Yeah. It's interesting that humanity had to spent so much time to make numbers "complete". There are quaternions, though, but they don't have some properties which imaginary numbers have.

Does been able tomae undefined numbers and singularities still keep it closed?

It is still possible to define further numbers; even natural numbers are “made up.” Check out the dual numbers, a separate number system where the new unit E^2 = 0

@@Kokurorokuko imaginary numbers also lack some properties of real numbers. Inequalities dont work anymore

Sets are only closed under particular operations, so we need to specify which operations and sets we're talking about for that statement to be meaningful.

Yeah, that's what I thought about right when I saw the thumbnail.

@@PunmasterSTP Because it is so much easier

This is quite neat.

Also my method. Had it in three steps.

So much uglier, but works

lmfaooo

LMAO

Perfection

welp he's definitely not asian

SAME. Everyone always thinks it's weird, but it is important to solving math entirely. Well, every number has a role. But I like 4's role.😊😊😊

Literally the same for me

I was studying complex numbers and this got recommended

@@thedarkslayer9475 same

Apple is listening through our phones I swear

Same for me. I think the video was recommended based on our search histories and overall interests.

before it was official, people used imaginary numbers to refer to any numbers that have an imaginary part

@@user-sb9ho5jz3e Official? Like what time? Cardano's time? Descartes time? Euler's time? Gauss time? Cauchy time? What is "official complex numbers"?

blackpenredpen found the √i using both this method and using polar coordinates.

Nice video but i is not equal to sqrt(-1) : suppose sqrt(-1) exist, so sqrt(-1)² = sqrt(-1)×sqrt(-1) = sqrt(-1×-1) = sqrt(1) = 1; wich is different of i², which is -1 So i is not a number that has a value, his only property is i²= -1, so your starting point is wrong, nice try though

@@themlaw9895 it is wrong √(-1)×√(-1)= √[(-1)×(-1)=√1=1 The property √a×√b=√ab holds if and only atleast one of the number is positive that is either a≥0 or b≥0

@@themlaw9895 The property of exponents you just used isn't true for complex numbers, but you enter that field when you talk about √-1. i is as real as real numbers, and nature works in complex numbers, not real ones.

I am excited to learn upper class math🤩, but I haven't books

Cool!

this, its a beautiful way to see it

Amazing

Nice 😊👍👍👍

What is the concept of rotating imaginary numbers called?

@@kaan8964 it’s really more of the general interpretation of multiplication in general. It’s a great interpretation. I recommend 3blue1brown’s video on complex numbers, he gives a wonderful way to rethink what addition and multiplication actually mean. The way to think of complex numbers falls right out of that.

@@kaan8964 complex numbers are in a form of a+bi but so is r*e^(i*theta), and theta is the rotation part. Example: i is equal to e^(i*pi/2) because it is rotated pi/2 radians from the +Real axis. -1 = e^(i*pi) because it is rotated pi radians, which is why when you multiply by negative numbers, you 'flip' to it's negative side.

No. Do you know what the definition of a function is?

@@angelmendez-rivera351 Oh boy. Look what we have here. A person who seems to correct every little thing just to feel superior because they want to demonstrate that they know what they're talking about. All you really demonstrate is that you're insecure and probably terrible at that which you boast to practice with such skill. There is something called a multivalued function. This is clearly what I meant by "function." This association is hardly needed when you're talking to people who are somewhat competent. In complex analysis, the definition of function is extended. If you knew that, you wouldn't be trying to correct me. It remains a fact that the nth root is a multivalued function on C, and every nonzero complex number has n different complex nth roots.

@@PubicGore *Oh boy. Look what we have here. A person who seems to correct every little thing just to feel superior because they want to demonstrate that they know what they're talking about. All you really demonstrate is that you're insecure and probably terrible at that which you boast to practice with such skill.* Being said by someone who is being outwardly and intentionally condescending, and putting their train of insults before any valid arguments, this is just meaningless bluster. Did you get it out of your system? Did that stroke your ego enough? *There is something called a multivalued function. This is clearly what I meant by "function."* I know what it is you are talking about, but your misuse of terminology was worth pointing out. "Multivalued function" is a contradiction in terms. *In complex analysis, the definition of function is extended. If you knew that, you wouldn't be trying to correct me.* No. I know you are wrong, and I have more than sufficient education to know that. Your ignorant self being in a state of denial does not change this fact. I know the study of Riemann sheets is important in complex analysis and projective geometry, and they are also an an important foundation for studying presheafs. It changes absolutely nothing about the fact that you are wrong. No textbook in complex analysis defines function differently than in any other context. If you actually were literate on the subject, you would never talk about "multivalued functions." Almost no one educated on the subject does.

@@PubicGore woah chill, you wrecked that guy-

How's 8th grade going?

pretty well so far, math score really good and I am going to be one of the 5 people in my school to do the mathematics national competition in my country (I hope that goes well)

​@@data50090Gl

dang howd that go?

@@kaauchpohtato it did not go well

that same guy: ...in real values? Me: 4throot(-1)

Hi there.

Ikr 1=2a² 1/2 = a² a = b= 1/√2

That's basically De Moivre's theorem

@@WaterMeetsLava Oh! Good to know! I'll look it up!

Think of the case of sqrt(-i)... then the roots are (1-i) and (-1+i) which one qualifies as positive?

@@simong1666 When dealing with b⁴ = 1 you only get 1 as answer, not -1 Because sqrt(36) = x x² = 36 Now we have created an extraneous solution x = -6 Because of principal root, sqrt(36) only equals 6. If talking about all roots, sqrt(36) = 6, -6 So because we squared the square root, when we do the inverse of x² = 36, we just do the step x = sqrt(36), and not x = ±sqrt(36) In the video he did ± which I think is wrong if we're talking about principal root, which we usually are

the whole point of complex numbers is to calculate all roots, not just the principal root. As Simon G pointed out, the concept of a principal root doesn't even make sense for roots of complex numbers (with a non-zero imaginary part). So I don't think we atre talking just about principal roots here.

The question is then which direction did you rotate. You can go the long way around the circle, or the short way (-1 had both paths be the same length) the path you took to get the destination determines whether the answer has a negative sign. Half of 360° is 180°, but 360° is the same as 0° and half of 0° is still 0°, and 0° is not the same as 180°.

@@angeldude101 It doesn’t matter because square roots can be positive or negative. sqrt(1) can be +1 or -1.

Of course, on a circle, π/2 radians is the same as -3π/2 radians, and half of that would be -3π/4. Plugging in the values gives you the negative answer. I will say, im not sure why it was surprising to the videomaker that there was two answers? Thats how the square root "function" works for every other number (that fits on the complex plane - i dont exactly know ALL math, so maybe there is a number that has only 1 square root?)

🧐

You miss a solution that way. If you start with e^(2nπi+iπ/2)=i, then take the square root, (to account for periodicity), then you’ll see you actually get two distinct solutions. The other is e^(5πi/4), which when you put into a+bi form gives -√2/2−i√2/2, which indeed gives i when squared.

Yes, and actually using the square root symbol is abuse of notation when dealing with complex numbers. A better way of putting the problem is "Solve the quadratic equation x^2=i" and that equation has....TWO (complex) solutions of the form a+bi.

In mathematics, a square root of a number x is a number y such that y2 = x So, by definition, you have two solutions

@@mathisnotforthefaintofheart How is it an abuse of notation? You can always just take the root with the smallest angle, even if you have the 5th root of i or any root of i, or any root of any complex number.

@@wiggles7976 Then that needs to be clearly mentioned in the question. Very often it isn't.

Glad you enjoyed it!

Your comment is √ i={cos(π/2)+isin(π/2)}^(1/2)=cos(π/4)+isin(π/4)=1/√2 +i1/√2, and -√ i =-{cos(π/2)+isin(π/2)}^(1/2)=-{cos(π/4)+isin(π/4)}=-(1/√2 +i1/√2)=-1/√2 -i1/√2 ? I certainly think it's more reasonable to think so. However, complex numbers that are different from real numbers are difficult for ordinary people to understand. If you have an explanation or material that is easy to understand, please let me know.

Nice! Thanks and thanks for watching!

Wait is i-th tetration definited? Because i did it by my own and i got ln(0) in eqution which is undefined (in limit -∞ )

@@aweebthatlovesmath4220 rip

also works of course!

One of mine too!

Fun fact: The German term is the misleading “Funktionentheorie”, “theory of functions”. To me it always was too rigid - every continuously differentiable function is infinitely continuously differentiable, where’s the fun in that? Look at all those real function that are differentiable but not even continuously differentiable.

@@magicmulder wait, don't get me wrong I'm just a high schooler but I thought only continuous functions can be differentiable throughout their domain?

You will realize it is not extraneous if you plug it into the original equation though!

The sqrt(-i) is 1-i and -1+i As can be seen there is no only positive or only negative rule to be followed

-i and i arent the same thing tho. One is -1 * sqrt(-1), the other is sqrt(-1) respectively. Yes, when multiplied by each other, they yield -1, but that doesnt mean they are the same

@John S. The square root is exactly as you logically think it should be. I mean it returns two values, one positive, one negative. For simplification we often use principal roots and because of our laziness (mathematicians can also be lazy) we often don't even bother to mention it (whether we use principal roots or just roots), although folks might have huge problems because of that. Actually, i is defined as +√-1 because √-1 = +i, -i. Nowadays we often define a complex number as an ordered set of two real numbers with a specific rule for multiplication, which means we don't have to deal with complex numbers on a formal level. And that's how it is done in more rigorous math books. Bottom line: there's nothing wrong with definitions. It's just authors neglect finer points and some explanations. *Actually, it's a malpractice in my opinion. All you said, all your critique is 100% valid, while other people just pretend to understand stuff they have no idea about, unfortunately.*

Hi Alex. If interested in math competitions, please consider kzfaq.info/get/bejne/gd1ndsZ9yKm5lWg.html and other videos in the Olympiad playlist. Hope you enjoy 😊

No, because (-i)²=-1 like i²=-1

@@Ostup_Burtik Which I realized after watching the video.