x-5}~=~\dfrac{x-3}{x-5}$

or

$-\dfrac{3-x}{x-5}~=~\dfrac{3-x}{-(x-5)}~=~\dfrac{3-x}{-x+5}~=~\dfrac{3-x}{5-x}$

If the last step in the above two examples surprised or puzzled you, my post on Negative Differences may help clear up the confusion.