@Patrycia: Awesome, really glad you enjoyed it!
I don’t have the words to express my gratitude to you the author. You relieved me from a deep hatred of complex numbers since school (I’m 50 now). And you truly enlightened me.
Thank you so much for this gift.
Hi JD, sorry about that! I wonder if a spam filter grabbed it. I’ve been having trouble with the false positive rate lately :(.
Very happy the guide helped – e, i, radians… until you get an intuition, it’s such a mess of symbols. I actually have a guide to Euler’s Identity here:
http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/
I think you’ll enjoy it!
@Manasa: Wow, really glad it clicked for you! I’m glad the analogies/visual approach was helpful, thanks for letting me know.
I agree with you, I wish people in high school / college focused on the intuition of truly understanding the concepts.
Absolutely incredible.
What an innovative and inspiring site you have here Kalid, I have just come back from writing an exam (I am actually a student taking maths as a major at Varsity) and literally thought I could not take in any information for the day but your articles have lead me from one to the other endlessly, and I am more in awe and get more excited after each one!
Keep up the good work, idea’s and creations like your’s motivate me to do my own bit for changing the way people view maths and physics and basically any of the sciences.
I look forward to hearing and reading more, especially as I plan to continue studying maths for the next couple of years!
@Brad: Thanks, I’m happy the article was able to stay palatable even after finishing the test! I think one of the keys is when we approach math with a curiosity to learn, it starts becoming invigorating/energizing on its own, and we just want to keep learning more. I really appreciate the support – definitely try your hand at helping people understand math better :).
@Bob: Thanks for the comment, I love hearing from other teachers! Great point about the rotation – I had wondered why quaternions needed 4 items (not 3) and I think that’s unraveling it. I completely agree about math being about exploring transformations in general, I’m starting to see that as I look back. Another nice topic to write about :).
@Ahmad: Awesome, glad it clicked. I really like having discussions with people, it helps cement understanding (my own as well – often I get questions that make me dive deeper for intuitive insights).
Kalid, Awesome page! I’m a math professor and have taught about complex numbers from almost this exact same perspective for about a dozen years, ever since I had the same revelation: Numbers can be thought of as transformations, and complex numbers as transformations in two dimensions. You’ve done a great job getting this message out to so many people.
One thing I’d like to mention for your readers: Lots of advanced math is about discovering transformation rules for more complicated kinds of data (like high-dimensional vectors and functions). For example, consider rotations in 3D - it matters what order you do them in! Try rotating a book about a vertical axis, then a horizontal one. Then try the other order!
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im in the last year of engineering and in quite a lot course i came across complex number and even before engineering. but bcz i never understood it, my interest was always very low. But now i know it would be different thanx to you.
And yeah the way you take your time to respond to the long questions in the comments section is simply great.
again thanx.
[…] to explain the concepts behind the maths…I've posted a link before, but here it is again, his explanation of complex numbers (if you remember them!) …well technically, this guy came up with it first, but his paper is […]
[…] A Visual, Intuitive Guide to Imaginary Numbers […]
[…] A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained – […]
Thanks Kalid for what you are doing, I know you know how much it is appreciated by many people.
With this tutorial, however, I was with you all the way to this:
“If we multiply them together we get:…” followed by the sums to get -1 + 7i (can’t copy n paste that part).
The maths here has lost me. Where has 3i and 4i^2 squared come from??? :S
Many thanks -
A tired brain, UK
Whoops, I actually wrote out ‘squared’ because I was thinking it lol! Did say I was tired.
I spent an hour mucking about the internet trying to find some sort of decent explanation and example of practical applications of complex and imaginary numbers, and not one of the other sites I visited made the least amount of sense. Especially in my sleep-deprived state. This on the other hand, this was simple and elegant. Thanks!
@sqlguy: Yep, you got it :). Raising i to a fractional exponent (like the square root, 0.5) will give a complex number halfway between real and imaginary, i.e. at a 45 degree angle. Basically, you need to get to i (90 degrees) in 2 steps, each of 45 degrees.