Hi Tom, glad it’s clicking! Great question – basically, it comes down to the effect we want to have on the result.
Multiplication gives the “properties” to the result. Starting with 3 and multiplying by -2 will infuse the size of -2 (which is 2) and the sign of -2 (negative) into the result. We’ll get -6.
Addition is more of a “slide” – starting at 3, we slide along -2 units, to end up at 1. It wasn’t really a combination, more an adjustment.
In the case of complex numbers, their angle is like the negative sign – it’s an intrinsic part of them. In order to “combine” the angles, we need to multiply the numbers together and infuse the result [this sounds a bit weird, but it’s just the analogy I use]. The actual reason this works is found here: http://betterexplained.com/articles/understanding-why-complex-multiplication-works/
If we just wanted to slide the endpoints around [follow a trajectory of 3+4i, then follow another trajectory of 1+i afterwards] then we would just add them. That could be useful in another circumstance, i.e. we want our final position after taking two shorter legs.
I’m a Biomedical Eng for a large research institution, my son just started 1st yr Engineering after taking a year off. When he asked me about Linear Algebra I kinda felt like I was having a flashback. My first yr of Engineering consisted of 48hr/wk ontop of a part time job, leaving me to have to make some sacrifices for time. Algebra was the victim (sorry), I only went to the first class, got the assignments and showed for the mid-term and final ( scored a B+ ).
after reading this blog and watching the video it has made this subject crystal clear, if you aren’t a Phd by now you should be.
i like to read what people like you, like you me writes about novel things and find (y)our satisfaction. i didn’t get c/x numbers but no worries, my brain will keep on working until it makes some sense out of it.
This is beautiful! I presented it to one of my Algebra 2 classes today, and I’ll show it to two more tomorrow. Eyes bugging out!!! Thank you for giving i meaning!!!
@sometimeszero: Wow, that is so great to hear! One of my goals for this site was arming teachers with the analogies that really helped things click for me, hoping they can modify and incorporate them into their own routines (just providing ammo in the fight against boring, ill-understood math). Thanks so much for the comment!
Ah of course! That makes perfect sense now. Many thanks for taking the time to respond and make that clear to me.
I’m looking forward to working through your other articles - all things that never made sense at school because of the way they were taught, unfortunately.
I really applaud your work to explain concepts in a way that makes sense. You are in a very real sense empowering people - great work, please keep it up.
Nice tutorials, Kalid, tyvm! I’m a math neophyte, but just enjoy numbers and came across your page and really enjoyed it. One thing I could not wrap my head around when you were discussing the heading problem was the fact that you MULTIPLIED the complex number (3+4i) by (1+i). I understood the multiplication process and even worked out the trig to make sure everything works, and it does. BUT it is not immediately evident to me WHY you multiply these two numbers rather than add them. Would you be good enough to point me in the right direction so I may understand, please? Thanks! -Tom
So I’ve been tutoring a student in precalculus and noticed that he’s starting complex numbers. I’ve never really engaged any of their applications, and this kid always needs to know how math relates to the real world.
It left me a bit puzzled, as I knew that just rattling off rules would shut him down and leave him hating complex numbers (as I did, too, for a long time). Luckily, I stumbled onto your blog and I’m shocked about how cool they are!
An intuitive understanding behind complex numbers may not be taught in classrooms, but I assure you that after today—thanks to your blog—at least one inquisitive high school student will have a better intuition for these fascinating numbers.
if
multiplying by i rotates the vector counter-clockwise
and
multiplying by -i rotates the vector clockwise
then
how about multiplying by +(1/i) or -(1/i)?
in what direction vector shall go?
and
will it change direction only?
or
magnitude as well?
You have not showed how to rotate a vector about an arbitrary angle.
Also, if the goal was simply to have a way of find negative numbers, why not something simpler, such as a flip flop?
You also have not defined Complex and imaginary numbers operations, even though you used them (I saw that you linked it but you should still touch on that).
@kalid, @Rich
Thanks guys! I guess I thought that since the square root of the negative unit in the real dimension was in a new dimension orthogonal to the real, that the square root of the negative of THAT unit might be in a new dimension mutually orthogonal to the complex plane. But actually raising i to any power has rotational effects in the complex plane, is that right?
But I guess I’m still confused. If raising i to a whole exponent gets you either a completely real or completely imaginary number, does raising i to a fractional exponent give you a complex number?
