I just discovered this absolutely insane identity, am I the next oiler? sqrt(1/2) vs sqrt(2)/2

I just discovered this absolutely insane identity. Am I the next Euler? Well, I know this is just for fun, so I'm gonna have some fun with you too. Firstly though, it's not spelled as O-I-L-E-R. I know this right here, you will just say as Euler. But what you really want to say is E-U-L-E-R. That's how you really spell Euler. But this right here does look like Euler. Yeah, but it's really, you know, you say it as Euler. But anyways, so let's take a look at the math. Firstly, we have square root of 1 over 2, and this is approximately 0.7071. And that's the same as square root of 2 over 2, which is also approximately the same as 0.7071. So how is this possible, right? Well, firstly, I'm going to show you how to go from here to here without a decimal. And secondly, I'm going to tell you guys why we do that procedure. Not just to make them equal, but it's actually much better if you do the computation. So here we go. If you start with square root of 1 over 2, what we can do is this is equal to square root of 1 over square root of 2. Because both of the 1 and 2 are positive, so this is totally okay. When you have a square root of a quotient, it's the same as the quotient of the square root, especially when both things inside, the top and bottom, are both positive. So this is totally okay. Now, can we work out square root of 1? Sure thing. That's the same as 1 times 1, which is 1. So square root of 1 is 1. And then square root of 2, what times what is equal to 2? Well, I don't know, right? Unless we use decimal. So I'm just going to keep it as square root of 2. Now, this does not look like that. How can we get from here to here? Well, here's the deal. Notice square root of 2 is what we call an irrational number because you cannot think about two numbers. They multiply to give you 2, right? So let's do this. I'm actually going to multiply square root of 2. And then we have to do the same thing on the top. Because this is like multiplying by 1. They cancel. It's like multiply by 1. When you have a fraction, multiply by 1, it does not change it. So you have to multiply the top and bottom by the same thing. Now, this step right here, notice we have now square root of 2 times square root of 2. What we can do is because both things inside are positive, we can multiply inside. So this equals square root of 4. Can we think about two numbers they multiply to be 4? Yes, the answer is just a 2. And now it's just a regular 2. So the bottom is just a regular 2 now. And this is actually so much better. And this step right here, when we clean up the irrational number from the denominator, this is called rationalizing the denominator. And let's finish it because 1 times square root of 2 is just square root of 2. It's like I sacrificed the bottom, but now the square root number, the irrational number is on the top. The question is like, why do we do that? Well, of course, firstly, I just show you like how this is equivalent to that. So we are done. But why do we do it? Here is the deal. Well, imagine that you don't have a calculator where you are trying to compute this by hand. 1 over square root of 2, which is this part right here, versus square root of 2 over 2, right? Let's do that. How do we do it? Well, I will have to tell you what square root of 2 is about. I'll just say not. And I can make another video showing you guys how to approximate square root of 2. But for now, I will just have to tell you. You guys have to trust me. Square root of 2 is approximately 1.414. What's the next digit? It's not 1.414. Continue. You let me know. But I'm just going to write down this much right here, right? So with this being granted, I will say this right here is approximately 1 on the top over the square root number. Square root of 2 is 1.414. Okay, how do we divide? Is a calculator? No, let's do long division. How though? Remember, you always put the top number inside and then the bottom number on the outside. And the truth is, this right here is not done yet. We have infinitely many digits. Okay, so how do we even start? Well, if I just put the three decimal places here, if you do long division, you will have to move this three decimal, like the decimal to the right three times. So that means you will have to put three zeros, right? We have one. Technically, it's like 1.00, but I will have to do the same thing, like move the decimal points three times to the right. This is like the traditional way to do long division or something like that. I know we don't do long division anymore once we have a calculator. Nowadays, it's even worse. We have charge EBT and AISO. But anyways, that's how you do it. And then you ask yourself, does 1,414 go into 1,000? No, technically zero times, right? So you will have to start with a zero, and then you line out this decimal point here. And once you put this down, you can put another zero and you continue. How many times does this go into 10,000? It's kind of hard. I will tell you seven times though. Seven? Seven times this, I will do it like this. Seven times four is 28. So F will have eight. Remember the two? Seven times one is seven. Add it with a two, we have nine. Continue. Seven times four is 28, which is two. And then seven times one is seven. So it's nine. And then you subtract. So many numbers. It's so hard. Yeah, not so fast. But however, if you did this procedure, which is called rationalizing the denominator, check this out. This now is approximately. This is 1.414, approximately speaking. And then divide it by two. Now, how do you work this out? I think many of you guys can do this in your head, right? Yes. Just think about this as like, if you have $1,414 and then divide it by two, you will have 707. And now just put decimal point right here. So the answer is 0.707. This is equal to that. This is approximately equal to this. So yeah. So as you can see, if we have a rational number, just like a regular number, like two here, it's much easier to go through the division process. And I believe. Yeah, that's it.