Understanding the multiplication of fractions

Question

I know this might sound like an easy question, but I was thinking about it the other day and I couldn't think of any good reasons (aside from the fact that it works). So, why is it that when we multiply fractions, we multiply numerator to numerator, and denominator to denominator?

Answer 1

Using the definition of a fraction we can reduce fraction multiplication to integer multiplication.

By definition $\ x = a/b\ $ is the unique solution of $\ \color{#c00}{b\,x=a}$

By definition $\ y = c/d\ $ is the unique solution of $\ \color{#0a0}{d\,y = c}$

Multiplying these equations $\,\Rightarrow\, \color{#c00}{bx}\:\!\color{#0a0}{dy} = \color{#c00}a\:\!\color{#0a0}c\ $ so $\ \color{#0a0}x\:\!\color{#c00}y = \dfrac{ac}{bd},\ $ i.e. $\ \color{#0a0}{\dfrac{a}b}\color{#c00}{\dfrac{c}d} = \dfrac{ac}{bd}$

Remark $ $ We implicitly used basic laws of the underlying ring of integers $\,\Bbb Z,\,$ (notably that multiplication associative & commutative, and $\,b,d\neq0\,\Rightarrow\, bd\neq 0)$

Analogous "reductionist" arguments apply elsewhere, e.g.

By definition $\ x = \sqrt2 \ $ is the unique solution $>0\,$ of $\ x^2 = 2$

By definition $\ y = \sqrt 3\ $ is the unique solution $>0\,$ of $\ y^2 = 3$

Multiplying these equations $\,\Rightarrow\, (xy)^2= 6\ $ thus $\ xy = \sqrt 6,\ $ i.e. $\ \sqrt2 \sqrt 3 = \sqrt 6,\,$ which reduces the multiplication of algebraic integers to that of integers, analogous to above, where we reduced the multiplication of fractions to the multiplication of (a pair of) integers above.

Generally the properties of the extended number systems follow from the fact that we desire (so require) it to have the same algebraic structure (e.g. satisfy ring axioms) as the base structure, i.e. we wish the essential arithmetical laws to persist in the extended number system. This was the principle that guided many extensions of number systems historically (e.g. see the Hankel or Peacock Permanence Principle), which nowadays is a basic constituent of the axiomatic method.

Answer 2

Going from first principles a fraction $\frac ab$ is equal to $a \times \frac 1b$. When you multiply this by another fraction $\frac cd = c \times \frac 1d$, you get: $a \times \frac 1b \times c \times \frac 1d$ which can be rearranged by commutativity and associativity into: $(a \times c) \times (\frac 1b \times \frac 1d) = \frac{ac}{bd}$