Complex Numbers, Part 5 - The Imaginary Unit

[realrockstar786]

thank u very much.it really helped lot.

[wingsofbala]

super....



[GearZNet]

My method is based on the cycle of real numbers i^2 = -1 and i^4 = 1. I call it " KEEPIN IT REAL!" :) If an EVEN exponent IS divisible by 4 then => +1 BUT If NOT => -1. Secondly, any ODD exponent is either -i or +i. EXAMPLE: [A] i^27 is i^26 * i => => 26 is NOT divisible by 4 so RESULT = -i. [B] i^29 = i^28 * i SO 28 IS divisible by 4 so result = +i ... SO No remainders are needed :)

Derrell Wiliams

[3bood1st]

nice jop keep it up :)



[Anonymystik]

... the imaginary parts canceled themselves and the only remaining part was the real one. Therefore, this has a point to count with these "unreal" numbers. Furthermore, lots of problems can be solved more easily by using them, instead of trying to avoid them. As for the example, in physics, complex numbers are used to deal with RLC circuits. But aren´t all the numbers just only the abstraction? You know, what means "three apples" or "three chairs", but what exactly means word "three" ?

[Anonymystik]

It is an interesting question, if we should take complex numbers just like the other numbers. I think, it is just an abstraction, maybe useful (not only in mathematics, but also in physics), but it is just only a different way how to solve some problems. Interesting is, that complex numbers were discovered, when Rafael Bombelli tried to solve a cubic equations, which had 3 real solutions. He found out, that the formula he was using lead to complex numbers, although at the end...

[fireluigi12]

what if the exponent of i is negative?

[derekowens]

@fireluigi12 The same pattern holds, going in the opposite direction.

i^0 = 1

i^-1 = -i

i^-2 = -2

i^-3 = i

and so on

[fireluigi12]

@derekowens is i^-2 really -2, or is it -1?

[derekowens]

@fireluigi12 Whoops, typing too fast. Yes, it is -1. Sorry, and thanks.

[fireluigi12]

geez, this is the first time i was ever interested in math.

[mgunar]

The way I do it is like this:

Given for example, i^400, this equals (i^2)^200 = (-1)^200 = 1

Basically i^2 raised to an odd number will give you -1 and i^2 raised to an even number will give 1. In other words, rewrite the expression - using laws of exponents - and just look at whether i^2 is raised to an odd or even power.

[f18viper]

You have truly excellent handwriting.

[SaurabhOKumar]

He is Just using the concept of VEDIC MATHEMATICS here, WITHOUT telling ANYONE OF US or Confessing about it, thats' how guys like HIM end up being ETHINICALLY GAY even to their forefathers; and BRAINWASH others to that THiNKING that things happen only through them, whose ansestors stole others' concepts without getting an authorization or rights. No Confession AT ALL in order to florish the names of his mENTALLY SICK FOREFATHERS!

[derekowens]

@SaurabhOKumar Sorry, but I'm not familiar with Vedic Mathematics. Nothing ethnic here, nor gay either, just some math theory and some intellectual contributions by Gauss. And yes, ideas come from a variety of places, but I'm not aware Gauss stealing any from anyone.

[mimoo90]

ohhhh... thanx god for creating persons like u...

[Anithas101]

great

[aztips]

7:45 I can't believe I've never seen it represented this way before (or maybe have and wasn't paying attention) - very intuitive!!

How about when i is raised to negative (integer) exponents?

[userpyromaster]

@aztips when you raise it to a negative integer, the answer will become a fraction. same rules apply.

[XDAdzXD]

what if it is rasied to to a realli big number but that big number is a prime number?

[thefritz123]

Thanks a lot! Well done!

[theimaginarynumber]

My brother, -i, is also the square root of -1. Be careful.

[derekowens]

Ah, you are correct! I stand corrected, by the imaginary number himself!

[derekowens]

-i is not the same as i. It is the opposite, just as -1 is the opposite of 1. I the complex plane, i is one unit up from the origin, -i is one unit down.

Hope that helps!

D.O.

[jonolollmao]

@derekowens Ohhhh I get it now! Hmm I was searching something yesterday, e ^ i pi + 1 = 0... seems interesting, though I have no idea what it all means hahaha

[derekowens]

@jonolollmao Oh, yes, that's an interesting equation. I'll have a video on that soon, too....

[frostwow]

previously said -i, is also the square root of -1 this suggests that (i) = +/- (i)

because i = sqrt(-1),

squaring both sides gives sqr(i) = (-1)

and sqarerooting it again, gives: (i) = +/- [sqrt(-1)]

of which the right side is +/- (i), and that gives us: (i) = +/- (i), and that +/- (i) = sqrt (-1)

it really is interesting

[drakecool2k]

loved this lesson........................­.....

[PresidentMotherLuva]

explained it well, thanks

[fuedebe]

i can just sit here and not go to class anymore. you lecture pretty well.

[whenultra]

imaginary units , can tahn be infinitely in motion with sine and cosine subsets, as (tahn)0+cosx+(tahn)non-0+cos x,,,,,,,,,,etc, tahn line+cos.

[uvermusic]

i find the circle part kinda similar to the derivatives of sine and cosine..is it coincidence? boy isnt maths just magical? =)

[derekowens]

Yes, it all ties together. In the Precalculus class, we use sine and cosine to express complex numbers in trigonometric form. It's great stuff.

[sayhellotomybeastie]

that's a revelation - cleared everything up, fantastic!

[brilliantfranz]

Hey pretty good tutorial... thanks for uploading!

Cheers from Venezuela