Visualisons Les Mathématiques Et Tout Le Monde Comprendra

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Transcript

00:00Did you have problems with maths at school?

00:02We didn't teach you in the right way.

00:05Today, I'm going to teach you some basics of mathematics in a visual way,

00:09and all these numbers and letters will finally make sense to you, I promise.

00:14Let's start with this.

00:16You remember this basic equation, a plus b squared.

00:20This equals a squared more than twice a multiplied by b plus b squared.

00:25We learned that as a rule,

00:27and now I'm going to show you that it's actually correct.

00:31Let's make a square with a side, a plus b.

00:35This is the left side of the equation.

00:37We have a side a plus b, and another side a plus b.

00:41So, the surface of the square is a plus b squared.

00:45Now, let's divide the square into four parts like this.

00:48There is a square with a side a and a surface a2,

00:52and another with a side b and a surface b2.

00:56In addition, two rectangles are formed with a side a and a side b.

01:01So, their surface is a times b.

01:04This is the right side of the equation.

01:07Thus, you can see that these surfaces are exactly equal.

01:12Now, let's visualize a similar equation, but with a minus.

01:17a minus b squared equals a squared minus twice a times b plus b squared.

01:23We start with a square with a side a.

01:25So, the surface of the square is, without a doubt, a2.

01:29Then, since we have a subtraction on the left side of the formula,

01:33we must reduce the side of b.

01:36We divide it in this way, this side being equal to b.

01:40Now, this square here has a side, ab, and its surface is ab2,

01:46that is, the left side of the equation.

01:49Now, we must know how we get that.

01:53We start with the big square a2.

01:55Then, we subtract this rectangle whose r is equal to ab.

02:00We still don't have the square we need,

02:03so we must subtract the r from the other rectangle ab here.

02:07But one piece is missing.

02:09So, before we can do that, we must add b2.

02:14That's it, and now we subtract another rectangle ab.

02:17So, as you can see, we now get the r of the square, ab,

02:21exactly as the formula tells us.

02:24One of the most famous theorems of geometry is Pythagoras' theorem,

02:28which states that, in a rectangle,

02:31the square on one side plus the square on the other side is equal to the square of the hypotenuse.

02:36Thus, a2 plus b2ac2.

02:40Now, let's prove that this is true for any rectangle.

02:44But, of course, we will do it visually.

02:47Here is our rectangle triangle.

02:49Let's name its sides a, b, and c.

02:52Now, let's duplicate the triangle and create three others exactly identical to the original.

02:58I will arrange them this way.

03:00They now form two additional shapes,

03:02this one around and this one in the middle.

03:05You guessed right, they are both squares.

03:08But how do we know that?

03:10Let's start with the most obvious.

03:11First, it has four right angles,

03:14which makes it a kind of rectangle.

03:17To be a square, the four sides must be equal.

03:20And, indeed, they are.

03:22Look, here is the side a of a triangle.

03:25And right next to it, there is another triangle whose side is equal to b.

03:29So, the side of this large figure is a plus b, right?

03:34Now, here is another a, and here is another b.

03:37Thus, this side is also a plus b.

03:40We can move on to the third side.

03:42Since the triangle is exactly identical, then this is a.

03:46The fourth triangle is also identical, and this part is equal to b.

03:51The fourth side is also equal to a plus b.

03:55We have established that the large figure is a square with a side, a plus b.

04:00So, the surface inside is a plus b dot 2.

04:04But what about the other shape, the one inside?

04:07As you remember, this is the C of the original triangle.

04:11But the other triangles are identical,

04:14so the four sides of the shape are equal to C.

04:16It is therefore another square, with an R of, yes, C2.

04:21Well, we remember it.

04:23We are going to make a copy of this square and put it aside for a moment.

04:28Now, let's rearrange the triangle this way.

04:31You see, there are now three squares formed.

04:33There is this one.

04:35This side is b, and this side is also b.

04:38So, it's a square with an R of b2.

04:41And there is another one.

04:42Here is the side a, and here is the side a.

04:45So, it's a square with an R of a2.

04:49Now, all together, they form a larger figure.

04:52Once again, it's a square.

04:54And we already know it.

04:56A square with a side, a plus b.

04:58And an R of a plus b2.

05:02Well, do you see where I want to go with this?

05:05We had this square, a plus b2, which was filled with four triangles and a square C2.

05:11Now, I have the same square, a plus b2, with the same four triangles,

05:17but two smaller squares, a2 and b2.

05:21So, it seems that C2 is the same surface as a2 plus b2.

05:26And that is precisely our proof.

05:28And it is only one of the more than 300 proofs that this theorem has so far.

05:33Maybe you can find another one.

05:37Well, we all know the formula to calculate the R of a circle.

05:40It is the radius squared multiplied by the mysterious number p.

05:44It is now time to show you the visual proof.

05:48Let's draw a circle and determine what its R is.

05:51We divide it into several equal parts.

05:53Let's say, to begin with, into 8 parts.

05:55We can arrange these 8 parts in a form similar to a rectangle.

06:00However, this does not help us a lot to calculate the R.

06:03Let's divide it into 16 parts, and once again, let's rearrange them by trying to form a rectangle.

06:10Well, we are getting closer, but that is not enough yet.

06:13We make more parts, and they become smaller.

06:16We now have 24 of them, and the rectangle is now more distinct.

06:20You see where I'm coming from?

06:22The more we divide the circle into parts, the smaller they are.

06:26And the closer they get to a perfect rectangle when we rearrange them.

06:30The smaller the parts, the closer we get.

06:33Look.

06:34Divide.

06:35Rearrange.

06:36It looks pretty good, doesn't it?

06:38We now have to find the R of the rectangle, and it will also be the R of the circle.

06:43This little side here, the height, corresponds to the size of the radius.

06:48The other side, the base, is equal to half the circumference of the circle.

06:53And the circumference is 2πr.

06:56So, the base is simply πr.

06:58Thus, we now see that the R of the rectangle and the R of the circle are R, πr, or πr².

07:06Now, let me clarify just one thing.

07:09The length of the circumference is the role of πr.

07:12Let's make 3 circles, with diameters of 1, 2, and 3.

07:16Here.

07:17The radius of each will be half the diameter, of course.

07:20Here is the first circle.

07:22Its radius is 0.5, and its diameter is 1.

07:25If we roll it on a rule, the length is approximately 3.14.

07:30Now, let's take the next circle.

07:32Its diameter is 2, its radius is 1.

07:35We roll it, and the length is about 6.28, which is 3.14 multiplied by 2.

07:41The last circle, with a diameter of 3 and a radius of 1.5.

07:46You know the procedure, we roll it and measure it.

07:49The length is now about 9.42, or 3.14 multiplied by 3.

07:54This mysterious number is called πr.

07:56In elegant terms, it is the ratio between the circumference of a circle and its diameter.

08:01And, as you have seen, it is always identical.

08:04That's it for today.

08:06I hope I have made mathematics a little more understandable.

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