Math Insights

Insights: Fraction Division

~Algebraic Proof of "Invert and Multiply"~

1. Rote Memorization: Why does "Division" become "Multiplication"?

Calculate $\frac{2}{3} \div \frac{5}{7}$.

In elementary school, we are taught a powerful rule for this calculation:
"To divide by a fraction, invert the second fraction (flip it) and multiply."

$$ \frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} $$

Following this procedure gives the correct answer.
However, this raises a question.
Why does the operator change to "multiplication" when we were supposed to be "dividing"? And why do the numbers flip?
We will prove the mathematical basis for this operation using an algebraic approach rather than intuitive explanations.

2. Division is just a "Fraction"

The key to solving this doubt lies in the definition of the division symbol "$\div$".
Essentially, $1 \div 2$ can be written as $\frac{1}{2}$.
In other words, the "$\div$" symbol is equivalent to the fraction bar (vinculum).

Following this definition, fraction division can be described as a "Complex Fraction" (a fraction within a fraction).

$$ \frac{2}{3} \div \frac{5}{7} = \frac{\frac{2}{3}}{\frac{5}{7}} $$

3. Identity Transformation: Making the Denominator "1"

Let's reorganize this complex fraction into a normal fraction.
Here we use a fundamental property of fractions: "Identity Transformation".

Identity Property:
Multiplying both the numerator and denominator by the "same number" does not change the value of the fraction.
(e.g., Multiplying $\frac{1}{2}$ by 5 on both sides gives $\frac{5}{10}$, which is still equivalent to $0.5$)

Using this property, we consider eliminating the denominator fraction ($\frac{5}{7}$) by turning it into "1".
To turn a number into 1, we simply multiply it by its Reciprocal ($\frac{7}{5}$).

However, to maintain equality, we must multiply the numerator by the same number as well.

$$ \frac{\frac{2}{3}}{\frac{5}{7}} = \frac{\frac{2}{3} \times \mathbf{\frac{7}{5}}}{\frac{5}{7} \times \mathbf{\frac{7}{5}}} $$

4. Conclusion: The Meaning of Multiplying by the Reciprocal

Let's proceed with the calculation.
The denominator is "original number × reciprocal", so it cancels out perfectly to become "1" and vanishes.

$$ \text{Denominator} = \frac{5}{7} \times \frac{7}{5} = 1 $$

Since the denominator is now 1, only the calculation in the numerator remains.

$$ \text{Result} = \frac{\frac{2}{3} \times \frac{7}{5}}{1} = \frac{2}{3} \times \frac{7}{5} $$

Compare this result with the initial expression.

Initial: $\frac{2}{3} \div \frac{5}{7}$
Transformed: $\frac{2}{3} \times \frac{7}{5}$

The operation performed as a "logical adjustment to make the denominator 1" simply appeared in the final form as "invert and multiply."
In other words, the rule "flip and multiply" is not an arbitrary change, but a shortcut for the logical operation of "multiplying by the reciprocal to eliminate the denominator."

5. Dr. WataWata's Note

"Inverse Element" in Matrix Algebra

"Division" is actually a somewhat awkward operation in mathematics.
In the world of "Linear Algebra (Matrices)" used in engineering and physics, the calculation $A \div B$ is not even defined.

Instead, we perform the operation of "multiplying by the Inverse Matrix $B^{-1}$ ($A \times B^{-1}$)."

By unifying operations into the form of "multiplying by the inverse (reciprocal)" rather than using a special "division" operation, mathematical formulas become consistent and dramatically more beautiful.
The "flip and multiply" rule learned in elementary school is actually your first entry into "Adult Mathematics (Product of Inverses)."