Ordinarily, multiplication is interpreted as repeated addition:

.

But what does it *mean* to multiply two arbitrary real numbers, such as ?

Well, if the second number is a positive integer like , our interpretation still applies no matter what the first number is: just add together five copies of it. So we can do multiplication of the form , or really (any positive integer). All that’s left is to decide what to do when instead of we have something like , with a possible minus sign and maybe digits after the decimal point.

For decimals, we can figure out what multiplying by *should* mean: since , we know that whatever is, the product should have the property that

.

(The validity of shifting parentheses around is called *associativity*.) It turns out that solving a problem of the form “What times 10 is a given number?” is easy: the unique solution is found by moving the decimal point in the given number one place to the left, and this is called *dividing by 10*. So we can calculate by first computing (which we know how to do: add together six times) and then divide the result by 10.

Similarly, since , we can calculate by first calculating and then dividing by 10 twice, and so on. This lets us calculate all the individual pieces , , , , etc., and we can add them up to multiply by a decimal with any finite number of digits:

Here we’re distributing the to each term in the finite sums using the law of *distributivity*. But distributivity by itself isn’t enough to handle an *infinite* number of digits after the decimal point; to pin down the value of we need something like *continuity*:

Since is the limit of the sequence , the product is the limit of the sequence .

Or we could use the *ordering* of the real numbers:

Since is between and , and between and , and between and , etc., we can deduce that is

- between and , and
- between and , and
- between and , and
- …
There’s only one number between all these pairs of products, so that’s the number we should call .

Either way, we now know what it means to multiply any real number by any *positive* real number. To handle the rest of the cases, we just have to know how to multiply by zero or by negative numbers, and for that we use distributivity again:

Whatever is, it has the property that . Subtracting one copy of from each side, we get .

(A more modern viewpoint is that the rule is really a *special case* of distributivity: just like is a sum of two numbers and is a sum of five numbers, we regard as the result when we add up *no* numbers, the “empty sum.” So if we start with , think of it as times the empty sum, and then distribute the across the (no) terms, we end up with another empty sum, so the result is again . In general, the “empty” version of a binary operation is the identity element for that operation, so the “empty sum” is the additive identity, , while the “empty product” is the multiplicative identity, .)

And for negative numbers:

If is a negative number and we want to calculate , start by multiplying by the positive number . Whatever is, its sum with must be zero:

.

Therefore and are additive inverses, so .

And there we go! To multiply , we first:

- Compute , , , , etc.
- Add up that infinite series to get .
- And then take the negative of that to get .

Now we can multiply any two real numbers!

To summarize the process we used to extend multiplication to all real numbers:

**Start with an interpretation for the operation.**

(In our case, “multiplication is repeated addition.”)**Write down some laws that this operation satisfies.**

(We used associativity and distributivity. We could also have added*commutativity*and*multiplicative identity*to the list.)**Extend as far as possible in such a way that the laws uniquely determine the result.**

(This is how we found the values for things like or times a negative number: if associativity is going to keep holding, needs to equal one tenth of , and distributivity forces to be the negative of . In general this tells us what any rational number must equal.)**Extend again in a “nice” way, if you can.**(We used continuity/order properties to extend from any

*rational*number to any*real number*.)