# An easy, alternative introduction to Imaginary Numbers

### 100 thoughts on “An easy, alternative introduction to Imaginary Numbers”

• October 9, 2013 at 4:36 pm

Nice introduction to visualising complex numbers. Bravo.

• October 12, 2013 at 8:50 pm

dude that was so informative and interesting! it's perhaps the first time complex numbers makes sense to me, thank you! and make more videos like this exploring other areas of math and making it so much fun!

• October 16, 2013 at 3:34 pm

so what you said as a whole is imaginary numbers are factors we know that are there but we cant explain what it is

• October 16, 2013 at 3:44 pm

um if it is not positive or negative feedback would it be neutral feedback?

• October 20, 2013 at 11:46 pm

(That sounds harsh but I didn't mean it as such – I just wanted in on the math puns!)

• October 29, 2013 at 9:26 pm

Imaginary numbers weren't created to deal with 2d rotations – it's just the kind of symmetry that they have and thinking about them as 2d rotations can help you get your head around them :).
There are neat extensions of this symmetry to algebras in higher dimensions – but to be honest, the i,j,k etc are easier to grasp. Though the i,j,k are vectors which don't come equipped natively with a way to deal with rotations (you need matrices/tensors/similar).

• October 29, 2013 at 9:31 pm

We take i to represent a counter-clockwise rotation is just a historical thing – everything would work if we chose it to represent a clockwise rotation instead :). As for the order of operations – the order doesn't matter – you get the same answer if you have positive first and negative second (this is known as commuting in maths).
To clarify, 90 degrees corresponds to i itself. The key idea is that multiplication of complex numbers is the same as *adding* their angles (not multiplying) 🙂

• October 30, 2013 at 10:58 pm

Might be a a dumb question: are imaginary numbers the same with different cultures? Math is universal in it's rules and laws. Do imaginary numbers work the same way?

• December 17, 2013 at 11:55 am

this is nice, good work.

• January 4, 2014 at 10:18 am

Oh why stop at 2 dimensions? What's wrong with a 1 dimensional number line?

• January 29, 2014 at 11:51 pm

If Marcus du had really made such a statement then it is high time that he Revisits Complex Numbers. For information there is nothing Imaginary about Imaginary numbers. The very fact that an Imaginary number can be Geometrically represented, rules out all Story telling associating them with myths.

• March 14, 2014 at 6:18 am

Fine job man a good illustration.  I wish I was thought the concept that way in my control engineering and advenced Calculus classes, Learnt it the hard way though…

• May 1, 2014 at 2:54 am

This is the coolest explanation I have seen on complex and imaginary numbers

• May 23, 2014 at 9:35 pm

Fantastic!

• June 24, 2014 at 11:40 pm

This imaginary comment is just as useful as any positive or negative comment

• June 26, 2014 at 6:17 am

Great vid! can you explain how this works in 3 dimensions for quaternions :3? Nice work man!

• July 24, 2014 at 12:54 pm

y u do dis to me math? 🙁

• August 7, 2014 at 1:28 pm

Wait so if I have understand this right does this mean "i" is just a value on another axis, doesn't that mean that "i" can be explained with normal numbers but with something that shows it's on another axis?

• August 14, 2014 at 10:14 am

Good approach and easy to understand  but can you slow your speech down a bit ?

• August 21, 2014 at 3:34 pm

Nicely explained man :)….great job….

• August 25, 2014 at 2:12 am

At last, a video which begins to make some sense of "i". If I understand this correctly, it's not so much the "value" of i which is the important thing to understand, but rather, that we wanted something which would produce this rotational effect in practise, and "Sqr root of -1" was found to be a value which, although meaningless by itself, produced the desired effect when plotted on the y-axis, like your video shows.
Also, it immediately strikes me that as a circle is produced, then i, (or rather, seeing as i by itself isn't a value, then some function of i) must have some relationship to pi?

• August 26, 2014 at 2:41 am

great vid

• September 3, 2014 at 9:01 pm

please do more videos : )

• September 18, 2014 at 7:35 am

You missed a golden opportunity at the end.  "Please leave me any feedback, positive, negative, or imaginary; below"

• October 29, 2014 at 10:44 pm

I'm one of those people who didn't and still don't undestand math… This explanation was hard to understand from the first minute…This is not for those who never understood math but now want to understand it a little bit…

And also because there is no visible structure of explanation, for me, at least, plus because the speed and velocity of speacking is growing all the time, I have no space in my mind to understand it. So sorry but for me is useless…

Can anyone tell me a video about complex no. but explained slower and more aware of detailes and language? For me are too many detailes that are not explained here… Thanks!

• January 17, 2015 at 12:11 am

…I'll watch this again tomorrow. Tomorrow I'll understand. …tomorrow.

• February 18, 2015 at 7:21 pm

When I was first introduced to imaginary numbers I would have big problems finding any applications for it, and it seemed to me like they were just introduced so that you could write down an answer for negative square roots. =P
But now when I am starting to study higher level mathematics (first years at university) I am starting to notice that they are in fact useful in many ways.
For examaple, it is used to calculate interferences of waves more easily, and also when adding various values of circuits.

• March 18, 2015 at 4:01 am

So by this logic you there exists an imaginary imaginary number which will rotate the vector 90 degrees away from the screen? Does such a thing exist?

• March 21, 2015 at 5:51 pm

Maybe slow down on the hard parts. Why use the "i" variable, what it represents, and how it can be applied to every day uses?

• March 28, 2015 at 5:11 pm

I like the explaination, but how can your velocity even be -1. Even though you are turning 180° which is a complete switch of direction, saying your velocity is changing -1 is like saying you turned -180 degrees. The problem gets more complicated if you look at the unit circle.

So if you were to turn 45° then and you called your velocity 1. Then you turned -45° then your velocity would not be -1. Becuase 45° or pi/4 does not equal -45 which is -pi/4 which is 315° which is also 5pi/4.

And even if you assume that you turn 225° which is the true oposite of 45° then the velocity would not be the same. If you cover 225° in the same time as you cover 45° then your speeds would be completly different making your velocity different as well.

In real life we dont refer to things negatively. For instance we dont travel 55mph negative north west. Or we dont say i turned -315° which is 45°. Or we dont walk -4ft.

• March 29, 2015 at 8:55 pm

i 'm not that good on maths but each time i see imaginary numbers.i see two dimensions
like an (x,y) plane.were x is the real part and y is the factor of the imaginary part.
my question is it really a good idea to define i as i x i =  -1. can't we use a
function? : squareroot(x) if x ≥ 0 then squareroot(x)= y
as y*y = x.
,if x ≤ 0
squareroot(x) = – absolute value of (y)
as y*y= x.
what is really useful in the i number.if we already can define a plane of two axis and define position of points with two coordinates.?

• April 19, 2015 at 3:29 pm

Hey can you explain how two completely imaginary numbers multiply into a real number using this complex plane…Algebraic calculation makes sense..but what does it really mean?..can u explain it using the complex plane..that would be really helpful

• May 3, 2015 at 10:04 pm

Hi There, really helpful exposition. Thank you. I used this video to supplement the explanation of the same topic by Kalid of better explained.com. The two videos bolt the concept in – one by introducing the concept intuitively and the other by adding a little more vigor and building on some of the concepts with a little more depth.

• May 5, 2015 at 6:55 pm

I like the instructor's methodology. We must remember that there are different learning styles. After forty years of instruction, I have learned to use different approaches to help the student understand.

Sir Walter Besant
“If a child can’t learn the way we teach, maybe we should teach the way they learn.”

• May 24, 2015 at 6:06 pm

Very nice video, finally properly understand what complex numbers are now, thanks.

• July 6, 2015 at 8:29 pm

I really like the fact that you introduce imaginary numbers and the complex plane all in one step, and the walking velocity is an excellent vehicle (no pun intended). I wonder if you have tried this on high school students and what problems, misconceptions, etc you encountered. I am thinking specifically of using it with students who have never heard of imaginary numbers before.

• August 31, 2015 at 7:24 pm

2:05 why is it something x something (x multiplying) not something x 2 when you need double the size? Because for example 9 x9 = 81 but 9 x 2 = 18.
I never understood this..

• September 1, 2015 at 10:29 pm

Can this analogy be extended to quaternions

• September 28, 2015 at 11:38 am

You said to quick. And also can not understand…..
Do you need to breathe…….

• October 7, 2015 at 7:56 am

What if i want to describe movements in 3d space?. Should i invent a new set of numbers for the purpose?

• October 7, 2015 at 3:24 pm

Thank you so much, IceDave, for sharing your knowledge.
With complex or abstract topics, many can understand but few can explain or teach in easily understandable terms to others. You have given us an awesome video and explanation!!
Much appreciated!

• October 8, 2015 at 9:46 am

Hey, +IceDave33
Have you seen +3Blue1Brown 's video? He introduces numbers as points, adders and multipliers. Anyway, its really cool! But you did it 4 years before! I wonder how long ago this idea was originally thought up, and how most students aren't explained this is math class..

• October 26, 2015 at 5:27 am

Thanks for the subtitles. This guy is more difficult to understand than the waiter at my fav Indian restaurant…

• October 30, 2015 at 12:35 pm

why not use the second axis with normal y?

• November 1, 2015 at 6:11 am

Very nicely done! The connection between trig and the complex plane is a tasty bite.

• November 20, 2015 at 12:06 am

Man, I had a orgasm watching this.

• December 11, 2015 at 1:32 am

not sure if someone has said this already, but Ive never learned it as sqrt(1/2)+sqrt(1/2)i

You can think of it this way: In trig, a 45 degree right triangle has an a and b value of 1, and the hypotenuse is root(2). Therefore You can apply sine to both 45 degree angles, to achieve 1/root(2) + 1/root(2)i , which is rationalized to root(2)/2+root(2)/2i (0.7071+0.7071i approx)

You will already have to know trig and polar form stuff though. I figured out the square rrot of i without looking it up, purely based on polar form and trig

• December 17, 2015 at 4:57 pm

It's really a very simple explanation. Thank you.

• January 10, 2016 at 1:09 am

Glad this vid has captions..narrator talks to fast and doesn't enunciate. Excellent explanation of complex numbers though!

• January 18, 2016 at 3:40 pm

5:29 what does it even mean to square that whole thing…? How does it even verify anything (without any explanation it's like you're just doing some random operation)?

• January 18, 2016 at 9:45 pm

This still doesn't disprove the imaginary part. Can you show an example of imaginary numbers in real life, like can be done with real numbers?

• January 20, 2016 at 12:14 am

Can you change directions without using complex numbers? This is probably what makes me uncomfortable with this analogy. It's cool tho. Better than solving x^(-1/2) which in itself explains complex numbers.

• January 21, 2016 at 5:43 am

this made me finally understand quaternions

• January 21, 2016 at 5:43 am

this made me finally truly understand quaternions, thanks.

• January 21, 2016 at 5:44 am

this made me finally truly understand quaternions, thanks.

• January 26, 2016 at 3:06 pm

i^1 = (√-1)

i^2 = (√-1) x (√-1)
= (-1)

i^3 = (-1) x (√-1)
= (- √-1)

i^4 = (-1) x (-1)
= 1
——————————–the pattern repeats
i^5 = (√-1)

i^6 = (√-1) x (√-1)
= (-1)

i^7 = (-1) x (√-1)
= (- √-1)

i^8 = (-1) x (-1)
= 1

• February 22, 2016 at 9:07 am

hi.
what to multiply with to go out of the plane
?

• March 11, 2016 at 5:57 am

• March 11, 2016 at 6:49 am

Please leave any feedback positive, negative, or imaginary below.

• March 20, 2016 at 1:57 pm

After decades, imaginary numbers and complex numbers make sense, thanks to this video. Great Job IceDave33! The video is a genius insight and provides a fresh perspective to the world of complex numbers which I know have always been very real to physicists (eg in the realms of electronics (where they are used to represent phase) and in quantum mechanics). I no longer see them as a trick to help solve problems and can now appreciate them for what they are.

• March 20, 2016 at 8:22 pm

Much in the same way that the idea of negative numbers extends the power of mumber system to permit the representation of debt; similarly, complex numbers, similarly, are an extension of the number system to permit us to capture information about rotation or phase or direction of a measurement in addition to its magnitude. A real world example being using complex numbers to represent the idea of electrical reactance (R+iXL) or representing vector quantities such as velocity or electric field.

Transformation of a complex plane using a complex function (w=k/z +c for example or w=z^2), allows us to work with changes in or, manipulate in those dimensions -for example, as we might do during digital image processing (https://www.youtube.com/watch?v=onMLujxxwug).

• April 1, 2016 at 9:25 pm

Keep it real! 😉

• April 3, 2016 at 2:26 am

i always thought of i as x, it is real, but it can't be simplified in any way, just as the quadratic formula can never be simplified to result in y=c for ax^2+bx+c, unless a=b=0, at which time you are simply saying 'i'm not looking at that number anymore, so i'll multiply by 0', which is why i is simple to me, it's just there. nothing else. (also, clockwise 90 degrees i always interpereted as velocity/i and since -1=i^2 and 1=-1^2, 1=i^4, and i^4/i by the law of exponents =i^3=i^2 * i=-1 * i=-i) (may be less intuitive, but people still need to learn how to divide by i) (you should totally do a video on everything needed in dealing with i, a/(1-i) x^i, all that.)

• April 3, 2016 at 8:15 pm

It's one of the simplest explaination thank you

• April 5, 2016 at 2:08 am

I came here after watching a video about Euler's Identity on BetterExplained. I see what you are doing with the cos x + isin x side of the equation, but still don't understand the e^(pi × i) side. Trying to get an intuitive grasp of that side so I don't always have to use the trig side.

• April 6, 2016 at 4:25 am

This is probably the best (intuitive) explanation of imaginary numbers I have ever seen.

• April 14, 2016 at 2:17 am

In electrical engineering "j" is used for imaginary numbers, because "i" is used for current (amperage, the flow of electricity). Complex numbers and their algebraic use only comes round when AC circuits are discussed, while "i" for current is used in learning the fundamental laws in DC circuits.

• April 22, 2016 at 7:02 am

you know what i just realized is going on with complex numbers thats supposed to be so great with this counterclockwise spiral thing thats happening? so take 1.. multiply it by i.. now it magically moves a quarter of a turn and is i.. now take i and multiply it by i and it goes another quarter turn, woah! what did we just do? we took 1 and multiplied it by -1… complex numbers… then we keep going and it does another quarter turn woah! -i… do it again! ok back to 1! cool! keep going!1*-1*-1*-1*-1… advanced math ladies and gentlemen… the obsession with this is called not being able to see the forest for the trees..

• May 5, 2016 at 8:12 am

What if i wanna backflip ? …let's put a z axes too to be fair….

• May 8, 2016 at 8:18 pm

Basically, calling them "imaginary" numbers is stupid. The better name is "complex" numbers.

• May 13, 2016 at 7:50 am

What class do you learn this in??

• May 13, 2016 at 7:52 am

I still don't get how they are useful in the real world? Don't we already have x, y and z? Isn't i = y?

• June 12, 2016 at 6:10 am

why cant people teach like you, you made it look so easy, thank you so much

• July 25, 2016 at 12:58 pm

Question for IceDave33: what software did you use to create the video?

• August 19, 2016 at 10:07 pm

Please slow down and enunciate so the audience have a chance to absorb and think about it.

• September 6, 2016 at 10:25 am

This was good! 🙂 Thankyou

• September 28, 2016 at 6:57 am

So, actually I think mathematicians make misconceptions about complex numbers and its relation with rotation. You can use a + bi (a complex number) to represent rotations, in the same way as
r cos(β) i + r sin(β) j, but use complex numbers for rotations only really make sense when you are handling problems with negative square roots. For standard rotation problems, is better forgetting about the imaginary number and focus in Linear Álgebra notations. I think 3D rotation with quaternions can be understand as a 4 dimensional vector in geometry problems, and as a 4 dimensional complex number in this category of problems. I really don't understand why people handle rotation with complex numbers in problems which doesn't involve imaginary numbers.

• January 12, 2017 at 9:45 pm

Hi! Interesting Presentation 🙂

You may also make imaginary numbers even more real by showing the link with phases (which was historically from were they came if I'm not mistaking) 😉

Thanks for Sharing 🙂

• January 16, 2017 at 1:20 pm

i^3=-i ,and the square root of a negative one is a plus or a minus i ,so -i = i

• March 4, 2017 at 10:09 am

Your presentation was nice (thanks!) but the very concept itself is still a bit eerie for me. Basically, what's happening is, math folks seem to be developing the theory (of complex number) from the applications (geometric rotation etc), instead of the other way round. Shouldn't a theory be able to stand by itself, without any reference to its applications?

• April 7, 2017 at 4:20 am

• April 7, 2017 at 4:22 am

Great idea, but I would say clock pin is a better example, or sling.

• July 17, 2017 at 12:07 pm

There seems to be a mistake in the end – rotating by 45 degrees is equivalent to multiplication by i^1/2 (square root of i) – not by i, as it follows from this video

• July 19, 2017 at 6:13 pm

Very Very Very Good. And I agree too fast for someone trying to learn this from scratch… also… I'd suggest slowing down on on the complex multiplication as well… I hope you have videos explaining how this relates to for example the mandelbrot sets, and complexity, etc.

• August 15, 2017 at 6:15 pm

Beautifully done, Clear and concise.
Now I just have to confirm if a Magnetic Spin 1/2 and is just in the opposite direction of a spin of -1/2.

• August 21, 2017 at 9:22 pm

complex numbers could be used for trig functions. I'm not sure if they are better or worse though. They weren't invented for that reason I guess and in fact were probably just given that purpose. Is the complex plain just the x, y plain arrived at via negative square roots? Or maybe they are polar coordinates.

• November 7, 2017 at 7:23 pm

Many thanks for this. I'm a university mathematics lecturer, and I will be sending this out to my students, as I think that it's a very clear way to picture what's going on. I may also post it to www.mathemafrica.org which is a mathematics blog run from South Africa.

• November 17, 2017 at 1:55 am

Thanks for adding the captions. I found that setting the speed of the video to 0.75 and reading the captions made it very easy to follow.
It is indeed a fantastic basic explanation of imaginary and complex numbers, how they work, and what they can be used for.
I wish I had this video when I was STRUGGLING in math classes back in high school and college.
Thanks much!

• November 29, 2017 at 3:34 pm

Could you open this vídeo for subtitles please?

• March 7, 2018 at 7:34 pm

A very helpful metaphor for any teacher of Maths

• April 8, 2018 at 2:35 pm

Just amazing explanation

• July 2, 2018 at 12:38 pm

Great explanation!

• July 12, 2018 at 6:45 am

if you want to rotate 90 degree then why don't you you add an extra speed in the Y direction? All you show me is basic 2D trigonometry equations, so why not use Y. I am still searching for why i is necessary

• October 2, 2018 at 2:13 pm

Why the hurry, mate???

• October 20, 2018 at 9:57 pm

Hm, I think it got a bit more confusing. If we introduce imaginary numbers to make sense of 2D velocity, then which numbers are used to describe velocity in 3D space? Magic numbers(?)

• December 31, 2018 at 1:16 pm

ممتع ومدهش جدا!!!!

• January 11, 2019 at 12:33 pm

Great video!

• April 24, 2019 at 8:45 am

NEW discovery explains all Mother Nature creations…. https://www.youtube.com/watch?v=EymBulzF0rg&t=28s