Lesson 2 · from §62 of the 1910 text
Changes of Sign in a Fraction
A fraction has three signs: the sign in front of the fraction, the sign of the numerator, and the sign of the denominator.
Division follows the usual sign rules (like signs give +, unlike give -), so +\frac{+8}{+2} = +4, \qquad +\frac{-8}{-2} = +4, \qquad -\frac{-8}{+2} = -(-4) = +4, \qquad -\frac{+8}{-2} = -(-4) = +4.
Principle. You may change any two of the three signs (numerator and denominator, numerator and fraction, or denominator and fraction) without changing the value of the fraction.
So every fraction can be written at least four ways: +\frac{2a}{x - 3} = +\frac{-2a}{-x + 3} = -\frac{-2a}{x - 3} = -\frac{2a}{-x + 3}.
Changing the sign of a polynomial means changing the sign of every term: the negative of x - 2y + 4 is -x + 2y - 4. +\frac{2x - 5}{x - 2y + 4} = +\frac{-2x + 5}{-x + 2y - 4} = -\frac{-2x + 5}{x - 2y + 4} = -\frac{2x - 5}{-x + 2y - 4}.
This trick matters a lot when adding fractions: \dfrac{3}{2 - x} is the same as -\dfrac{3}{x - 2}, and spotting that can save a lot of work.
Typing tip. In the exercises, the sign in front is shown for you; type the fraction that follows it, like (-2a)/(-x + 3).
Quick check
Write \dfrac{2a}{x - 3} with the denominator 3 - x. The sign in front will be -; type the fraction that follows it.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Write each fraction in three other ways. The sign in front of the fraction is shown; type the fraction that comes after it, like x/(-y) or (-x + 3)/(2x + 4).
\dfrac{-x}{y}
\dfrac{x}{-y}
\dfrac{a}{a-b}
\dfrac{x-3}{2x+4}
-\dfrac{c^2-d}{-2c+d}
-\dfrac{x-y}{x+y-3}
\dfrac{x^2-5x+6}{x^2-7x+12}
-\dfrac{-x+x^2-2}{x^3-x^2y-1}