@Osama: Great question, I’ll be covering that in a separate post.
@Jaloopa: Thanks for the info! Yep, I agree on ‘complex vs complicated’ (http://betterexplained.com/articles/combining-simplicity-and-complexity/)… but the pun may not flow as well when we write “complex numbers aren’t complicated” :).
[…] To most of us, it’s “the number in the middle” or a number that is “balanced”. I’m a fan of taking multiple viewpoints, so here’s another interpretation of the average: […]
Great explanation. Byt it begs the next question.
Negative numbers complete the “real” numbers in a one-dimensional number line. Imaginary nu,mbers open that out into a two dimensional complex number space. So what is in the three, four and higher dimesnional number spaces?
Hi Alec, great question. There are ways to consider i, j and k to handle more degrees of rotation (called quaternions, I don’t have much experience with them). At this point, it’s probably easier to use linear algebra (matrices) to keep track of multi-dimensional data. Any set of x, y and z coordinates can be represented in a matrix, and other matrices can represent transformations like rotation and scaling.
But why were imaginary numbers first used? I understand they have many uses today, but what were they used for in 1572 when they were first discovered?
Hi Jane, take a look at comments #4 and #5, they may help answer your question :).
Actually, what makes to think of an imaginary axis. Where can be this imaginary number stuff be applied?
I’m not getting the right image on how you were looking at the number.
@maheshexp: Take a look at the example in the article – imaginary numbers help deal with rotations, without having to use trigonometry.
In general, imaginary numbers are good for things that move in cycles (since i can be seen as rotations about a center point). In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns. It can often make the math much easier.
I feel smarter now.
Awesome, glad you found it useful :).
Khalid, now I could make a pretty difference between ‘Complex Number’ & Trigs.
Trig -> Given your position or distance (eg: 4N,3E), what angle should you move from the current point.
Complex Number -> Given an angle, solves what would be the new position
Hi maheshexp, that’s an interesting observation. Yes, trig mostly deals with the raw angles, while complex numbers have you think about distances.
i never understood math in school (geometry being the exception). but just recently it clicked, and now i think math is beautiful. i spend more time reading these sorts of pages than on actual schoolwork, and i don’t even have any math classes this semester!
Thanks, Kalid (is that “Khalid”?) for making maths so easy. I’m 38, very interested in Science, Tech, Electronics, Computers et al but maths has always been my “Achilles’ heel”. I had an almost-aha moment when I was studying remedial maths at the South Tyneside College back in '93. That was calculus.
Sorry for intruding between all the other ‘Math’ (why not Maths?) wiz kids(?) - I’m still just learnin
I always had I hard time understanding eulers formula. After all we are used to think expotential functions as something that grows.
But then I thought about the multiplication rule, and that e^i is really just a point on the circle with angle 1.
And when you do e^ix = (e^i)^x with x being an integer, you are really just multiplying this point by itself, and thereby adding angle 1 each time.
Just like you could use the formula i^x=cos(pi/2x)+isin(pi/2*x) as the complex number (0+1i) lies on this circle 
I’m very happy with your article.
@Anonymous: Thanks, glad you enjoyed it! That’s a great insight about moving around on the unit circle, I like that interpretation too :).
Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
Thanks.
Hi Peter, great question. Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).
If you want even more dimensions, you can use linear algebra (vectors & matrices) to represent data with n dimensions. For example, Google uses matrices to represent the multi-dimensional relationships between web pages in its ranking algorithms, when a single number (or quaternion) won’t do it justice :).
Hope this helps,
-Kalid
[…] Imaginary Numbers […]
Thank you for this great explanation. I have been reading about vectors for the first time recently, and your article makes me wonder whether the “ship’s heading” example could be solved similarly with vectors. More generally, can imaginary numbers be seen as a specific case of wider vector-based concepts?
Thanks for your thoughts!
Weston