For all numbers $a$, $b$, and $c$,

- Commutativity: $a\times b = b\times a$
- Associativity: $(a\times b)\times c = a\times(b\times c)$
- Distributivity over Addition:
$$a\times (b+c) = a\times b + a\times c$$

- The properties are sometimes called axioms.
- That is, statements or propositions that are regarded as being established, accepted, or self-evidently true.
- Or, as Voltaire put it:
I recommend you to question all your beliefs, except that two and two make four. As quoted in

The Number Sense (2011)

For every number $a$,

- $a\times1 = a$
- There is a number $A$ such that $a\times A = 1$

The number $A$ is often called the reciprocal of $a$ and is denoted by $1/a$.

- When we write $a/b$, we mean $a\times(1/b)$.
- That is, by
definition,
$$\frac{a}{b} = a\times \left(\frac{1}{b}\right).$$
- Division is not commutative.
- But,
$$a\times \frac{1}{b} = \frac{1}{b}\times a$$.

- There are a few more axioms. Wikipedia has a complete list of axioms for the real numbers.
- The axioms actually appear in the Florida state standards!

I suppose $6/3$ better equal $2$.

\begin{align}\frac{6}{3} &= 6\times \frac{1}{3} =
(2\times 3)\times \frac{1}{3} \\
&= 2 \times \left(3\times\frac{1}{3}\right) \\
&= 2 \times 1 = 2.
\end{align}

How about something like $\frac{1}{2}\times \frac{1}{3}$?

I guess it oughtta be $1/6$, since

\begin{align}6 \times \left(\frac{1}{3} \times \frac{1}{2} \right) &=
\left(6 \times \frac{1}{3}\right) \times \frac{1}{2} \\
&= 2\times \frac{1}{2} = 1.
\end{align}

Now let's try $\frac{5}{2}\times \frac{4}{3}$.

\begin{align}\frac{5}{2}\times \frac{4}{3} &=
\left(5\times\frac{1}{2}\right) \times \left(4\times\frac{1}{3}\right) \\
&= \left(5\times4\right)\times\left(\frac{1}{2}\times\frac{1}{3}\right) \\
&= \frac{5\times4}{2\times3} = \frac{20}{6}.
\end{align}

When you multiply/divide a long string of numbers, you can move things around via commutativity and associativity, as long as you express the division in terms of multiplication.

Addition and subtraction have a similar relationship. Thus, to understand how subtraction works with respect to the properties, it helps to think in terms of addition of negative numbers.

How do we add fractions with common denominators? Something like:

$$\frac{2}{7} + \frac{4}{7} = ?$$I guess we add the numerators and put the result over the same denomominator.

$$\frac{2}{7} + \frac{4}{7} = \frac{2+4}{7}=\frac{6}{7}.$$

I wonder why that works, though?

We're wondering if:

$$\frac{a}{b} + \frac{c}{b} = \frac{a+b}{b}?$$
We just gotta factor the $1/b$ out:
\begin{align}
\frac{a}{b} + \frac{c}{b} &=
a\times\frac{1}{b} + c\times\frac{1}{b} \\
&= (a+c)\times\frac{1}{b} = \frac{a+c}{b}.
\end{align}

If the denominators aren't the same, we can always multiply top and bottom by the appropriate things to make them the same.

Conventionally, $5\frac{1}{2}$ means $5+\frac{1}{2}$.

Thus, if we want to multiply this kind of number by another, we can use the distributive property.

\begin{align}2\times\left(5\frac{1}{2}\right) &=
2\times\left(5+\frac{1}{2}\right) \\
&= 2\times5 + 2\times\frac{1}{2} = 10 + 1 = 11.
\end{align}