Common Denominators and Complex Fractions, foundations for calculus lesson

Many calculus computations start with something like a fraction of fractions. The slope of a secant line on the graph of $y = \frac{1}{x}$ is a difference of two fractions divided by a change in $x$ , and the first job is always to turn that stack into a single clean fraction. The arithmetic is the same as with $\frac{1}{2} + \frac{1}{3}$ . What is new is keeping track of every input that was forbidden along the way.

Key terms

Equivalent fraction

Definition If $B \ne 0$ and $C \ne 0$ , then

\frac{A}{B} = \frac{A \cdot C}{B \cdot C}.

Multiplying numerator and denominator by the same expression $C$ , which is multiplying by $\frac{C}{C}$ , preserves the value only at inputs where $C \ne 0$ . Where $C = 0$ the new form is undefined even if the old one was not.

In plain words Multiplying by $\frac{C}{C}$ is multiplying by $1$ , and that is only true when $C$ is not zero.

Example For $x \ne -2$ and $x \ne 1$ ,

\frac{1}{x + 2} = \frac{x - 1}{(x + 2)(x - 1)}.

At $x = 1$ the left side equals $\frac{1}{3}$ , but the right side is undefined. So the two are equal only away from $x = 1$ . A good multiplier is one whose zeros the original expression already excludes, so that no new restriction appears.

Numerator

Definition In a fraction $\frac{A}{B}$ the expression $A$ is the numerator. When two fractions share a denominator $C \ne 0$ ,

\begin{gathered} \frac{A}{C} + \frac{B}{C} = \frac{A + B}{C}, \\[0.4em] \frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}. \end{gathered}

In a difference the whole numerator $B$ is subtracted, so it belongs inside parentheses.

In plain words With a shared denominator you combine the tops and keep the bottom. A minus sign in front of a fraction applies to every term of its numerator.

Example For $x \ne 3$ ,

\begin{aligned} &\frac{x + 1}{x - 3} - \frac{2x - 5}{x - 3} \\[0.4em] &\quad = \frac{x + 1 - (2x - 5)}{x - 3} \\[0.4em] &\quad = \frac{6 - x}{x - 3}. \end{aligned}

Dropping the parentheses would give $x + 1 - 2x - 5$ , and the sign of the $5$ would be wrong. At $x = 0$ the original is $-\frac{1}{3} - \frac{5}{3} = -2$ , and $\frac{6}{-3} = -2$ agrees.

Common denominator

Definition A common denominator of the fractions $\frac{A}{B}$ and $\frac{C}{D}$ is an expression that both $B$ and $D$ divide. Once every denominator is factored completely, the least common denominator is the product of every factor that appears in any denominator, each to the highest power with which it appears. With $B \ne 0$ and $D \ne 0$ ,

\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}.

In plain words Rewrite each fraction over the same denominator, built from the factors you already have, then add the numerators.

Example For $x \ne 0$ and $x \ne -1$ ,

\begin{aligned} \frac{2}{x} + \frac{3}{x + 1} &= \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)} \\[0.4em] &= \frac{5x + 2}{x(x + 1)}. \end{aligned}

As a check at $x = 1$ , the left side is $2 + \frac{3}{2} = \frac{7}{2}$ and the right side is $\frac{7}{2}$ as well. Each multiplier here, $x + 1$ or $x$ , is already nonzero on the original domain, so no new restriction appears.

Division by a fraction

Definition Dividing by a nonzero quantity means multiplying by its reciprocal. If $B \ne 0$ , $C \ne 0$ and $D \ne 0$ , then

\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}.

The condition $C \ne 0$ is needed because $\frac{C}{D}$ is the divisor and must not be zero.

In plain words Flip the fraction you divide by and multiply. You may only do it when that fraction is not zero.

Example Consider

\begin{aligned} \frac{x}{x + 1} \div \frac{x^2}{x + 1} &= \frac{x}{x + 1} \cdot \frac{x + 1}{x^2} \\[0.4em] &= \frac{1}{x}. \end{aligned}

The original needs $x \ne -1$ for its denominators and $x^2 \ne 0$ for its divisor, so the result is $\frac{1}{x}$ with $x \ne 0$ and $x \ne -1$ . The formula $\frac{1}{x}$ alone would allow $x = -1$ , but the original does not.

Complex fraction

Definition A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction. One standard way to simplify it is to multiply its numerator and denominator by the least common denominator $M$ of all the inner fractions. This is allowed at every input of the original domain, because $M$ is a product of factors of the inner denominators, each of which is nonzero there, and hence $M \ne 0$ there too.

In plain words Clear all the small fractions at once by multiplying top and bottom by something that every small denominator goes into.

Example The difference quotient of $f(x) = \frac{1}{x}$ between $2$ and $x$ is

\frac{\frac{1}{x} - \frac{1}{2}}{x - 2}, \qquad x \ne 0, \ x \ne 2.

Multiply top and bottom by $2x$ , which is nonzero here.

\begin{aligned} \frac{\frac{1}{x} - \frac{1}{2}}{x - 2} &= \frac{2 - x}{2x(x - 2)} \\[0.4em] &= \frac{-(x - 2)}{2x(x - 2)} \\[0.4em] &= -\frac{1}{2x}. \end{aligned}

So the quotient equals $-\frac{1}{2x}$ for $x \ne 0$ and $x \ne 2$ . The value $x = 2$ remains excluded even after its factor cancels. At $x = 1$ the original gives $\frac{1 - 0.5}{-1} = -0.5$ , matching $-\frac{1}{2}$ .

Restriction

Definition The restrictions of an expression are the conditions its inputs must satisfy for the expression, as originally written, to have a value. For fractions this means every denominator is nonzero, including the denominators of inner fractions and every quantity you divide by. These restrictions belong to the expression and are carried unchanged to every equivalent form.

In plain words Write the list of forbidden inputs at the start, from the original, and keep it attached to every later line.

Example The expression

\frac{1}{1 - \frac{1}{x}}

needs $x \ne 0$ for the inner fraction and $1 - \frac{1}{x} \ne 0$ , that is $x \ne 1$ , for the outer one. Multiplying top and bottom by $x$ gives $\frac{x}{x - 1}$ with $x \ne 0$ and $x \ne 1$ . The new formula gives $0$ at $x = 0$ , but the original has no value there.

Check yourself

Combine into one fraction and state the restrictions.

\frac{x}{x - 3} - \frac{3}{x + 3}

Show the answer

Over the common denominator $(x - 3)(x + 3)$ ,

\begin{aligned} &\frac{x(x + 3) - 3(x - 3)}{(x - 3)(x + 3)} \\[0.4em] &\quad = \frac{x^2 + 9}{(x - 3)(x + 3)}, \\[0.4em] &x \ne 3, \ x \ne -3. \end{aligned}

Simplify the difference quotient of $f(x) = \frac{1}{x}$ below, with $x \ne 0$ , $x + h \ne 0$ and $h \ne 0$ .

\frac{\frac{1}{x + h} - \frac{1}{x}}{h}

Show the answer

Multiplying top and bottom by $x(x + h)$ gives

\begin{aligned} \frac{x - (x + h)}{h \, x(x + h)} &= \frac{-h}{h \, x(x + h)} \\[0.4em] &= -\frac{1}{x(x + h)}, \end{aligned}

still with $x \ne 0$ , $x + h \ne 0$ and $h \ne 0$ .

Simplify the expression below and list every excluded value.

\frac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}}

Show the answer

The inner fractions need $x \ne 0$ , and the outer denominator $\frac{x^2 - 1}{x^2}$ must be nonzero, so $x \ne 1$ and $x \ne -1$ . Multiplying top and bottom by $x^2$ gives

\begin{aligned} \frac{x^2 + x}{x^2 - 1} &= \frac{x(x + 1)}{(x - 1)(x + 1)} \\[0.4em] &= \frac{x}{x - 1}, \\[0.4em] &x \ne 0, \ x \ne 1, \ x \ne -1. \end{aligned}

Common Denominators and Complex Fractions

Key terms

Equivalent fraction

Numerator

Common denominator

Division by a fraction

Complex fraction

Restriction

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus