I have problems,
I’m a mathematician (Laughter)
Mathematicians have problems, many problems.
Well I know that others also have problems.
But mathematicians like problems.
A good problem is a gift for a mathematician.
An excellent problem is a gift for the entire mathematical community.
And a classic problem is a gift, I believe, for all humanity.
And because?
Well, I’m going to tell you a story about a problem, its solution,
and a couple of messages that I bring
on the part of mathematicians for mortals.
Let’s see … a little thing … a little something before.
In this talk there is mathematics.
That is, equations, formulas, numbers … all that.
So, if there are impressionable people here,
that is going through a nervous breakdown, minors …
In the hard moments of the talk, cover your eyes.
Support one another, feel the warmth of those around you,
there is medical equipment available; everything will be fine, do not worry.
We are prepared?
I start.
A while ago, a few months ago, I posted on the bulletin board
from the math department where I work,
I placed a problem, a problem, this problem:
The question is whether the perimeter of that square is larger or smaller
than the length of the circumference, of that circumference.
Think about it Look a bit at the problem.
It is not difficult, it is not a difficult problem.
I see restless people, as if wanting to cry …
I am going to give the solution, do not worry.
If someone does not want to hear it, this is a party pooper of my talk.
I will give you the solution.
To solve this problem, I asked people to send me solutions.
Anyone who had a solution that would make it come to me.
Let me get the solution I had.
It was simpler, more complex. The one they wanted.
And this is what I did. I solved it, too.
Suppose we can trace from the center of the circle …
– Now the mates begin, huh? Watch out. Impressionable people: the mates begin –
From the center of the circle we can draw a line to the corner,
or to the midpoint of that square.
Suppose that the radius of the circumference is one.
For this problem it is not relevant.
So in that triangle, that side measures one.
This other, as it is half the side of the square, measures L means.
And this other one measures L minus 1. Why L-1?
Because, if this is 1, that’s also a radio, right?
That other radius is one and all this square is L.
For what 1 lacks to be L is L minus 1, and there it is.
We have a right triangle with its hypotenuse and its legs.
The words hypotenuse and legs are only used for one thing in life.
In all your life, you only use them for one thing:
The Pythagorean Theorem.
Out of there, those words do not exist, we never use them.
It is your moment of glory to say:
“I know the Pythagorean Theorem.
The square of the hypotenuse is the sum of the squares of the legs. “
This is the magic of Pythagoras, which allows you to translate those lines,
the geometry, the circle, the triangle … in equations, in algebra.
With that we can work,
and doing those manipulations that mathematicians do with algebra,
Draw conclusions from the geometry, from the lines.
Then one puts the equation that gives us the Pythagorean Theorem,
What about the squares, there you have it.
Makes those manipulations that mathematicians do with accounts
and it reaches that the side of the square is eight fifths.
Eight fifths Very good.
You stay at ease, and you say: “eight fifths”.
So the perimeter is four times that: 32 fifths.
The length of the circumference is 2 pi.
OK, and what is bigger? 32 fifths or 2 pi?
One can take the calculator, and one can also divide.
I divide this between two: 16 fifths on the one hand, pi on the other.
And 16 fifths, how much is it?
16 fifths is 3.2, which we all know is bigger than pi.
Because pi is 3.14 and peak, right?
So, dear friends, finish your worry,
You can relax, not too much, but you can stop crying.
The perimeter of the square is greater than the circumference.
And one feels relieved, well, goes back to sleep at night.
Good.
Well this is the problem, this is the solution.
I received 25 solutions for 25 people for this problem.
They wrote to me: professors, classmates of the university, students,
children of some companions …
Everyone wrote me with solutions to this problem.
I received 25 different solutions, and 11 were essentially different.
They used different mathematics.
There were those who used ancient mathematics:
power of a point with respect to the circumference,
similarity of triangles, areas of triangles.
There were people who used more modern mathematics:
they used Cartesian coordinates, the equation of the circumference …
There were even those who used complex numbers to solve it.
Mathematics still more modern.
But all the solutions, all …
All had in common that in the end they depended on 3.2 is greater than pi.
With all this I decided to organize a talk at my university,
and tell the people in my department and everyone who wants these solutions.
The different ways to solve this problem.
It seemed beautiful to me.
Because different solutions had been given,
each according to their baggage, according to their preferences.
We could even say,
According to his personality they had solved that problem.
It seemed like a nice subject for a talk.
I organized a talk in my faculty,
I prepared this talk at the University of La Rioja.
The teachers came, those who had sent solutions, students came,
Teachers came from the institutes, my high school teacher came!
The one who gave me my math!
With which I took my first steps in real mathematics.
Surely he would be proud, because of the kid he met at 15 years old,
who introduced in mathematics, then look!
Now giving talks at the university, researching, a doctor of mathematics …
I’m sure he was proud.
Well, the fact is that the talk went well.
It was very entertaining.
In the end we were discussing, the discussion was interesting,
We were discussing if that problem could not be solved
without knowing how much pi is worth
Not knowing that pi is smaller than 3.2.
Not knowing if it’s worth 3, or 4, 15, or 16 …
A purely geometric solution,
that will only explore
the relations between that square and that circumference.
We had talked about it before, some colleagues.
We also talked about it there and the truth is that we did not get it.
But hey, we would say it would be a nice solution, that.
Three days later I received an email
with an answer to that problem, a solution that did not use the value of pi.
That only used the relations between the square and the circumference.
A purely geometric solution.
It was sent to me by my high school teacher.
So many years later, and so many maths later,
He is still my teacher.
He is still my teacher. He keeps teaching me.
There are teachers who are for life and I am lucky to have met one.
I believe that we can all be teachers for life.
We just need to be willing to always learn.
Because education is many things.
Teaching is many things.
And if there is one thing that is to teach, it is to share what has been learned.
And if you are not willing to continue learning all your life,
he can not be a teacher all his life.
So here I bring you these two messages, from the mathematicians for you.
They are two very obvious messages.
They are messages that we all know.
What happens is that sometimes we forget,
and mathematicians, by our profession, we deal with them every day.
One: “A problem is always an opportunity to learn”.
Always always.
Not only mathematical problems, which are exercises to learn.
Every problem is an opportunity to learn.
The problems are coming, sure, that is inevitable.
But one can decide if he learns from them or not.
So the first thing I bring from the mathematicians is:
One: “A problem is an opportunity to learn”.
And the second, to reach the solution to a problem, there are different ways.
Many different ways.
They will probably depend on your baggage, your preferences,
even your personality.
If that happens for a mathematical problem, how is it not going to happen for the others?
So there are many ways, it is the second message:
“There are many ways to get to the solution of a problem.”
I think that after these minutes you get a little better idea
why mathematicians like problems so much.
Although it is only a problem of knowing
if the perimeter of a square is more or less large
than a circle.
So, we keep learning.
Happy problems
Thank you very much.
(Applause)
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