I really want to congratulate you to this very nice explanation.
My point is that the imaginary numbers as well as quaternions are just half the truth. Your explanation of i is so well anticipated because it uses the 2D space and rotations that are intuitively easy to understand. One of your first posters (Chick on 21st December 2007) wrote about the original problem that led to imaginary numbers and in fact it is a 2D problem as well.
I have finished a masters degree in physics, and I have written i about 10.000 times in my life. I have solved hundreds of problems in my professional career using i and I always suffered that we just define it like that because istâs so practical. And I had to become 45 years now when I finally stumbled above the really big thing and thatâs so awesome and mindblowing that I really want to share this here. Decovered already about 130 years ago it starts from the real side - multi dimensional problems and finds a super elegant, consistent way to describe much more than complex numbers, quaternions, and all that stuff. If you do it this way, you can handle 3D spaces as well as 5D spaces as well as Maxwell equations as well as Pauli Matrices without the need of i at all but with an intuitive and consistent methodology. And best of all, when you are using this you get komplex numbers and quaternions for free. They are included, but you get an intuitive interpretation and you will see that they are just half the truth. And by the way, you will get rid of the x-product that only works in 3D space and is inconsistent. Itâs so mindblowing. We really do NOT need i to describe nature in physics. So if anyone is curious - look for âGeometric Algebraâ!
Thank you for your explanation, it makes a lot of sense. But i still am not able to grapple with the idea of some number raised to the power of i. could you explain that?
Thanks Oliver, thatâs a great point. x and y are identical, whereas i implies a rotation (so indeed, i*i = -1).
@Vegard: Thanks!
@Brajabasi: Thanks, Eulerâs formula is a great application.
@Karl: Very glad it helped!
@Stan: The problem with that reasoning is youâve excluded the possibility of numbers beyond positive and negatives from the start. Similarly, someone might argue that a decimal like sqrt(2) doesnât exist because 11 = 1, and 22 = 4, and therefore no number can be squared to get 2. Why must we limit ourselves to numbers that can be shown on our hands, or along a single dimension?
X² canât be a negative number. X²=9, when X is either 3 or -3.
33=9
-3-3=9
thank you 
This was really helpful, Kalid. Thank you for helping me to get a much better understanding of imaginary and complex numbers, something that has puzzled me for a long time. You are doing a valuable thing here.
Vegard, Norway
Iâm pretty horrible with math in general, I tend to think its in part due to my brain shutting down because of the way its taught (rules to memorize of short-cuts that are sometimes counter-intuitive taught by teachers who say âwhy? because thatâs the way it works-- thatâs whyâ). Iâm pretty horrible as in have at best a feeble grasp on algebra, and while doing some signal processing coding I of course came into fast Fourier transform which I had to sit down and read into in order to help wrap my brain around them, which like a lot of this stuff became a recursive process for me and started at complex numbers.
In trying to understand the âwhyâ of it I came across this explanation, which for me, was well ahead of anything else I had read and of course, the humor kept me smiling through the process. Iâd be lying if I said I totally got it still, but the idea of rotation and additional dimensions cemented the very beginnings of it.
Thank you quite a bit, I look forward to reading your other writings. Please keep it up!
This makes me wonder why we are taught to use âyâ in the coordinate system instead of âiâ. Y just seems extraneous since i basically means y. Is it just because people are avoiding the use of imaginary numbers or is there some reason to still use x-y instead of x-i?
[âŚ] Imaginary numbers are another dimension, and multiplication by i is a 90-degree rotation into that dimension! Two [âŚ]
This is a good explanation .You have used the Argandâs Diagram for representation of complex numbers.
Another use of âImaginary Numberâ , is to reduce the computation involving Infinite Power Series . Try to find out " Sin^2(x)+cos^2(x) = 1 " , using Infinite series representation of Sin(x) and Cos(x) .It becomes much easier to represent by the Complex FormâExp(ix) = cos(x) +isin(x) and Exp(-ix) = Cos(x) - iSin(x)
The Product Exp(ix-ix)=exp(0) = 1 = cos^2(x)+ Sin^2(x) ,gives the desired solution in a compact manner.
just want to say thank you
Hi Harmony,
an important difference between y and i is. yy=1 in classical vector algebra like xx = 1 while i*i = -1. x-y diagrams are way more intuitive to the unexperienced pupil like x-i diagrams.
Hi Harmony, great question. âxâ and âyâ are the generic descriptions of two different dimensions (such as distance vs. time, income vs. time, etc.), and âiâ is a specific interpretation. In fact, the interpretation of i as a new dimension wasnât discovered until many years later. But, it would definitely help to have i used as the âyâ dimension in a lot of problems â then it wouldnât seem so strange!
very nice article, exploring hidden concepts,
[âŚ] until after college that I started to take an interest in mathematics. It all started when I read an article about imaginary numbers and how to think about them. When it was explained to me that multiplying a [âŚ]
[âŚ] A Visual, Intuitive Guide to Imaginary Numbers [âŚ]
@jamal if u find out 1/(-i)=i & 1/i=-i( i.e by conjugation multiplying & dividing numerator & denom by i) u will be the same magnitude but only the direction of rotation is opposite to that what is mentionedâŚ
Thanks for that,
You have saved me a lifetime of boring stuff!
Wow. This was amazing. Iâve just subscribed. Iâm in love with maths but I often have no idea where to start; this is now my favorite resource right next to MathIsFun.com and Khan Academy on YouTube. Thanks!
(and oops I was using the question/feedback form up above to post this comment. Ignore that!)