Dividing Rational Numbers
Before discussing division of rational numbers, we should look at the reciprocal of a number. The reciprocal of a number is [latex]1[/latex] divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. An important feature for a number and its reciprocal is that their product is [latex]1[/latex].
reciprocal
The reciprocal of a fraction [latex]\frac{a}{b}[/latex] is [latex]\frac{b}{a}[/latex], where [latex]a[/latex] and [latex]b[/latex] are non-zero. Multiplying a fraction by its reciprocal always results in [latex]1[/latex].
If you multiply two numbers together and get [latex]1[/latex] as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:
| Original number | Reciprocal | Product |
|---|---|---|
| [latex]\frac{3}{4}[/latex] | [latex]\frac{4}{3}[/latex] | [latex]\frac{3}{4} \times \frac{4}{3}=\frac{3 \times 4}{4 \times 3}=\frac{12}{12}=1[/latex] |
| [latex]\frac{1}{2}[/latex] | [latex]\frac{2}{1}[/latex] | [latex]\frac{1}{2} \times \frac{2}{1}=\frac{1 \times 2}{2 \times 1}=\frac{2}{2}=1[/latex] |
| [latex]3=\frac{3}{1}[/latex] | [latex]\frac{1}{3}[/latex] | [latex]\frac{3}{1} \times \frac{1}{3}=\frac{3 \times 1}{1 \times 3}=\frac{3}{3}=1[/latex] |
| [latex]2\frac{1}{3}=\frac{7}{3}[/latex] | [latex]\frac{3}{7}[/latex] | [latex]\frac{7}{3} \times \frac{3}{7}=\frac{7 \times 3}{3 \times 7}=\frac{21}{21}=1[/latex] |
Sometimes we call the reciprocal the “flip” of the other number: flip [latex]\frac{2}{5}[/latex] to get the reciprocal [latex]\frac{5}{2}[/latex].
When dividing two rational numbers, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication.
dividing rational numbers
Dividing rational numbers involves a process where you multiply by the reciprocal of the number you’re dividing by.
Symbolically, we write this as: If [latex]b[/latex], [latex]c[/latex] and [latex]d[/latex] are non-zero integers, then
Any easy way to remember how to divide fractions is the phrase “keep, change, flip.” This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.
Division by Zero
You know what it means to divide by [latex]2[/latex] or divide by [latex]10[/latex], but what does it mean to divide a quantity by [latex]0[/latex]? Is this even possible? Can you divide 0 by a number? Consider the fraction
[latex]\frac{0}{8}[/latex]
We can read it as, “zero divided by eight.” Since multiplication is the inverse of division, we could rewrite this as a multiplication problem.
[latex]\text{?}\cdot{8}=0[/latex].
We can infer that the unknown must be [latex]0[/latex] since that is the only number that will give a result of [latex]0[/latex] when it is multiplied by [latex]8[/latex].
Now let’s consider the reciprocal of [latex]\frac{0}{8}[/latex] which would be [latex]\frac{8}{0}[/latex]. If we rewrite this as a multiplication problem, we will have
[latex]\text{?}\cdot{0}=8[/latex].
This doesn’t make any sense. There are no numbers that you can multiply by zero to get a result of [latex]8[/latex]. The reciprocal of [latex]\frac{8}{0}[/latex] is undefined, and in fact, all division by zero is undefined.
Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number [latex]a[/latex], [latex]\frac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\frac{0}{a}[/latex] will always be undefined.