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(Spanish) Buenas noches.

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Welcome to math class!

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

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(Laughter)

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OK, that was a joke.

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But raise your hand
if you love mathematics.

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

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

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

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The Babylonians were not only good,

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were not only producing
one of the first literary works,

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The epic of Gilgamesh,

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they were actually
quite good at mathematics.

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

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but they were good
at mathematics, as I said,

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because they already knew
the Pythagorean theorem,

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and that is quite remarkable,

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because Pythagoras wasn't even born yet.

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(Laughter)

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They also could handle
quadratic equations,

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they could solve them,

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they had a general formula
for quadratic equations.

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They could even handle
some cubic equations.

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

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and negative numbers are not that easy.

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Let me give an example.

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

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then I give away one, two...

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and then what?

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

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

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Minus one imaginary ball.

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(Laughter)

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Well, the Greek mathematicians

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were working with length,
and area, and volume,

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

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What they did was
eliminating the negative numbers.

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

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Think about the amount of money
in your bank account

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if we could only...

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eliminate the negative numbers,
that would be great.

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Yes.

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

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And that was because scholars
were translating and studying

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Islamic and Byzantine sources.

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

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and much, much more,

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

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

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and he had a great idea.

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

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Just as simple.

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Then it became quite clear
what a negative number was,

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because if you have two
and you subtract three,

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you end up in minus one.

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So that was quite clear.

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But what about complex numbers?

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

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and he had a great idea

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

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

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

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

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

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Yes, the Greeks were
very inventive with numbers.

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(Laughter)

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They still are.

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(Applause)

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Maybe, maybe, maybe...

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

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I don't know.

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But if we continue the story
about complex numbers,

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

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There was a guy named Tartaglia,

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and he won a mathematical competition.

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He wrote about the solution
of a cubic equation,

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and that was really great

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because other mathematicians at that time
thought it was impossible,

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

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

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

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It goes something like this:

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

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se agguaglia à qualche numero discreto."

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It was a long poem,

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and he made this in order to prevent

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that other mathematicians
could steal his solution.

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

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

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But he'd promised not to do so.

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

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but he didn't agree, so...

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Tartaglia engaged Cardano
in a decade-long fight

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over the publication,

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and the real problem was
that this Cardano guy

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

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because he called these imaginary
numbers 'mental tortures.'

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

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and he was the first one

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who really understood something
about complex numbers.

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He could make the link
between the real numbers,

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

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and the complex, imaginary numbers.

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So he was the first one.

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He introduced the symbol i
that we are using today,

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and he made also
some rules for calculating.

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In the 17th and 18th century,

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

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but nobody really understood
what was going on.

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And then, another guy came,

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and he made a geometrical interpretation
of this complex number.

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

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so I will spare you the details,

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

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(Laughter)

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What he did was, he gave
a geometrical interpretation,

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

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so this vertical axis
that is the imaginary axis.

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And then it became
quite clear what it was.

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

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Then, everybody understood
what was going on.

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By analogy, it can be said

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

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until someone gave
a geometrical interpretation.

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

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

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

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

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A few years ago,

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

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

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and I know for sure

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that the author thought
long and hard about the structure.

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

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

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It's written by Alfred North Whitehead
and Bertrand Russell

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who also won
the Nobel Prize in Literature.

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It took them over 360 pages

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in order to prove
that one plus one equals two.

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So that's not so easy.

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Now, mathematics and literature
have something in common.

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

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They are more interrelated
than you might think,

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and I think mathematics can learn
something from literature.

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Instead of giving you
the definition of a complex number

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and giving some rules for calculating,

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I told you a story.

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

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instead of endless algebra exercises.

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Without stories,

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mathematics become maybe boring,

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and without stories,

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

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

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and think about the applications
of mathematics.

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

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because of the way we teach the subject.

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And this, ladies and gentlemen,

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can only be improved by telling stories.

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Thank you.

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(Applause)