(Spanish) Buenas noches.

Welcome to math class!

The coming 9,000 seconds you'll be mine.

(Laughter)

OK, that was a joke.

But raise your hand
if you love mathematics.

Oh, that's a lot. Mmm. (Laughter)

Mmm, that will be a tough one. (Laughter)

Let's take you back 
to 2600 B.C. to Mesopotamia.

The Babylonians were not only good,

were not only producing
one of the first literary works,

The epic of Gilgamesh,

they were actually
quite good at mathematics.

The epic of Gilgamesh was written
in cuneiform on clay tablets,

but they were good
at mathematics, as I said,

because they already knew
the Pythagorean theorem,

and that is quite remarkable,

because Pythagoras wasn't even born yet.

(Laughter)

They also could handle
quadratic equations,

they could solve them,

they had a general formula
for quadratic equations.

They could even handle
some cubic equations.

Now, when you solve any equations,
you often get negative solutions,

and negative numbers are not that easy.

Let me give an example.

If I have two tennis balls
and if I have to give away three,

then I give away one, two...

and then what?

Well, let's create an imaginary ball,
- this is an imaginary ball -

and I give it away,
so what do I have left?

Minus one imaginary ball.

(Laughter)

Well, the Greek mathematicians

were working with length,
and area, and volume,

so they didn't need negative numbers,
they only kept the positive ones.

What they did was
eliminating the negative numbers.

Now, that's a great way
to deal with problems, isn't it?

Think about the amount of money
in your bank account

if we could only...

eliminate the negative numbers,
that would be great.

Yes.

Negative numbers only began to appear
in Europe in the 15th century.

And that was because scholars
were translating and studying

Islamic and Byzantine sources.

Even the great Euler, the genius Euler,
who invented the number e

and much, much more,

didn't quite understand
negative numbers as we do today.

Finally, there was a guy John Wallis,
an English mathematician,

and he had a great idea.

What he did was extending
the number line to the left.

Just as simple.

Then it became quite clear
what a negative number was,

because if you have two
and you subtract three,

you end up in minus one.

So that was quite clear.

But what about complex numbers?

Well, there was a Greek mathematician,
Heron of Alexandria,

and he had a great idea

because in his work, the number,
the square root of minus 63 appeared,

and what he did was replacing it
by the square root of 63.

So, he replaced a minus by a plus.
Now that's even better, right?

Think about the amount of money
in your bank account now,

if we could only replace
a minus by a plus; well, that's great!

Yes, the Greeks were
very inventive with numbers.

(Laughter)

They still are.

(Applause)

Maybe, maybe, maybe...

Maybe, I don't know, maybe, that's part
of their current financial problem,

I don't know.

But if we continue the story
about complex numbers,

we have to time-travel to Bologna,
Renaissance Italy, 16th century.

There was a guy named Tartaglia,

and he won a mathematical competition.

He wrote about the solution
of a cubic equation,

and that was really great

because other mathematicians at that time
thought it was impossible,

because it required an understanding
of the square root of a negative number.

He even encoded his solution
in a form of a poem,

and my Italian is not good,
but let me try the first two sentences.

It goes something like this:

(Italian) "Quando chel cubo
con le cose appresso,

se agguaglia à qualche numero discreto."

It was a long poem,

and he made this in order to prevent

that other mathematicians
could steal his solution.

But unfortunately, it was leaked
to the other guy, Cardano,

and he published this proof
in his book "Ars magna" in 1545.

But he'd promised not to do so.

Tartaglia was mentioned in the book,
he was acknowledged in the book,

but he didn't agree, so...

Tartaglia engaged Cardano
in a decade-long fight

over the publication,

and the real problem was
that this Cardano guy

didn't even understand
what he had written down in the book,

because he called these imaginary
numbers 'mental tortures.'

Later on, there was another guy,
Bombelli, who is below,

and he was the first one

who really understood something
about complex numbers.

He could make the link
between the real numbers,

- the normal numbers, 1, 2, 3, 4, -

and the complex, imaginary numbers.

So he was the first one.

He introduced the symbol i
that we are using today,

and he made also
some rules for calculating.

In the 17th and 18th century,

there were a lot of mathematicians
working with the complex numbers,

but nobody really understood
what was going on.

And then, another guy came,

and he made a geometrical interpretation
of this complex number.

I will spare you the details,
- that's homework -

so I will spare you the details,

you figure out yourself when you come home
tonight or tomorrow, I don't care.

(Laughter)

What he did was, he gave
a geometrical interpretation,

and he didn't create this imaginary ball,
no, he created an imaginary axis,

so this vertical axis
that is the imaginary axis.

And then it became
quite clear what it was.

A complex number was
a 2-dimensional number: a plus i b.

Then, everybody understood
what was going on.

By analogy, it can be said

that complex numbers were
not only complex, but also absurd,

until someone gave
a geometrical interpretation.

Now, I'm a math teacher and an author,

and that may sound like a rare
or strange combination, but it isn't.

I like to read stories,
and I like to write stories,

I like doing math,
I like to imagine the imaginary.

A few years ago,

I read this proof,
this beautiful poem, isn't it?

If you read it aloud,
you can really hear the rhythm,

and I know for sure

that the author thought
long and hard about the structure.

And every word, and every sign
is written down with the highest care.

It is taken from "Principia Mathematica",
beginning of the 20th century.

It's written by Alfred North Whitehead
and Bertrand Russell

who also won
the Nobel Prize in Literature.

It took them over 360 pages

in order to prove
that one plus one equals two.

So that's not so easy.

Now, mathematics and literature
have something in common.

They've been a part of our human culture
for thousands of years.

They are more interrelated
than you might think,

and I think mathematics can learn
something from literature.

Instead of giving you
the definition of a complex number

and giving some rules for calculating,

I told you a story.

In my talk, I made the case for telling
stories in mathematical education

instead of endless algebra exercises.

Without stories,

mathematics become maybe boring,

and without stories,

some important aspects of mathematics
are left out of the curriculum.

Think about the history of mathematics,
think about the philosophy of mathematics,

and think about the applications
of mathematics.

I've seen too many students
that don't follow mathematics

because of the way we teach the subject.

And this, ladies and gentlemen,

can only be improved by telling stories.

Thank you.

(Applause)