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:


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:


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


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.

If there are two fractions being subtracted, and you move the subtraction sign into the numerator or denominator of the fraction being subtracted, put a plus sign where the subtraction sign was (subtracting is the same as adding a negative):




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:




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!

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.


    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. yes, it very much does! ( thank you very much( It also surprised me how quickly you answered that))

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

    2. 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

  1. Fractions are of the form p/q where p and are natural numbers and q is not equal to zero. If so, why negative numbers is used in fractions

  2. Thanks! Negative sign on a fraction help. Very short and simple. 50 years old, studying for college Accurate test. God Bless!

    1. I’m not sure I understand your question. Do you mean something like -3/4? Or are you asking about a situation that has multiple terms, one of which is a negative fraction?

  3. The equation is y = 7 – 3/5x, find slope. Obviously slope is -3/5 but in the question on the paper, the (-) is in front of 3/5 making the negative to be able to be either on the 3 or the 5. I think this would affect the slope majorly but I am not sure, I was wondering if it would affect it and how would one know which number the negative sign is on whether the 3 or the 5.

    1. Think of the slope as Rise / Run.

      If you start from the origin (0, 0), Rise by -3, then Run by +5, you will end up at (5, -3) and the slope from the origin to that point is -3/5.

      Now start from (5, -3), and this time put the negative sign by the 5, so Rise by 3, then Run by -5… you are back at the origin. This tells you that the two slopes are equivalent, since they both keep you on the same line.

      So a a slope of (-3)/(5) = (3)/(-5) = – (3/5). No matter which version you use, if you move from one point on the line using that slope you will end up at another point that DOES lie on the line… although perhaps in the other direction from the starting point.

    1. Debra,
      What you typed can be simplified, but not solved, as it does not include a variable or an equal sign with something on the other side.

      I am unsure whether you intended the expression to be a single fraction, as in
      Or interpreted as your calculator would
      4 + \dfrac{3^2}{-1}\cdot (4)(2) - 1
      They will have different results

    1. Zoey,

      There are several ways of thinking about your question, all of which will lead to the same answer. So take your pick as to which makes the most sense to you:
      1) Since the numerator and denominator have exactly the same value, the answer is 1
      2) You can re-write the fraction as (-1)(3) / (-1)(3), then separate it into a product of two fractions: (-1/-1)*(3/3). Negative one divided by negative one equals positive 1. Three divided by three equals 1. One times one equals 1.
      3) Since both the numerator and the denominator contain negative values, the result will be positive (pairs of negatives cancel each other out when multiplied or divided). Now that we have decided on the sign of the result, ignore the negative signs and focus on 3/3, which equals 1. So the answer is positive one.

  4. what is negative 5 as a fraction in a rational number line ranging for negative five to positive five?

    1. Camden,
      Negative 5 can be written as
      a: -5
      b: -(5/1)
      c: (-5)/1
      d: 5/(-1)
      e: -(10/2)
      and the list goes on…

      (b), (c), or (d) are the “cleanest” ways of representing a -5 as a fraction. They all evaluate to be a -5 on a calculator.
      To answer your question, I would use (b).
      5/1 is a rational number because it is a ratio of two integers.
      -(5/1) makes it clear that you are dealing with a negative number, which will therefore have to be plotted to the left of the origin on a number line.
      And, since your number line ends at -5, you can put a dot or a vertical line through that point on the number line, and label it as -(5/1) to indicate that is where the value you were asked to plot lies.

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