Complex Numbers, Part 7 - Why We Need Them, Continued

[realrockstar786]

this has broaden my concept of maths n its importance.nw i certainly will love doing maths. thanks lot



[MrAlexGTV]

If the universe is digital as quantum theory seems to suggest (energy quanta, Planck length, Plank time etc.) the real "real" numbers are only integers.

[derekowens]

@MrAlexGTV Or rather, integer multiples of the Planck constant.

[derekowens]

@7PunkCyber Yes, thanks. Tell me, though, in the AC circuit analysis, doesn't the imaginary part typically get dropped at the end, leaving only the real part as the actual meaningful value?

[3bood1st]

Do u have a video about changing from complex to polar ??

[derekowens]

@3bood1st In my Precalculus class, I do cover some topics on complex numbers in polar form. The videos are very old and low quality, though. I need to re-record them before I post them for the public to see.

[nigellimjr]

helped alot. sweet

[oHeyitsthatguy]

hah, make a box pi units by pi units exactly - infinite space ._.

Problem? :)

Very good video thanks.

[13Septem13]

this is great explanation

[WarringGod]

if you ever feel like we don't appreciate this. we do you sir are a LEGEND

[deeFUNKtify]

Thank you derek, this was a greatly done video, with ample history to explain the systems we use in mathematics. much better job of it than uni lecturers for sure! =)

[radutheskater1]

thank you

[frankr29]

A revealing statement is made at 6:00 of this video: "... irrational numbers show up in the real world, so we need to add those to our system of numbers." The same is true for the rational numbers, etc. that came before, but it is NOT true for the imaginary/complex numbers that come after. The ONLY rationale for these numbers is algebraic closure. This is why, in my view, using the word "numbers" for so-called imaginary numbers is fundamentally misleading. Any comments, Derek?

[derekowens]

Wow, you are a very astute listener, and that's a great comment. When people ask about the practical, real world application of imaginary numbers, the common example given is from electrical engineering - phase diagrams for alternating current. Perhaps it is revealing that the imaginary part of the answer gets discarded in such problems. It ends up being only the real part that counts.

[kfk4life]

@derekowens It doesn't _always_ get discarded... They're essential in quantum mechanics.

[mbarkhau]

@derekowens While we are nit picking. I have a problem with using the word "exists" in the context of numbers. Numbers don't exist, they are concepts which can be meaningful or meaningless. A tree exists, an apple exists, you and I exist. Numbers don't exist, neither do laws or rights or society. But just because they don't exist, doesn't mean the concepts aren't useful to make sense of the world.

[derekowens]

@mbarkhau Okay, I'll grant you that. Existence can refer to material objects, not concepts. One might argue the point, though. Also, my understanding of complex numbers has changed since I recorded these videos, and doing a little editing is on my ToDo list. They don't seem to be actual numbers in the same sense that real numbers are. If real numbers are called "real" because they can be used to represent real things, then imaginary numbers are called imaginary because they cannot.

[Mejiera12]

than you

[DrBPhD]

In physics complex numbers frequently show up.

[nifected]

i enjoy watching your sessions very much

thank you :-)

[AluminumStudios]

Elegant, clear, brilliant explanation!

[jaakkokuu]

Are you sure it is totally completed? what if we are just too dum to see that there is even wider set of numbers than the complex numbers? Like, you would just add another dimension so that you would have x as real, y as Im and z as the extra super_new_set!!! ;) You're probably going to say that there are no problems that can't be solved if using complex numbers, but maybe our math hasn't just evolved to high enough level. We would be as the jungle ppl who didn't know about the other numbers.

[derekowens]

That's a good point. Some mathematicians have expanded the field of numbers further. Abraham Robinson, for example, in the 1960's, proved some work with "hyper-real" numbers that is, in my opinion, very important. It deals with infinitely small numbers (and their reciprocals, which are infinitely large) but which are not quite zero (or not quite infinity), and these are important concepts in the development of calculus. I don't know of any numbers on a third axis, though.

[AnotherChessPlayer]

You're looking for Vectors. Linear Algebra.

[derekowens]

Thanks!

There actually are some answers to that question. I can name two:

1) I really like this stuff. In other words, I'm pretty internally motivated.

2) I'm externally motivated also. Most teachers teach in a school, in which their pay is set by the bureaucracy. Even if they do a great job, they get the same pay. I work in the free market. My compensation depends on the job I do for my clients. So I'm motivated to improve, to innovate, and to do the best I can.

[knownbeforetime]

Maybe we should go back to the system we had a couple hundred years ago or so. When you got a job by challenging the person whose job you want and see who knows more or who's doing higher level work. It would certainly keep everyone on their toes. :D

[rmayhew53]

So only people with money can buy quality teachers for their kids?

[moltencheese123]

You know when you said that after we include rational and irrational numbers then our number line has no "holes2 in it...

How about numbers such as "e" or "pi", which i beleive are called "transcendental" numbers...(numbers which are not the root of any polynomial) would these numbers not be "holes" in the number line of just rational and irrational numbers?

[derekowens]

e and pi are, as you say, transcendental numbers. They are also irrational. Irrational and transcendental are not mutually exclusive categories. From what I understand, the number line is "complete" if you consider all the irrational and the rational numbers, and the transcendental numbers would be included with the irrationals.

I believe it was the German mathematician Dedekind who developed the important theories in this area. Interesting stuff.

[moltencheese123]

An excellent point! I didn't really think, but yes transcendental numbers must be a subset of the irrationals, haha, woops :)

...and YES it is very interesting stuff, explained particularly well by the way - great video

[dannyman2000]

in the last minute, you sketched +/- sqrt(2) i

you only mapped it in the positive direction. is there suppose to be one in the negative direction?

[derekowens]

Yes, good point there. I could have (and should have) plotted a point in the negative direction as well. There are two solutions.

[antianticamper]

you are really good at what you do. i bet your videos are going to have a much further effect than you ever guessed...

thnx!

[martymilan]

Great man.

[dr00fd]

you know what , i really really liked the way of your explanation , and believe it or not i was thinking about why do we need complex numbers and what is the use of it but i didn't care about it since i can solve it , but thanks for making it clear ..

and please keep uplaoding your lessons , it is so much easy over here .