Thank you, from Iran 
thnks. a lot. ive been looking for a website like explaining math intuitively. i hope that one day you will come up with an article explaining fourier transform in a similar fashion. again thank a lot.
@Luis: Thank you for the comment, really glad it was helpful. I think the Fourier transform would be a great topic, I need to study it more to move beyond an “academic” understanding into an intuitive one. But once that happens I’ll be writing about it :).
In the division portion, why is (3+4i) divided by the magnitude squared of (1+i) and simply just the magnitude?
@Steve: Great question – I should make this more clear.
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We can approach it intuitively:
* Rotate by opposite angle: multiply by (1 - i) instead of (1 + i)
[Note: When we multiply by (1-i) we cancel out the angle just like dividing by (1+i) would. However, we end up scaling the number by the size of (1-i).]
* Divide by magnitude squared: divide by |sqrt(2)|^2 = 2
[We need to divide twice: first, by the size of (1-i), because we multiplied by it above, and second by the size of (1+i), which was part of the original division. Both have the same size (just reflections of the same angle)].
So, we have to divide twice: once for the original (1+i), and again to cancel the “side effect” of multiplying by (1-i) in the first step.
If we wanted, we could just multiply by (1-i)/sizeof(1-i) to remove the angle and keep the same size, all in one step. Then we could divide by the sizeof(1+i) as we intended in the beginning.
Hope this helps!
Thanks for the explanation - very easily understood. Anyone have any ideas on how to visualize taking a number to the power of i? I’m trying to better understand Euler’s identity (e^(i*pi)+1=0). Thanks again,
Micah
@Micah: Thanks, glad you enjoyed it. Euler’s identity is a great use of complex numbers – one way to visualize it is seeing e^i*x as saying you need to change by x% in the “i” (perpendicular), vs in the real dimension. Constantly changing in a perpendicular direction will move you in a circle, leading to the full formula. I’d like to write more on this in a future article :).
[…] Complex numbers: visual introduction, intuitive arithmetic […]
Kalid, Delicious work…
When calculating the magnitude |a + bi|, why is the “i” term left out of SQRT(a^2 + b^2)? I know it would give the wrong answer, but isn’t “i” part of the original complex number?
@John: Great question! The “i” in the original number is more of a direction designation, and says you’re going b units in the i direction.
If I said I walked 3 blocks East and 4 blocks North, I might write it as:
3E + 4N
With our normal numbers, we assume everything is a real number (East/West) so can just write
3 + 4N
And if I wanted to find the total distance traveled, using the pythagorean theorem I’d do sqrt(3^2 + 4^2) = 5.
Note that N (or i) is just a way to designate that the quantity is in a different, 90-degree direction. The pythagorean theorem assumes this (in fact, requires it; it only works in right triangles) so we don’t need i in the calculation. I hope this helps!
Haha! This is just great! I’ve used Cnum’s when coding fractals, but never really understood the concept. I’m a better man now. You explain really, really well. And i love your humour.
GR8 man! Vivid explanations!!!I wish u 'd been my Math tutor @ college…I 'd taken up math as my career! Nevertheless…I m enjoying it now!!! Thanks man!!!
The traditional difficulty in understanding the complex numbers is a man-made one. It reflects intellectual shortage in the course of definitions and thought construction. The term “imaginary” component in the definition of complex numbers is misleading. Furthermore, the definition of radicals or square roots in the field of Algebra has its own flaw where “negative” numbers become hard to define a radical for. Much of that is due to the historic and gradual accumulation of mathematical knowledge which characterizes mathematicians more as a cult—a culture with rich inheritance of thought, history, and terminology. The term “imaginary” for instance should have been called “rotational” instead. Imaginary things are understood as things that has no real reflection and exist only in the human mind. This is not a true case with complex numbers and their “imaginary” components. Complex numbers have real applications in physics when rotational physical phenomena arises—things that has a cyclic nature such as an alternative current in electricity. All what an imaginary component in a complex number means is that the magnitude of the complex number is specially aligned at a certain angle relative to the conventional positive real line axis. Again the real line is a misname for real numbers. So really, the complex numbers are nothing but real numbers in two-dimensional space with some closure field property pertaining to the roots of negative numbers. With this idea of rotation in mind, the study of complex numbers can be much easily understood.
Thank you for your explanations. Can you do an article on imaginary exponentiation? I get that multiplication by an imaginary is like rotation, but what about multiplying a real by itself an imaginary number of times!?
Err, quite elegant for the most part, however I’m hung up on the division. Why do we take the conjugate of the denominator and not the numerator (obviously, algebraically this is obvious, but intuitively?) And why do we WANT to shrink the denominator by its modulus, why not shrink it by 56 or something arbitrary?
Here is a crisp and dynamic word for complex numbers, hand both as a verb and a noun: twirl.
I’ve used it with my eight year old son, thinking of a video camera over a plane that can be both zoomed and rotated. It did not take him long to convince himself that zooms and rotations can be combined, in arbitrary order, without affecting the result. The combination of a zoom and a rotation is a twirl (imagine the twirling trajectory of the point 1+0i).
Unfortunately, it is a useful term only for multiplication, not adding. But perhaps that is a feature rather than a bug, helping keep focus on the operations rather than the numbers.
May I say that I love this site, and wish I had discovered it before today!
@Largo: Oh, I like “twirl” and the video camera analogy – it gets the idea across! (I love, love, love seeing how other people see these topics). And well, adding is like sliding the camera up and down (moving left/right/up/down is adding numbers in the real or complex plane).
Thanks for the kind words, happy you’re enjoying the site!
Dear Kalid,
Your blog is really mind-bending! I am sure one of your readers will be the next Great! (of course I am included in that set of potential ones ) Well, on sincere lines, I am really grateful to you for making this intuitive work available to all of us readers.
When it comes to naming, one thought came to my mind - people call non-existing things like ghosts and spirits as supernatural things and they call numbers which are as real as Real ones imaginary ! Just coz they are kinda intangible for the time being and in another dimension!
Well, even great mathematician Fibonacci called zero as ‘sign zero’ ! So, I think its only a matter of time before we realize the realness of these so-interpreted ‘non-real’ numbers