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** Index**

You have been working with numbers for most of your life,
and you know hundreds of facts about them. The first numbers you encountered
were probably the positive integers . You learned that
multiplication was repeated addition, and that
, because
3 groups of apples, each of which contains 4 apples combine to
give a group of 12 apples. Later you learned that
. This did not mean that groups of apples,
each of which contained apples combine to give a group
of apples. The definition had changed, but
whatever it meant, you knew how to get the ``right answer''. At some point
you met decimals.
Since
, you knew
that
.

However was more of
a problem. Although was close to , the two
numbers were not equal. Perhaps you considered *infinite decimals*,
so

Then

But
, So
must be equal to .
They don't look equal, but was probably close enough.
At first the fact that
was probably rather puzzling,
but you got used to it after a while.
You may have encountered the *imaginary number* such that
and
when is positive. Then you found that

and

This may have been unnerving.
At some time numbers became identified with points on a line.
Addition is straightforward: to add two numbers, you just
slide the lines so that they share a common end, and combine them into
one line.

but what does multiplication mean?

** Next:** The Goal of the
** Up:** Introduction
** Previous:** Introduction
** Index**