@Dave: No worries, this is a tough concept :). “1 + i” is 45 degrees the same way “1 Mile East + 1 Mile North” is 45 degrees – you’ve moving the same amount in each direction, so are “diagonal” from your starting point. Hope this helps!
@Pandaroux: Great question! Yes, I’ve come to expand my understanding of basic arithmetic, and the idea that addition and multiplication are transformations to some base.
“Plus 3” really means “slide something over”. And how much? Well the number 3, seen from an addition standpoint, is starting at 0 and sliding over 3 units (0 to 1, 1 to 2, 2 to 3).
“Times 3” means “make it 3x larger” when using a multiplication standpoint. In this perspective, you start at 1 and scale to 3x the size.
Breaking arithmetic down like this helps understand how funky numbers like i can work. There’s some more here:
http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/
I didn’t understand i until I started thinking of it as “What transformation, when applied twice, could turn 1 to -1?”
@sqlguy: Great question! Let’s think about what it means: What transformation, when applied twice, turns 1 into -i?
Well, we want to get from 1 (“due East”) to -i (“due South”) in 2 steps. Each step should be a negative 45 degree turn, or a trajectory of (1, -1).
But… we need to account for the scaling effect, so need the 45 degree turn to be normalized to the unit circle (length sqrt(2)), so the trajectory is (1/sqrt(2), -1/sqrt(2)).
So ONE square root of -i is
1/sqrt(2) - i/sqrt(2)
The other square root can be found by getting to “due South” using two positive rotations :).
[…] all that approximations belong to my imagination. For a better expression you can visit http://betterexplained.com/articles/...inary-numbers/ You are right i have some missing points but your explanation explain nothing. Just like the […]
@Alonzo: You’re more than welcome, I really enjoy these types of conversations. There’s no time limit on when these aha! moments come, I’m coming back to topics I studied in school 10 years ago and still learning new things. I’m thrilled if you’re able to share your aha! moments with others, that’s what learning is truly about.
Thanks again for your kind words – I love the interaction of sharing ideas, getting feedback on what works (or doesn’t) and polishing explanations into things which really click for us. Till next time!
@Classof1: Wow, thanks for the kind words!
What’s the square root of -i ?
Thanks Charles – really glad the article was able to click for you. Heh, I might like to go back to school one day, right now I’m having a lot of fun just studying math on my own :).
I have just one word to appreciate your article: Beautiful!
And you inspired me to think about that zero. I have tried to share my understanding here http://niket-kumar.blogspot.in/2012/07/zero.html
Thanks a lot! 
[…] an example: I can casually describe i (the imaginary number) as the square root of -1 and you can blindly accept […]
Hi Kalid,
I already emailed you once thanking you for the very clear explanations that certainly helped me. It now strikes me that besides for finding roots of equations, complex math is used almost exclusively for wave-like phenomena, and the reason is because they occupy a closed circular space, where if you advance one full cycle you arrive back where you started, and yet it also has a magnitude dimension that multiplies the conventional way. A LOT of confusion could perhaps be avoided by re-naming the “imaginary” dimension to the “rotational” dimension! The weird thing about complex math is that it merges the linear and the rotational in a unique way, to create a linear/rotational hybrid with unique properties that mirror the amplitude and phase properties of wave phenomena.
Hi Steven, thanks for dropping by – I must apologize, I believe I have an email response I owe you on some of your other work as well!
Yes, exactly, a huge application for complex numbers is anything which cycles (since they seem to do so naturally, and are a great way to model it). They’re almost a way to use rectangular coordinates in a polar way (i.e., I want to rotate my x,y coords, so I multiply by i, or another complex number, instead of converting everything to polar).
I’d be hesitant to rename to something as specific as “rotational” because there may be other (yet undiscovered) applications, but almost anything is better than “imaginary”! Perhaps “alternate” or “supplementary” numbers.
@nuur leina:
You simply carry out the multiplication on the right side (as you would do in multiplying two polynomials: term by term) and then equate real and imaginary parts of both sides.
(3+i)(2-3i) = 6 - 9i + 2i - 3(i^2)
= 6 - 7i + 3
= 9 - 7i
Then x + yi = 9 - 7i.
Then x = 9 and y = -7.
(actually, just realised this is the same answer as above (225):
1/sqrt(2) - i/sqrt(2) = (1-i)/sqrt(2) = ((1-i)^2) / 2
= (1-2i+i^2) / 2 = -2i/2 = - i !!
(you wouldn’t have thought it!)
Would that be:
Sqrt(-i) = Sqrt(-1 x i) = Sqrt (-1) x Sqrt ( i) = i x Sqrt( i) = i^(3/2).
Check: ( i^(3/2) ) ^2 = i^3 = i^2 x i = –1 x i = –i.
[…] 帖子的下面，很多人给出了自己的解释，还推荐了一篇非常棒的文章《虚数的图解》。我读后恍然大悟，醍醐灌顶，原来虚数这么简单，一点也不奇怪和难懂！ […]
Kalid, many thanks for your patience and explanations to the crazy questions I have asked. Although I have expressed frustration in some of my previous threads with trying to grasp things - it has all been worth it! As I read back over the discussion, things really started to sink in. I truly have a perspective that was not available to me before. I simply don’t know why this perspective has comes so late in life. But, it will help me to help others with their frustrations. In fact, I plan on using our correspondence for the basis of conversations with folks that I think may be stuck on similar issues.
But again, thanks for your time and insight. The education your providing people on this website probably has more of an affect than you know, so excellent job on making an impact !
Until next time, take care.
Alonzo
how to calculate this question…
x +yi=(3+i)(2-3i)…
find the value of x and y…
please help me…
Simply brilliant. One of the best explanations for any mathematical concepts I’ve seen. Kudos.
Wow. This is just phenomenal. This, coupled with the videos from Khan Academy really helped me truly understand this. I’m not completely comfortable with it, but its a huge improvement from where I was an hour ago. The chart comparing negative numbers and complex numbers, the idea of rotation and the example of the boat were amazing. I just realised my school did not even teach us any real life applications of this chapter, and that’s quite dissapointing. Its one of the reasons why some of my friends still struggle with this chapter. For me, If I don’t see any application, I usually show little interest in it, and this was a life-saver. Thanks!!