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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[br]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 [br]to 2600 B.C. to Mesopotamia.

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The Babylonians were not only good,

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were not only producing[br]one of the first literary works,

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The epic of Gilgamesh,

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they were actually[br]quite good at mathematics.

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

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but they were good[br]at mathematics, as I said,

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because they already knew[br]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[br]quadratic equations,

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they could solve them,

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they had a general formula[br]for quadratic equations.

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They could even handle[br]some cubic equations.

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Now, when you solve any equations,[br]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[br]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,[br]- this is an imaginary ball -

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and I give it away,[br]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,[br]and area, and volume,

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

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What they did was[br]eliminating the negative numbers.

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

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Think about the amount of money[br]in your bank account

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if we could only...

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eliminate the negative numbers,[br]that would be great.

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Yes.

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

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And that was because scholars[br]were translating and studying

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Islamic and Byzantine sources.

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Even the great Euler, the genius Euler,[br]who invented the number e

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and much, much more,

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

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Finally, there was a guy John Wallis,[br]an English mathematician,

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and he had a great idea.

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What he did was extending[br]the number line to the left.

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Just as simple.

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Then it became quite clear[br]what a negative number was,

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because if you have two[br]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,[br]Heron of Alexandria,

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and he had a great idea

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

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

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

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

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

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Yes, the Greeks were[br]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[br]of their current financial problem,

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I don't know.

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But if we continue the story[br]about complex numbers,

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we have to time-travel to Bologna,[br]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[br]of a cubic equation,

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and that was really great

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because other mathematicians at that time[br]thought it was impossible,

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

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He even encoded his solution[br]in a form of a poem,

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and my Italian is not good,[br]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[br]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[br]could steal his solution.

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But unfortunately, it was leaked[br]to the other guy, Cardano,

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and he published this proof[br]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,[br]he was acknowledged in the book,

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but he didn't agree, so...

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Tartaglia engaged Cardano[br]in a decade-long fight

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over the publication,

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and the real problem was[br]that this Cardano guy

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

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because he called these imaginary[br]numbers 'mental tortures.'

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Later on, there was another guy,[br]Bombelli, who is below,

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and he was the first one

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who really understood something[br]about complex numbers.

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He could make the link[br]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[br]that we are using today,

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and he made also[br]some rules for calculating.

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In the 17th and 18th century,

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

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but nobody really understood[br]what was going on.

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And then, another guy came,

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and he made a geometrical interpretation[br]of this complex number.

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I will spare you the details,[br]- 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[br]tonight or tomorrow, I don't care.

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(Laughter)

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What he did was, he gave[br]a geometrical interpretation,

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

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so this vertical axis[br]that is the imaginary axis.

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And then it became[br]quite clear what it was.

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

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Then, everybody understood[br]what was going on.

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By analogy, it can be said

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that complex numbers were[br]not only complex, but also absurd,

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until someone gave[br]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[br]or strange combination, but it isn't.

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I like to read stories,[br]and I like to write stories,

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I like doing math,[br]I like to imagine the imaginary.

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A few years ago,

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

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If you read it aloud,[br]you can really hear the rhythm,

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and I know for sure

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that the author thought[br]long and hard about the structure.

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

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

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It's written by Alfred North Whitehead[br]and Bertrand Russell

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who also won[br]the Nobel Prize in Literature.

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It took them over 360 pages

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in order to prove[br]that one plus one equals two.

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So that's not so easy.

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Now, mathematics and literature[br]have something in common.

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

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They are more interrelated[br]than you might think,

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and I think mathematics can learn[br]something from literature.

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Instead of giving you[br]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[br]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[br]are left out of the curriculum.

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Think about the history of mathematics,[br]think about the philosophy of mathematics,

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and think about the applications[br]of mathematics.

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I've seen too many students[br]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)