Introduction to i and imaginary numbers | Imaginary and complex numbers | Precalculus | Khan Academy

Introduction to i and imaginary numbers

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[rubenlinares]

I better get an imaginary 100 on the test.

[petergriffin]

If there are imaginary numbers I should be able to make an imaginary answer

[LuisHernandez-mvfy]

khan academy= 5 minutes

teacher= 1+ hour(s)

[Visualizingmathandphysics]

Complex numbers actually indicate rotation.
i = rotation by 90 degrees.
i^2 = rotation by 180 degrees = -1 = negative of that number.
.
i^3= rotation by 270 degrees = -i
i^4 = rotation by 360 degrees =1.

.
Take any object in your hand and rotate it by 90 degrees 4 times. You will see that it comes back to as it was before rotation.


There is a whole video on this on my channel if you would like.

With Love
Binnoy

[sevi]

i^4 could be more easily explained as i² * i², dont you think? -1 * -1 = 1 is obvious

[prettytwistedd]

I came here to learn and I'm feeling so attacked right now

[Visualizingmathandphysics]

'i' is just a way by which mathematicians could denote that the object is rotated or inclined. For example if a stick is kept horizontal its length say is '5'. Now if the same stick is kept vertical, can we measure it on the same number line? No right.
For this reason, mathematicians used complex numbers to just denote that the object is just rotated now, so they denoted it as 5i.
Hope this helps,
Binnoy
Visualizing Maths
A book beyond formulas.

[just]

I'm glad to see you explain this in such an approachable way. Surely this topic invites some seriously complicated topics but I feel like you explain it in a way that anyone could understand. Does anyone else feel like this needs to be taught to kids much earlier so that they have more time to sort through the topic and come to terms with it? I tell my younger relatives all the time that they need to be taking calculus in high school, but admittedly I never mention that they need to understand this topic as well. I hope in the coming years we see some real attention given to such an important topic for budding engineers.

[I-did-not-ask-for-a-handle]

"Aye times aye!" Sounds like Pirates math.

[HiddenTalent_]

wow.

*dials up khan academy*

Me: Excuse me, may I speak to Mr. Khan? Yes, I'm a student and I'd like o request
that he come as a guest speaker to my pre-calculous class. Why? Oh, because he's
ten times better than any math teacher at my school.

[Visualizingmathandphysics]

i = rotation by 90 deg

-1 = rotation by 180 deg

-i = rotation by 270 deg.

-1 x i x i = rotation by (180+90+90) = rotation by 360 deg.

Also -i x i = rotation by (270 + 90) = rotation by 360 deg.


'1' means after a complete rotation, the object has now come back to start .
This happens at 0 deg, 360 deg, 720 deg, 1080 deg... so on.
Hope this helps

Binnoy
Visualizing maths
A book beyond the formulas

[DasoPanda]

Thank you for putting this up. Imaginary numbers always confused me in high school.

[theprocessionist]

These videos help me a lot. thank you. I do have a question, on this one.

What if the exponent is a real number? What happens between the integer exponents?

[oTheBlackHawko]

stoped the video just in time before my head exploded

[antonytolstunov]

This stuff is so easy thanks to you khan...u simplify it

[Re-lxmd]

can I make up my own number?
w= the number of 0's you have to add to get 1.

0*infinity =/= w.

[purpleturtle]

Why am i watching this at 3am I don't even go to school anymore

[Visualizingmathandphysics]

Lets say you are facing towards the East.
Now suddenly you shift by 90 degees and start facing the North.
You just term this position as "i" which indicates rotation by 90 degrees from the start point.
Now lets say again you rotate by 90 degree. You are now facing the west. You term this as -1 since your direction is negative as compared to the start point.Hence i*i = -1 which means rotating twice by 90 degrees makes you point in the opposite direction.
Binnoy
Visualizing Maths.

[abaetol]

Oh my goodness this is so cool!! Thank you Khan Academy

[Frutpunch]

Oh khanacademy math man you are a god send