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[latexpage]

So far, we have no single model that makes sense of fractions in all contexts. Sometimes a fraction is an action (“Cut this in half.”) Sometimes it is a quantity (“We each get 2/3 of a pie!”) And sometimes we want to treat fractions like numbers, like ticks on the number line in-between whole numbers.

We could say that a fraction is just a pair of numbers $a$ and $b$, where we require that $b\neq 0$. We just happen to write the pair as $\frac a b$.

But again this is not quite right, since a whole infinite collection of pairs of numbers represent the same fraction! For example:

\[\frac23 \ =\
\frac46 \ =\
\frac69 \ =\
\frac8{12} \ =\
\dots.\]

So a single fraction is actually a whole infinite class of pairs of numbers that we consider “equivalent.”

How do mathematicians think about fractions? Well, in exactly this way. They think of pairs of numbers written as $\frac a b$, where we remember two important facts:

• $b\neq 0$, and
•  $\frac a b$  is really shorthand for a whole infinite class of pairs that look like $\frac{xa}{xb}$ for all $x\neq 0$.

This is a hefty shift of thinking: The notion of a “number” has changed from being a specific combination of symbols to a whole class of combinations of symbols that are deemed equivalent.

Mathematicians then define the addition of fractions to be given by the daunting rule:

\[\frac{a}{b} + \frac{c}{d} = \frac{ad +bc}{bd}.\]

This is obviously motivated by something like the “Pies Per Child Model.” But if we just define things this way, we must worry about proving that choosing different representations for $\frac a b$ and $\frac c d$ lead to the same final answer.

For example, it is not immediately obvious that

\[ \frac23 +\frac45 \quad\text{ and }\quad \frac46 + \frac{40}{50}\]

give answers that are equivalent. (Check that they do!)

They also define the product of fractions as:

\[\frac{a}{b} \cdot \frac{c}{d}
\  = \
\frac{ac}{bd}.\]

Again, if we start from here, we have to prove that you get equivalent answers for different choices of fractions equivalent to $\frac a b$ and $\frac c d$.

Then mathematicians establish that the axioms of an arithmetic system hold with these definitions and carry on from there! (That is, they check that addition and multiplication are both commutative and associative, that the distributive law holds, that all representations of 0 act like an additive identity, and so on.)

This is abstract, dry, and not at all the best first encounter to offer students on the topic of fractions. And, moreover, this approach completely avoids the question as to what a fraction really means in the “real world.” But it is the best one can do if one is to be completely honest.