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.

If you have other questions (like the one at the top of this post), please ask them in a comment!

Published by

Whit Ford

Math Tutor in Yarmouth, Maine

7 thoughts on “Negative Fractions”

    1. If I am interpreting your question correctly, you are asking where the (-1) came from in the second line of the first example above. There are a couple of principles involved:

      a) 1 is the multiplicative identity, which means that any quantity can be multiplied by 1 without changing it. Therefore, a factor of (1) can be introduced into ANY product without changing the result, or any number can be represented as the product of 1 and itself. For example, 3 = (1)(3) or (2)(3) = (2)(1)(3).

      b) The additive “opposite” of a number is (-1) times the number. Additive opposites always add to zero. So, (3) + (-3) = 0 … the first number (3), multiplied by negative one (-1)(3), produces its additive opposite (-3). Added together they result in a sum of zero. Or, (-5) + (5) = 0 … the first number (-5), multiplied by negative one (-1)(-5), produces its additive opposite (5). Added together they result in a sum of zero.

      So, the first example in the post above starts with a negative fraction, which using the ideas from the explanation above, can be described as (-1)(3/5) (the additive opposite of positive 3/5). A negative sign in front of the fraction can be interpreted as “multiply the positive fraction by (-1) to make it a negative number” if you wish, since (-1)(3/5) = – (3)/(5) = (-3)/(5) = (3)/(-5).

      I hope that answers your question!

    1. Subtraction is a pain, because it is not commutative or associative like addition:
      3 – 2 does NOT = 2 – 3
      4 – 3 – 2 does NOT = 4 – (3 – 2)
      To get around this problem, and make subtraction problems easier to re-arrange without changing their result, we can rewrite subtraction as the addition of a negative:
      3 – 2
      = 3 + (-2)
      = (-2) + 3 using the commutative property of addition

      Negative 2 is the “additive opposite” of 2. When “additive opposites” are added together, the result will always be zero. In other words, if I add a number (any number) to its negative, the result MUST always be zero. Thinking in terms of a number line, a number and its negative will always be on opposite sides of zero, and at equal distances from zero. “Negating” a number will flip it from one side of zero to the other on the number line. If I negate a number twice, it flips to the other size of zero twice, which puts it right back where it started.

      So, to answer your question using a number line, 9 is located nine units to the RIGHT of zero. Negating it, by putting a “-” in front of it, flips it to the other side of zero so that (-9) is nine units to the LEFT of zero. Negating that quantity, -(-9), flips it back so that it is now nine units to the RIGHT of zero, or equal to the original quantity: +9.

      5 – (-9)
      = 5 + (- (-9)) After changing subtraction into the addition of a negative
      = 5 + 9 After using the reasoning above to change a double negative into a positive
      = 14

      I hope that helps!

      1. The parentheses are only there to help clarify things. Two negative signs in a row cancel each other out, with or without parentheses, as long as no other variable or number is between the two negatives:
        -~-3~=~3\\* -(-3)~=~3\\* (-(-3))~=~3

        5-3~=~2

        5-~-3~=~8\\* 5-(-3)~=~8

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