Are Imaginary Numbers Real?
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- Опубліковано 22 бер 2022
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The Imaginary number i was much debated long ago, yet is widely accepted now. But the question is: are imaginary numbers real? I mean, they're "imaginary" right? They don't actually exist....do they?
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Squarespace, nice coincedance in a video that starts with squared numbers being a problem :p
What's Greatest integer Value of [0.9^Bar] ??
@@Games_Era44 1.
can you tell me why this give a contradiction here
Well we have : -1 = (-1)^2/2 =[(-1)^2]^1/2 = 1^1/2 = 1 ?!
Can you tell me what is the wrong here
@@zaidamvs4905 (-1)^2/2 = [(-1)^2]^(1/2) is false.
square space? Nah, I'll stick with complex space
I am sorry for making 2 jokes
ua-cam.com/video/l_qWp_ON_1E/v-deo.html
triangle space
Lololololol
This deserves a heart
How to win internet arguments:
1:42
that's BORING + what if you're wrong?
Great video, only thing is that "complex" doesn't mean complicated.
Complex: having multiple parts or aspects that are usually interrelated
Complicated: involving a lot of different parts, in a way that is difficult to understand
It's like "apartment complex" or "vitamin B complex".
It’s like objectively complex and subjectively complex
He says that
I mean, it's also a gramatical error, since the adjective and the substantive of the word have different meanings, so it's kinda wrong to use it as adjective expecting people to interpret it with its meaning as a substantive.
He literally said those words damn near verbatim in the video tho...
short answer: it depends on what your definitions of "real" and "imaginary" are.
depends on what you concieve to be a 'number'
But is your Definition "real" or "imaginary"?
imaginary = your girlfriend
@@LukasDrakon I guess you can say it's a... complex relationship.
ello rex
I remember as a yoot always being confused when the teacher would clarify that this only applied to "real" numbers. I couldn't understand what a "non-real" number was. Learned i in algebra 2 in high school, and complex analysis was one of my favorite classes in college. Complex analysis is still one of my favorite topics.
When I talk to my kids about it nowadays, I tend to go back to that number line representation, and ask, what happens if you step above or below it? They might object, but I will remind them what when they started, they only had the positive values, so they learned to step to the left of 0. Now think about stepping above or below.
I think conceptually it helps them to grasp the idea, although still not the significance, and definitely not the features.
Basically the same thing happened to me. 6 or 7 years ago, I was in the second year of high school and asked the teacher why he always wrote "x∉ℝ" (x isn't a real number) instead of "∄x" (x doesn't exist at all) whenever a quadratic equation ended up having a negative discriminant. He got visibly nervous and dodged the question, as if I was asking for classified information. When I started researching on my own and discovered complex numbers, I felt sad: why didn't he want to know about such a beatiful topic? Aren't teachers supposed to like when someone wants to learn more about the subject they teach? To this day, I literally get tense whenever someone ignores the complex solutions to a problem just because they're not real.
To be fair, that very same teacher did awnser me, only one year later, what a derivative was, and why the derivative of sine was cosine (my school didn't have calculus: i had stumbled upon derivatives online). He wrote a neatly organized proof using the definition of derivative, that even had some diagrams on the side. I'm really thankful to him for doing that, to the point that I keep that proof in a special folder. His kindness inspired me to not only teach myself calculus, but also love the subject. Calculus, alongside Linear Algebra and Complex Analysis, is still some of my favorite subjects, even though I've completely given up on having a career on STEM (Until a few years ago, I studied Physics, but switched majors to History, my lifelong passion).
Depending on how old the kids are, this idea works:
There are two islands, positive island and negative island, and on positive island there are 1, 2, 3, 4, et cetera, and on negative island there are -1, -2, -3, et cetera. Theres a bridge between the islands, and on that bridge there are imaginary numbers
They told them imaginary because of their way of solving the equations, they drawed literal squares to do the equations and it was illogic to think of something with negative area.
Good answer. Yes they would solve questions geometrically back in the day before imaginary numbers and the concept of a negative area was assumed to be imaginary. But it now appears that "imaginary" numbers are more real than REAL numbers. and I hate the term imaginary and would take off marks if my students used the term. I would joke to my engineering students that if they thought imaginary numbers where imaginary they should stick their tongue in an electrical outlet this weekend and tell me on Monday how imaginary it felt.
@@johnnychinstrap That's horrible. You take points off because you don't like the correct terminology? What the hell kind of teacher are you? "Imaginary" whether you approve or not, is a mathematical term with a specific, well defined, mathimatical definition. Just as irrational number has nothing to do with the mental state of a number and negative number has nothing to do with the emotional disposition of a number, imaginary has nothing to do with the existance of a number.
@@herbie_the_hillbillie_goat I was teaching engineers not physicists. And was very well respected by my students. I taught them how to apply math and make stuff. The term imaginary arose because it creates a negative space and that was considered imaginary back at that time. Teaching involves more than conveying data and formulas. engineers need to be critical thinkers or new advances in technology do not take place. I will be posting a paper a Colleague and I wrote to show why this video is wrong and it was because of critical thinking and being open to new ideas that we figured this out.
@@herbie_the_hillbillie_goat I think another point you are missing is that i is not the square root of -1. That is a symptom of it. It is a complex vector and carries way more information than the lay person understands. I can not let my engineers follow the same misinformed understanding of i based on centuries old math. It is this reason that this folklore answer is accepted as fact.
@@herbie_the_hillbillie_goat I missed the most fundamental flaw to your statement. In the context of my story I was teaching them power math and imaginary power is actually incorrect lexicon wise too. It is reactive power and I would joke with them if they called it imaginary power they should go home and stick their tongue in an outlet and tell me on Monday how imaginary it felt because some of that imaginary power is getting stored in you. It was actually a very good way to convey to them the concept I was trying to get across. Complex numbers are real even though they are not REAL.
The better question is "are real numbers real?"
Agreed. And in many ways, the real numbers are actually more of a made-up abstraction than the complex numbers. Just take a look at the formal definition of the real numbers: they are a totally ordered Dedekind complete lattice, a field, and addition and multiplication are monotonic. Just what is that?! A Dedekind complete lattice? That is just as much an abstraction as the complex numbers being algebraically closed.
Exactly, when I tell people real numbers are actually sets with an infinite number of rational numbers inside them (whether you use cauchy sequences or dedekind cuts), they start thinking real numbers aren't as "real" as they originally believe.
Quantum mechanics uses complex numbers to describe quantum phenomena, not just to calulate things in between.
And yes, you can't make math grounded to reality. There is nothing wrong with coming up with anything as long as it's logically consistent with the rest.
Real numbers are made up too
@@anshumanagrawal346 All of mathematics is made up. That is the very nature of mathematics. We start with certain conceptual axioms about abstract object descriptions that we arbitrarily declare to be true, and then we work to understand the formal logical implications of those axioms, using logic that is itself made up.
@@angelmendez-rivera351 Exactly
That doesn't make me think i is real as much as it makes me doubt quantum mechanics, if what you are saying is fundamentally accurate.
@@angelmendez-rivera351
The thing is that everything comes from counting. Like 5 apples + 7 apples, everything else is defined using that, so it's not fully made up. Even logs like
log2(8)= 3 means 2^3 = 8
2^3 = 2 * 2 * 2
2 * 2 = 2 + 2
2 + 2 + 2 + 2 + 2 + 2 = 8
Everything is related to addition, and there is no disbelief in addition
And schools treat algebra like a totally new subject when in reality you do basics of algebra since like grade 3. Equations like 5 * _ = 15 and then they (to me) poorly describe algebra with things like 5a = 15
Complex numbers are also used in ac circuit analysis like adding up two sinusoidal signal with phasor.
Of course they are, because no matter what corner of mathematics you're talking about, electrical engineering will wind up using it, leaving no doubt of its status as the most difficult of the STEM fields.
Imaginary numbers are more intangible than imaginary. Can “i” stand for that?
And I recall working with Quaternions and having i, j, and k along with a “real” part, because sometimes traditional complex numbers aren’t complex enough. Though it did teach me t think of imaginary numbers as a perpendicular phase shift- which makes it easier to imagine in my head.
what the fluck
I love quarternions: they're my personal "exhibit a" of situations where it's easier to dive into complex space than it is to work everything out using the real numbers. Most 3d modeling programs allow you to set your coordinates or describe your animations through quarternions.
In Poland we have even worse. Imaginery numbers are in polish called ,,urojone" witch means like something imagined by somebody with mental illness. And for complex we have ,,zespolone" (compound) witch is correct name i guess but it sounds a little akward to me
Thanks for the information Jakub
@@abhigyakumar3705 is that sarcasm?
@@jakubpacua2351 no I really meant that
"compound" seems more appropriate to me than complex.
I was also wondering
Ok
@@csicee okay
@@kaylenpluymers2747 ok
@@csicee ok
@@Krakabraka ok
1:40 There is another problem with not having no square roots of negative numbers: the cubic formula requires them when there are 3 solutions. It's possible to solve cubic equations without using complex numbers explicitly, but in this case you have to consider that a separate case
This is actually the historical reason that complex numbers were first studied.
As concise as it seems to build up the number systems by talking about solving more polynomials, there are a few inaccuracies in it.
1) Not all real numbers are solutions to polynomials
2) It messes up the historical motivation
I think the story about cubic polynomials is much better to tell than wanting to solve all quadratics. It makes perfect sense why some quadratics wouldn't have roots (if you only stuck to real numbers)!
@@MuffinsAPlenty I agree about the cubics. But which numbers are you referring to in 1) ?
@@clementrimagne4053 They're called "transcendental numbers". Usually, transcendental numbers are contrasted with "algebraic numbers".
A complex number is an algebraic number if it is the root of a polynomial with integer coefficients. So, for example, sqrt(2) is an algebraic number since it is the root of x^2−2, which is a polynomial with integer coefficients. The idea is that sqrt(2) can be described _algebraically_ using the polynomial x^2−2. An example of an algebraic number which is imaginary is i. That's because i is a root of the polynomial x^2+1, which has integer coefficients. But sometimes people make a distinction between real and non-real complex algebraic numbers.
A transcendental number is a complex number which is not algebraic. In other words, it is not the root of _any_ polynomial with integer coefficients. It's called transcendental because it transcends the tools of algebra. The tools of algebra cannot distinguish them. It turns out that π and e are examples of transcendental numbers - they are not the roots of _any_ polynomial with integer coefficients.
Abstract algebra is able to prove that the collection of algebraic numbers is all you need for _algebra._ If you add two algebraic numbers together, you get an algebraic number. If you subtract an algebraic number by an algebraic number, you get an algebraic number. If you multiply two algebraic numbers together, you get an algebraic number. If you divide an algebraic number by a nonzero algebraic number, you get an algebraic number. But even better than that (and relevant to my first comment) - if you take a polynomial with _algebraic numbers_ as coefficients (not just integers as coefficients), all of the roots will still be algebraic numbers. So if you are just interested in solving polynomials, there is never a need to go further than the algebraic numbers. If all anyone ever cared about was solving polynomials, no one would care about numbers like π and e. But we care about more than just solving polynomials - we care about geometry and calculus, which is where numbers like π and e show up. If you want to be able to do _calculus,_ you need the full set of real numbers. But if you're just doing algebra, you don't need all of them.
@@MuffinsAPlenty Your explanation is fantastic.
I really wish Bri would go on to the topics of quarternions and what other more complicated number systems are possible or impossible and why
I believe in you, Bri! Thanks for the video.
Another nice video; thanks for making and sharing it!
Glad you enjoyed it!
@@BriTheMathGuy i absolutely did.
In the history of numbers, humans took longer to accept negative numbers (thousands of yrs), which are just an equal distance from Zero in Opposite World, or better yet, a 180-degree rotation around Zero, than it did for Mathematicians to accept 90-degree rotations (a halfway stop along the way to Opposite World) - which we now called (i). Once you accept negative 4 bananas, 4 imaginary bananas (a quarter rotation away from 4 'real' bananas), isn't as big a jump. Humans innately understand rotating - it's the math of rotations
Damn this made me understand so much. It seems similar to the concept of spin, with quantum particles. There can be 0 spin, 1 spin, and then in between which dont make sense.
Also the negative and imaginative bananas also make sense. Imaginary bananas are bananas that arent there. Negative bananas are bananas that are even more not there, that they cause things that are there to go away.
@@o_sch Exactly - you got it. Humans have an innate, from birth, understanding of rotating/spinning around. We also have innately, an understanding of owing - i.e. that person saved my life, I owe him. If you asked a child, holding an icecream to give their icecream to someone else, and if they do that, they will get another one - even a child understands owing - which we have invented a word called negative to describe - but it took a thousand years to have the courage to give it a number (-1) being 'owing' 1 - but it really is the opposite/a reflection in the other direction of a 'real' icecream. With further thought, we could then put down in words/symbols, that between a real icecream, and owing an icecream - was ZERO icecreams (that also took a LONG time for everyone to agree was a real thing - Zero being a number was not appreciated and rejected) - but eventually this created a 2-dimensional feeling of numbers - i.e. 1 was on 1-side of zero and owing 1 was on the other side of 1 (a sort of reflection). We as humans still understood rotations - and we had to deal with them in real life - so the seed was planted, long ago, and it was waiting for someone to connect Rotations to numbers - someone had to be the person to step off flatland of the number line, and ROTATE, above or below, the 'numbers' to get to the Negative world. That insight would have then lit so many lightbulbs about angles, how a 90-degree rotation + another 90-degree rotation gets you 180-degrees (which gets you to negative land) - and then BINGO - the intuition bombs would not stop going off and that was how we get i^2 is (-1) or the sqrt(-1) is (i) - it's a halfway stop, above the flat land world in the world of rotations, from 1 going to (-1)
Imaginary numbers absolutely related to rotations very intimately. I often call them "spherical numbers" specifically because they're so closely tied with spherical geometry. To say that i^2 = -1 is really to say that two 90 degree rotations gives a 180 degree rotation, which in the context of real numbers is also a reflection. (Naturally there are also numbers that behave the same way but for hyperbolic and euclidean geometry.) Quaternions exist to bring the 2D rotations of complex numbers into 3D. Since there are now so many more planes that can rotate, you need 2 additional imaginary units to distinguish between them.
More infinite series please I need this right now!
the best way that helped me in understanding complex numbers is seeing them as motions
because every complex number can be associated with a vector and a vector represents moving a point in the direction of the vector and with speed equal to its magnitude.
but now I have no idea what numbers actually are !!!
Yeah they are just like vectors especially when you learn the geometrical analysis of complex numbers like rotation, locus of argument,ect..
Anyone who dislikes imaginary numbers doesn't know the glory of the Laplace Transform and frequency domain solving all your problems. Laplace is the easiest way to solve differential equations and Euler's equations are the basis for making calcutions for anything involving AC electricity not trigonometry hell. (When 120cos(2π60t + Φ) becomes 1
"What exactly is imaginary about a number like i, and what exactly is real about a number like pi"
That was literally so clean
Great video 🎉🎉🎉
Just learnt that chapter yesterday !
I'll go with Euler on this one.
The issue isn't that imaginary numbers aren't real. The issue is that negative numbers are not real. We say they're real, but we have made up negative numbers because they're useful when talking about _taking away_ number (i.e. subtracting). But subtraction can really be thought of as addition in reverse. For instance, when we take away three oranges from a number of oranges, we think of it as subtracting three from a number, but we can also think if it as _adding_ three oranges to the universe of things that _aren't_ the remaining number of oranges. It's this dual, mutually dependent and proportional nature of addition and subtraction that creates the bizarre property of negative numbers, and, thus, imaginary numbers.
speaking of math teachers: my teacher in year 8 went on a rant once about how it were utterly incorrect to put arrow heads on both ends of a number line. i see there is no agreement on this subject in academic circles
No, there definitely is an agreement. The vast majority of mathematicians agree. Your one math teacher disagreeing is irrelevant.
if both ends go off to their own version of countable infinity, i really see no issue with putting arrow heads on the number lines. heck, I would argue that if only one section of the number line had an arrow, it would actually represent the bounds of [x,inf) rather then (-inf, inf)
Maybe we should call them planar numbers since, you know, they exist in a plane
This would be quite a good idea, especially since it can be proven that the field of complex numbers is the only 2-dimensional division algebra over the real numbers.
I prefer spherical or elliptical numbers, due to the intimate relation they have with spherical geometry. It also highlights the existence of their often forgotten hyperbolic and flat cousins.
I'm gonna go to the bank and just tell them my debt is not real and show them this video 😤
As the great mathematician Paul Bernays wrote about the objective reality of mathematics described by Plato and their analogs in today's mathematical world - "This application is so widespread that it is not an exaggeration to say that Platonism reigns today in mathematics."
I'm right at the beginning of imaginary numbers in uni right now. I don't know very much of anything about them. But until now, I've always just considered "i" to be basically the same as the "-" in front of negative numbers. Thats it. The only difference being, that, in contrast to negative and positive numbers, imaginary numbers can't be directly added or subtracted from "real" numbers. Thus needing a a variable instead "z". Is that about how it works? If yes, its much less complicated than I initially thought before starting uni.
That's pretty much right. To be fair, it took mathematicians about 300 years from when they first needed square roots of negative numbers until they got a geometric view of complex numbers as points in a plane where adding "imaginary" numbers moved laterally to the "real" number line.
I love how you said both types of number are imaginary! I've been waiting for someone to say that.
I had a diff eq test today and when you showed that you would get cos2x+isin2x I was freaking out that I forgot the i until you showed what we are supposed to put
3:29 Of course! That’s Master Chief himself!
Hey! Could you do a video on solving for "n" in
this equation. A=P*(1+r/n) ^nt
how i would do it:
i = e^ipi/2
rewrite i^1/i as (e^ipi/2)^1/i
use properties of exponents: e^ipi/2*1/i
i and 1/i cancel out so you are left with:
e^pi/2 (same as 1/e^-pi/2)
3:27 Printed on Master Chief’s armor
4:00
me: hey! i'm poor and have 0 dollars, can i borrow 100 dollars? oops! just dropped it down the drain! well now i have -100 dollars! anyways... gotta go!
Another random thought is that, in some ways, i is more real than e.g. Tree(3) (you can concisely describe i in far more detail than you can Tree(3), where we don't even know a single digit of its base-10 representation).
I once tried to imagine how maths would work if there was some absolute limit to the natural numbers (i.e. if ℕ={0,1,...,G} for some big number G). It drove me a little crazy. But the inspiration came from computer programming, and the 'naive C programmer's assumption that malloc() will never fail.
So one idea was, following the Monad idea of functional programming, and the exception mechanism of e.g. C++ or Python, to have more logical states than True and False, namely True, False and Exception. So e.g. sometimes n = n + 1 was False, and other times n = n + 1 was Exception (basically if your proof tries to access numbers beyond G in any way, you deduce neither True, nor False, but Exception instead.) I wonder what a more talented logician would do with this.
I heard that the word "imaginary" was derived from the word "image" because the imaginary number line was projected from the real number line at a normal angle.
It's a nice cutesy untruth to tell people if you think the word "imaginary" is bad, but it's false. The terminology "imaginary number" dates back to René Descartes in the 17th century. The view of complex numbers as being part of a plane with the imaginary axis perpendicular to the real axis dates first came about 200 years later, in the early 19th century.
Descartes called them "imaginary" numbers because his view of numbers was related to geometry in the form of length, area, and volume. There was no length which could be the side length of a square of area -1, so in order to use square roots of negative numbers, you had to "imagine" they existed. You had to use them as pure abstract concepts in your head, rather than as something representing a geometric notion (at least, to the extent that Descartes understood geometry at the time).
I don't think we've been lied to, as your title suggests. In fact, I recall my math teachers saying things like "we can't take the square root of a negative number... yet" or "for now, we just say there are no real solutions". So that was a bit click-baity. It's not "being lied to" when teachers introduce ideas in a simplified form.
Infach my teachers told me that there are no solutions.
But i agree, quite clickbaity.
Good video this.
3:56
I know how to show negetive
A wide filed is here and we brought 1 blocks to the filed
But it has copyright
4:08 People used to think that the _i_ in quantum mechanics was of this nature; a mere 'side adventure' that was used to get to a 'real' answer; their goal was to remove _i_ entirely, since if that were true, then it would be expressible in terms without an _i;_ turns out, the _i_ is absolutely essential in quantum mechanics. You cannot remove it, and it is not just a mere 'stepping stone' or 'side adventure' to get to the real answer.
Anyway, the name _imaginary_ and _complex_ are not bad names at all; their names sound cool and are cool, and they increased the 'awe' factor when I was learning. And that 'awe' factor has never left; what is learned is truly more amazing than the awesome aesthetics of a good name
5:06 Dude really just went "Complex numbers? I find them quite simple"
Now after watching this video, I can say that I don't have an iota of doubt!!! 😐🤷🏻♂️
BriTheMathGuy: "show me negative nine ( -9 ) things"
Me: *shows the marks I scored in my exam.
You can count your marks even if you can't really see them or touch them.
Unit bivectors in geometric algebra are isomorphic to imaginary units, and are a good way to conceptualize imaginary numbers as real, geometric objects
The number line models continuous measurements like distance and volume; similarly, the complex plane models models continuous measurements on, well, a plane. That means that complex numbers are useful when you want to model behaviors in more than one dimensions.
Not necessarily. You could consider R² instead, which is of course a two-dimensional vector space.
The complex numbers allow to view the plane as a one-dimensional vector space instead. Plus you obtain a multiplication. As nice as this is, it doesn't work in higher dimensions. From dimension 3 onwards a multiplication of this sort ceases to be abelian.
The crucial point about complex numbers is really the factorization of polynomials. The fundamental theorem of algebra. That is why complex numbers are _the_ numbers (and everything before are mere subsets and everything beyond mere compounds) and that is what entails all the other stuff.
Imaginary numbers were created by Heron of Alexandria, but they were coined by René Descartes to take the piss out of them. Hence the name "imaginary". As in unreal, pretence.
Dude that is the Wafflest waffle I have ever heard.
I found a problem to discuss with the maths teacher to waste class time
thanks!!
bro really didn't mention the Schöringer equation
“Show me negative 9 things”
My bank account
I think when people when they ask this question mean if the numbers have a real world intepretation. Real numbers can be used to represent a quantity or removal of quantity such as i have 5 apples and ate 3 of them so now i have 5-3 apples. But is there a phyisical interpretation of having "i" apples?
But how does some numbers like 10^999999 have a real life usage? By your logic shouldn’t that be imaginary?
@@Firefly256 I'm not trying to focus on usage but rather on having a physical interpretation. and i can interpret what having 10^9999999 apples would mean even if it can't happen. But i dont know what having i*10^9999999 apples would even principally mean in the real world.
@@christian1233211000 That is because your notion of "physical representation" is ridiculously naive. I mean, by that argument, what is the physical representation of "TREE(3)"? There is no such a thing, not in practice, anyway, but no one calls "TREE(3)" as "imaginary," except for the very few ultrafinitist mathematicians and conspiracy theorists in the world.
Complex numbers are represented in physical phenomena all the time. And I am not saying that they are convenient for simplifying the language with which we describe the phenomena. No, the complex numbers are fundamental for even _defining_ what those phenomena are to begin, and there do not exist ways of describing those phenomena using real numbers only. If that is not enough for you to consider it "real," then it certainly is not enough to consider negative integers to be "real" either.
@@angelmendez-rivera351 "What is the physical representation of "TREE(3)"?" Three groups of one tree. Or one group of three trees. Either way, a total of 3 trees.
@@angelmendez-rivera351 but i never said representation i said interpretation. I don't think the number 3 exists in nature, but again i understand what 3 apples mean and not i*3 apples.
They are called "imaginary numbers" because it was meant as an INSULT to your (mathematical) opponent. In today's terms, it is equivalent to called your opponent's numbers as "bullshit numbers"
I'm taking a high school final exam in one month and probably the examiner would be astonished if I wrote such a thing
Me:(reads) are imaginary numbers real?
Me:(says) What an imaginary thought is this.
Bro that was legit instantaneous wow(I mean the heart)
Let’s be for real, which one is the BEST sponsee? Square space or Brilliant?
I wonder why nunber 117 was chosen.
As soon as you conceive of an unlimited successor function (such as the natural numbers), you open the gate for numbers to exist that don't model any known quantity in the universe.
Ditto.
10⁸¹
Funny thing about the name "imaginary," it was meant to be a dismissive name but it's what got me interested in them in the first place...
I like to call C the planar numbers, and R the linear numbers, and i the lateral unit.
help me im stuck in an infinite loop of the same two imaginary numbers videos
"Show me -9 things."
*steals 9 of your things*
3:54 it is out side of circle 9small circle
Him : Are Complex Numbers Real ?
Me : Are Dead People Alive ?
Why on earth do they consider √-1 only an imaginary number.. coz there are many imaginary stuffs like inverse sine or inverse cosine of N where |N|>1,, i mean the solutions of sine or cosine function greater than 1 could have also be taken as imaginary, but why did they only consider √-1 as an imaginary ??
An other example of imagining number being: a positive number raised to any power equals to a negative number: eg. ln(-a) = b, where a & b are positive numbers!!
I wondered just these but there could be many more imaginary numbers which have not been taken into consideration, hence I wanna know the reason what should be done abt the above two?? Should it be treated as same as "i" or have a separate role in the field of maths?
I always thought about them as "two-dimensional numbers".
They are two-dimensional but the dimensions are not interchangeable. They are for addition but not for multiplication
every time we're introduced to new numbers, we found ourselves searching for new intuitive way to understand them.
everything seemed like it made sense with the real numbers,
but I always find myself surprised by the results in complex numbers...
for example:
the integral in the Real numbers was so friendly, and everything made sense with it if we understood it as an area of some sort..
but this way of understanding it makes no sense in the complex numbers (the integral of complex valued functions)
like How on Earth is the integral of 1/z along a closed curve counting the number of times this curve is circling the point (0,0)???????!!!!!!!
How on Earth if a complex function is differentiable one time, it will be differentiable infinitely many times??!!!!!
It's magic, isn't it ;)
The point about the integral is that it never, fundamentally, described an area in the first place. It just happened to be one of the things you can do with it. What it really is is a summation taken to the limit. An additional limit beyond infinite series that is, where the number of terms becomes uncountable.
And the thing about differentiable complex functions? That's simply how functions are supposed to work. It's not the holomorphic functions which act strangely, it's the real functions that are broken. The two notions of differentiation (real and complex) are fundamentally incompatible and there's good reason to argue that the real one is actually false. We just cling to it out of sentimental reasons (and because it works for many applications, of course).
Sometimes if we want to get real, we must get imaginative first.
Why is this video deleted?
It looks live and running to me - what is it saying on your end?
It's an abstract construct like "U substitution"
I was expecting math, instead I got the semantics of math. Oh well
A great question! Conventionally defined as 2*real, same sqrt(-1) can be applied to more *real* (as our real universe) sequences, such as 2*naturals, right? Because our *real* universe is quantum, consisting ultimately of natural numbers only. It is!
1+1=1, because they joined together
Well quantum wave functions rely on complex numbers so it’s not just natural numbers involved.
@@zemoxian What I've said is that in reality there appear to be different forms of complex numbers based on how you build your perception of space at each new moment - may be complex real, may be complex integer. You know, rational is +- natural / natural. I've also got a lot to say about combinatorial spaces that change at faster-than-light speeds, but that's the edge of global comprehension of physics.
@@zemoxian and quantum pathways seem to a product of a huge number of many different sources
I didn't understand a single word from what you just said.
I like to think about complex numbers as the last puzzle piece needed to make algebra complete, i.e. making sure every polynomial equation (which is the most fundamental type of equation) has as many solutions as it‘s degree (with multiplicity). With real numbers you can always factor polynomials into at least quadratic factors. (For example a degree 5 polynomial can be factored as 5 linear, 1 linear and 2 quadratic, or 3 linear and 1 quadratic factor). But only with complex numbers you can decompose any polynomial of degree n into n linear factors.
Making all quadratics solvable was the last thing necessary to make algebra complete.
Saying that polynomial "equations" are the most fundamental type of equation is definitely not accurate.
@@angelmendez-rivera351 Not accurate but he got the message across I believe. Polynomials of any degree are the most complicated an algebraic equation can get, it's not a crazy idea but it's true.
@@angelmendez-rivera351 The way I see it is that Polynomials are fundamental because they involve just basic arithmetic: Addition and Multiplication (the exponents are natural numbers so it's just multiplication). So Polynomials can exist in a Ring or a Field if you want rational coefficients.
@@SeeTv. Technically, you need a *commutative* ring for the ring of polynomials to be well-defined. But yes, your description is otherwise correct. And I agree that the concept of polynomials over a ring is definitely fundamental in some aspects, but to say that it is the _most_ fundamental concept is an exaggeration, in my view.
Also, I think using algebraic closure as the explanation is insufficient. For example, the algebraic closure of the rational numbers is the set of algebraic numbers. Why do we not work with that? Why do we work with the algebraic closure of the real numbers instead? Education on the Internet typically fails to address this. What exactly are the real numbers? Why do we care about them? I know the answer to these questions, but I only know these answers because I have a lot of exposure to very high level mathematics. But most media available do not cover these questions at an introductory level.
who else has been looped back and forth between the two videos?
“Are imaginary numbers real?”
GREAT JOB SHERLOCK
What about a new scaling type of numbers? We know that we have negative and positive but what if it was another scale out there that results into negative x ? A new simbol? An already known simbol? Lets say an >
well, technically, if you number is negative (let's say -1 in this case) and since -x^2 /= (-x)^2 , you can say that you're doing -1^2, and technically you can calculate this equasion in real numbers ( -1^2 + 1 = 0 = -1+1 = 0 )
don't think I explained what I was thinking of well but you get it
So does math allow you to invent any number system you like or are there any restrictions? Like, could you just invent a number that satisfies |x|=-1, or the limit of sin(x) as x goes to infinity?
There are definitely restrictions. You cannot have a number system in which there are two statements that contradict each other. As long as you avoid contradictions, though, anything goes. And in fact, extremely bizarre number systems do exist out there. For example, there is a number system with only three numbers: {p, r, s}, and they add in the following way: p + r = r + p = p, p + s = s + p = s, r + s = s + r = r, and p + p = p, r + r = r, s + s = s. This number system has the bizarre property that there is no 0, and there is no subtraction, and addition is commutative, but not associative. But here is the interesting thing: this number system actually models something in the real world. "What?!," you are probably asking yourself. How could such a ridiculous system model anything? Well, it does model something, and of relatively high interest: it completely models the game of rock-paper-scissors. Indeed, p stands for "paper," r stands for "rock," and s stands for "scissors." The addition operator is the versus operation: in other words, p + r reads "paper versus rock," and the result of the addition is whichever one wins the round. So p + r = r + p = p is true because "paper beats rock." Are there any contradictions in this number system? No. So then it is a perfectly valid number system, regardless of how bizarre it is.
Now, your other questions are also interesting, but a lot harder to answer. See, when you talk about creating a number system where |x| = -1 has solutions, you run into trouble. Why? Because not only are we talking about numbers, we are also talking about functions, and the meaning of symbols. What is the meaning of the symbol ||? What is the absolute value function? People say the absolutely value is "the distance from 0." But that is not a definition, just an intuition, and it is very unclear, since it then raises the question of "what is distance?" In mathematics, everything needs to have a precise definition. Otherwise, we are going to confuse ourselves and end up with contradictions and mistakes everywhere. An absolute value function |•| is a function that takes a ring as its domain (technically, the ring is an integral domain), and the real numbers as their codomain, and the function satisfies three properties: |x·y| = |x|·|y|, |x| >= 0, and |x + y| =< |x| + |y|. If one of the three properties is not satisfied, then by definition, the function is not an absolute value. And since |x| = -1 contradicts the above properties, it follows that it is impossible. So you cannot make a number system where |x| = -1 can happen, and that is due to how absolute values are defined.
Defining a number system where sin(x) converges as x -> ♾ is even trickier. You need to define what you mean when you say "converge," and you also need to come up with a definition of what it means to take the sine of an object.
@@angelmendez-rivera351 Such a nice comment. Thanks bro, for knowledge.
@@angelmendez-rivera351 Wow, that was a fast reply for such a long answer. I had a feeling that limit example I gave was tricky, but my understanding of how number systems work isn't perfect.
However, I'm interested in them. I've heard of some really crazy one's before, but I never thought Rock, Paper, Scissors could be represented that way. I had always wondered how to do it mathematically, since using nothing but inequalities obviously didn't work. So thanks, now I know!
@@FLS96 The difficulty with the example involving limits is that it requires understanding topology. The reason you can do calculus using real numbers is because the real numbers can be given a topology that works nicely with our applications. You can extend the real numbers to another number system, and give that number system a topology that extends the topology of the real numbers, but this notion is extremely complicated, and it is not clear whether it will be free of contradictions or not. Also, you want your topology to be such that sequences have a unique limit if they do have one. We want to avoid the situation where sequences converge to multiple points at once. Introducing the concept of a topology is already difficult to begin with as is. Just explaining what it is requires some mental labor.
When mathematicians think of inventing new number systems, they typically focus on dealing with addition and multiplication, more so than dealing with with complicated concepts such a limits or trigonometry. As a good introduction to abstract algebra, you want to start thinking of only one binary operation, and think, what are the possible number systems I can create using only this binary operation? For example, you are familiar with the integers already. You know that the integers have addition, and that addition is associative and commutativity. Plus, you have 0 with 0 + x = x + 0 = x, and you also have that addition can always be inverted, i.e, subtraction is well-defined. And the last defining characteristic of the integers is that you have some object, which we label with the symbol "1," which is different than 0. In principle, which symbol you use does not matter: the properties will not change. So you have 0, and you have 1. But you can add integers, so you also have 1 + 1, and 1 + 1 + 1, and these are all different in the integers. And then you have -1, with the property that (-1) + 1 = 1 + (-1) = 0. And then you have (-1) + (-1), (-1) + (-1) + (-1), and so on. And these combinations are all the integers. We come up with special symbols for each of these combinations. These combinations are all different from one another. But now, coming up with new number systems is easy: what if we decided that 1 + 1 = 0? Including this extra rule gives us a completely different number system. In fact, this number system has only two numbers: 0 and 1. Because (-1) = 1, and 1 + 1 + 1 = 1 + 0 = 1. Why would we care about a number system like this? Well, because it allows us to study parity. 1 + 1 = 0 is a way of encoding the fact that Odd + Odd = Even. An odd permutation, combined with an odd permutation, is an even permutation. What if 1 + 1 is different than 0, but 1 + 1 + 1 = 0 instead? Then that gives us yet another number system, with three numbers: 0, 1, and 1 + 1 = 2. This number system is used to model arithmetic modulo 3. And so on. These small modifications give you different number systems, and they get you familiar with what it means for new number systems to be created.
Here is another one. These are the triplex numbers. Take the real numbers, and now throw in a new number, and call it ω. ω is a triplex not equal to any real number. ω·ω = ω^2 is not equal to any real number either, and it is also not equal to ω or to any real number multiplying ω. However, ω^3 = 1. So the triplex numbers are numbers of the form a + b·ω + c·ω^2 where ω^3 = 1. And you can get some rather interesting things with this. For example, (1 + ω + ω^2)·(ω - ω^2) = (ω +ω^2 + 1) - (ω^2 + 1 + ω) = 0, even though neither 1 + ω + ω^2 nor ω - ω^2 are 0. These numbers can be represented by 3by3 matrices, much like how complex numbers can be represened by 2by2 matrices.
@@angelmendez-rivera351 The spacetime metric does actually yield proper time distances in Minkowski space whose "length" are imaginary so there is some merit in extending our idea of length to incorporate those lengths that aren't non-negative.
So is a game of concepts?
"show me -9"
alright then have a look at my bank acc
0:50 hidden Ψ
in 3rd grade i just learned adding a billion times while i knew square roots so 3rd grade was a piece of cake
ok
If you can edit this you should use y= R cos(theta) + i sin(theta) and R is the vector in the reals and i is a vector in the up or complex dimension. Leaving out the R is not technically complete then my engineers will know that it really means cos(theta) is the scalar in the real direction and sin( theta) is the scalar in the "imaginary" direction. But use complex because "imaginary" numbers a actually more real than the Real numbers
You know, I came to the realization that if i'm fine with accepting negative numbers, who don't have any meaningful representation in the real world, I mean there's nothing in the universe that's a negative amount of something, then I should also be fine with complex numbers, who also don't have any meaningful representation in the real world, since you can't have a comple amount of something either, it just doesn't make any sense. I'm kinda pissed about this, because I still don't think that complex numbers are actually in any way like real numbers or a different kind of number, we've just defined some sort of new algebra and said this is how adding,subtracting,multiplying and dividing works when you're dealing with numbers in a certain way. There's still no way to actually express something like sqrt(-1) with the real numbers.
In nature, there is positive and negative charge. You could count that as a meaningful representation if you wanted to.
@@lonestarr1490 We really could have called those anything we just chose to name them positive and negative but that doesn’t mean they have anything to do with positive and negative numbers
But this wasn't the motivation behind inventing the imaginary number, just to create a solution for x^2 +1= 0. It was a cubic equation which motivated people to create imaginary numbers. (I think you know this)
But the video was an excellent explanation.
Add one => x^2+2=1=> 2=1-x^2=(1+x)(1-x).
More in my comment above.
2:00 if we square any real number it's non-negative
squarespace? More like complex space
I suppose that i is imaginary as much as sqrt(2) is irrational as in loony toons. People were killed over the idea that sqrt(2) wasn’t rational.
What’s fun is that mathematician kept their most important techniques secret. The early mathematicians would often solve problems and publish their conclusions but not their techniques. It was all very competitive.
So when they stumbled onto imaginary numbers and were able to solve new classes of problems they became rock gods of the mathematical world. I’m guessing these were all of the sort of problems where the imaginary parts canceled out.
Once the techniques became more widespread I guess it still felt a lot like voodoo.
Today _I’m_ like quaternions are so last century so why aren’t *geometric algebra multi-vectors* more popular?
"Complex" numbers? I find them quite simple, really.
Gotta like a math channel that states a problem with something is that it is boring!
I love complex numbers!
Right now I could say: "x / 0 = xi. And 0 / 0 = z. And 5 / 0 = 5i so 5i * 0 = 5"
cool video
My girlfriend is like the square root of -100.
A perfect 10 but also imaginary.
Someone : "It would be so awesome if we knew the value of i"
Me : "It is the square root of -1!!!!"
😹
“Numbers” are supposed to be used to count or measure. Complex no.s dont help us to actually count or measure anything; therefore, they are not numbers.
My fourth form math teacher (Year 10 teacher) once said, 'Do you have an issue with "i" representing the imaginary unit, the square root of -1?' Well, consider this: I conjecture that even -1 is imaginary concept itself!
Oh no