Prêt à mettre votre cerveau à l'épreuve ? Rejoignez-nous pour une séance amusante et captivante où nous résolvons des énigmes mathématiques à l'aide de la visualisation : comprendre va devenir super facile. C'est un moyen fantastique de vous mettre vous-même au défi, d'affûter vos compétences en résolution de problèmes et peut-être même d'apprendre une astuce ou deux en cours de route ! Que vous soyez un as des maths ou simplement à la recherche d'un exercice mental amusant, ces énigmes visuelles sont parfaites pour tout le monde. C'est parti ! Voyons à quelle vitesse vous pouvez résoudre ces casse-têtes et montrez-nous vos compétences ! Vous pourriez même vous découvrir une nouvelle passion pour les mathématiques ! Animation créée par Sympa.
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Musique par Epidemic Sound https://www.epidemicsound.com
Pour ne rien perdre de Sympa, abonnez-vous!: https://goo.gl/6E4Xna
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Nos réseaux sociaux :
Facebook: https://www.facebook.com/sympasympacom/
Instagram: https://www.instagram.com/sympa.officiel/
Stock de fichiers (photos, vidéos et autres):
https://www.depositphotos.com
https://www.shutterstock.com
https://www.eastnews.ru
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Si tu en veux encore plus, fais un tour ici:
http://sympa-sympa.com
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FunTranscript
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.