Math in a Visual Way Everyone Will Understand

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Ready to put your brain to the test? Join us for a fun and engaging session where we solve math riddles using visuals that make it super easy to understand. It’s a fantastic way to challenge yourself, sharpen your problem-solving skills, and maybe learn a trick or two along the way. Whether you're a math whiz or just looking for a fun mental workout, these visual riddles are perfect for everyone. Come on, let’s see how quick you can crack these puzzles and show off your skills! You might even discover a new love for math.  Animation is created by Bright Side. ---------------------------------------------------------------------------------------- Music by Epidemic Sound https://www.epidemicsound.com   Check our Bright Side podcast on Spotify and leave a positive review! https://open.spotify.com/show/0hUkPxD34jRLrMrJux4VxV   Subscribe to Bright Side: https://goo.gl/rQTJZz  ---------------------------------------------------------------------------------------- Our Social Media: Facebook: https://www.facebook.com/brightside  Instagram: https://www.instagram.com/brightside.official TikTok: https://www.tiktok.com/@brightside.official?lang=en   Stock materials (photos, footages and other): https://www.depositphotos.com https://www.shutterstock.com https://www.eastnews.ru ----------------------------------------------------------------------------------------  For more videos and articles visit: http://www.brightside.me

Transcript

00:00If you had problems with math in school, maybe they weren't teaching you the right way.

00:04Today, I'll teach you some basic math visually,

00:08and Latin letters and numbers will finally make sense to you.

00:11I promise.

00:13Let's start with this one.

00:14Do you remember this basic equation, a plus b squared?

00:17It's equal to a squared plus 2 times a multiplied by b plus b squared.

00:24We learned it as a rule and accepted that it's true.

00:27But now, I'll make you see that it's indeed correct.

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

00:34That's the left part of the formula, right?

00:37One side is a plus b, and the other side is a plus b.

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

00:45Now let's break the square into four pieces like this.

00:48There's a square with a side of a in the space of a squared,

00:52and another one with a side of b in the space of b squared.

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

01:01So, their space is a by b.

01:04This is exactly the right side of the formula.

01:07So, now you can see that these spaces are exactly equal.

01:11Now let's have a visualization of a similar formula,

01:14but with a minus.

01:16a minus b squared is equal to a squared minus 2 by a by b plus b squared.

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

01:25So, the area of the square is, without a doubt, a squared.

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

01:33we need to make the side smaller by b.

01:36So, we divide it like this, with this side being equal to b.

01:40Now, this square right here has a side of a minus b,

01:44and its area is a minus b squared,

01:47which is the left side of the equation here.

01:50Now we need to track how we get it.

01:53We start with the big A squared square.

01:55Then we subtract this rectangle whose area is equal to a times b.

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

02:03so we need to subtract the area of the other a times b rectangle here.

02:07But it's now missing a piece.

02:09So, before we can do that, we need to add the b squared.

02:14Yes, and now we subtract another a b rectangle.

02:17So, as you see, now we got the area of a minus b squared,

02:21just like the formula tells us.

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

02:28which states that, for a right triangle,

02:30the square of one side plus the square of the other side

02:33is equal to the square of the hypotenuse.

02:36So, a squared plus b squared equals c squared.

02:40Now, let's prove that it's right for any right triangle.

02:44But, of course, we're going to do it visually.

02:48So, here's our right triangle.

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

02:53Now, let's duplicate the triangle

02:55and make three more that are exactly like the original one.

02:58I'll arrange them this way.

03:00They're now forming two additional shapes,

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

03:05You guessed it right.

03:06Both of these are squares.

03:08But how do we know that?

03:10Let's start with the big one.

03:11First, it has four right angles,

03:14which makes it some kind of a rectangle.

03:16To be a square, all four sides must be equal.

03:19And, indeed, they are.

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

03:25And right next to it,

03:26there's another triangle with the side equal to b.

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

03:33Now, this is another a and this is another b.

03:37So, this side is a plus b, too.

03:40Off to the third side.

03:41Since it's the exact same triangle,

03:44then this is a.

03:46The fourth triangle is the same, too.

03:48And this part is equal to b.

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

03:54Now, we established that the big figure is a square

03:58with a side of a plus b.

04:00So, the area inside it is a plus b squared.

04:04But what about the other shape,

04:06the one that's inside?

04:07As you remember,

04:08this one is c from the original triangle.

04:11But the other triangles are the same.

04:13So, all four sides of the shape are equal to c.

04:16So, it's another square with an area of,

04:19you guessed it, c squared.

04:22Okay, we remember it.

04:23In fact, let's make a duplicate of that square

04:25and put it aside for a little bit.

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

04:30You see, now there are three squares formed.

04:33There's this one.

04:34This side is b and this side is b, too.

04:37So, it's a square with an area of b squared.

04:41And there's another one.

04:42This is side a and this is side a.

04:45So, it's a square with an area a squared.

04:48Now, all together, they form a bigger shape.

04:51Once again, it's a square.

04:53And one, we already know,

04:55a square with a side of a plus b

04:57in the area of a plus b squared.

05:01Okay, so do you see where I'm driving at?

05:03We had this a plus b squared square

05:06that was filled with four triangles

05:08and one c squared square.

05:11Now, I have the same a plus b squared square

05:14with the same four triangles,

05:16but two smaller squares, a squared and b squared.

05:20So, it seems like c squared is the same area

05:23as a squared plus b squared.

05:26And this is exactly our proof.

05:28This is just one of over 300 proofs

05:31this theorem has so far.

05:33Maybe you can come up with some other one.

05:36Now, we all know the formula for the area of a circle.

05:40It's the squared radius multiplied by the mysterious number pi.

05:44Now, it's time to show you the visual proof of it.

05:48Let's draw a circle,

05:49and we want to figure out what its area is.

05:52We divide it into several equal parts.

05:54Let's say into eight parts, for starters.

05:57We can arrange these eight parts into a rectangle-like shape.

06:00Still, this doesn't help us much to figure out the area.

06:04Let's divide it into 16 pieces then.

06:06And once again, rearrange them trying to form a rectangle.

06:10Now, it's already closer, but still not enough.

06:14We make more pieces, and they turn smaller.

06:17It's 24 now, and the rectangle is more distinctive now.

06:20Do you see what I'm getting at?

06:22The more pieces we divide the circle into,

06:24and the smaller they are,

06:25the closer they are to being a perfect rectangle

06:28when we rearrange them.

06:30The smaller the pieces, the closer it gets.

06:33Look at this one now.

06:34Divide, rearrange.

06:36Looks good enough, huh?

06:37So, now we need to find the area of the rectangle,

06:40and it'll be the area of the circle too.

06:43This smaller side right here, the height,

06:46is the size of the radius.

06:48The other side, or the base,

06:49is equal to half of the circumference of the circle.

06:52And the circumference is 2 pi r,

06:55so the base is just pi r.

06:57So, now we see that the area of the rectangle

07:00and the area of the circle is r times pi r,

07:04or pi r squared, 2.

07:07Now let me clarify just one thing,

07:09the length of the circumference and how pi comes into play.

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

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

07:20Here we take the first circle.

07:22The radius is 0.5, and the diameter is 1.

07:26If we roll it on the ruler, the length is approximately 3.14.

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

07:32The diameter is 2, the radius is 1,

07:35we roll it, and the length is approximately 6.28,

07:38which is 3.14 multiplied by 2.

07:42The last circle with the diameter of 3 and the radius of 1.5.

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

07:49The length is now approximately 9.42,

07:52or 3.14 multiplied by 3.

07:54This mysterious number got called pi.

07:57In fancy words, it's the ratio of the circumference of a circle

08:00to its diameter.

08:01And as you've seen, it's always the same.

08:04So, that's it for today,

08:05and I hope I made math a bit more understandable for you.

08:08That's it for today.

08:09So, hey, if you pacified your curiosity,

08:11then give the video a like and share it with your friends.

08:14Or if you want more, just click on these videos

08:16and stay on the Bright Side.

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