$\div \times \div \,\, \& \, \div \times \, \div$


Properties of multiplication

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)

Two more axioms

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,
    $$a\times \frac{1}{b} = \frac{1}{b}\times a$$

Two more comments

The properties in action

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}

Multiplying denominators

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}

Multiplying more denominators

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.


$$2\times3\times\frac{3}{4}\times\frac{1}{3}\times\frac{2\times3}{5\times6} =\frac{2\times3\times3\times2\times3}{4\times3\times5\times6}. $$


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.


$$2-3 = 2+(-3) = (-3)+2.$$

Adding fractions

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?

More generally

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}

Different denominators

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


\begin{align} \frac{2}{3} + \frac{3}{2} &= \frac{2\times2}{3\times2} + \frac{3\times3}{2\times3} \\ &= \frac{4}{6} + \frac{9}{6} = \frac{4+9}{6} = \frac{13}{6}. \end{align}

This and a that

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}