Imaginary i exists

The imaginary number i exists and the square root is -1

Category:

Science & Technology

Tags:

• imaginary
• number
• numbers
• math
• complex
• trig
• trigonometry
• square
• root
• -1

[blizzard477]

all u did at 1:29 was show that e to the io power equals itself

[camgere]

@blizzard477 Thanks for watching. I was just trying to emphasize that e^(i*theta) has a magnitude of 1. Good for you if this seems obvious.

[blower05]

I still don't have the insight how e can be introduced into the complex number. Does it matter it does need "e" or some other symbols or variables to denote the cos(wt) + isin(wt)?

[camgere]

Congratulations! You stumped the professor!. Honestly, I don't understand the proof, either. Wikipedia has a good explanation of "Euler's Formula". Apparently you can take a Taylor Series expansion of all three terms and prove an identity. It has to be "e". The result is very useful.

[camgere]

The fact that the polar form of complex numbers obeys the rules of exponentiation is quite wonderful. Good mathematical notation is a joy forever. It will lead to the correct result for the third, fourth and fifth roots of -1 as well. I have a whold video on this topic. Two is the square root of four because if you square it, you get four back. Same thing. 2.1 isn't the square root of 4.

[CogitoErgoCogitoSum]

Time stamp 2:43. This is not a valid proof. Its a demonstration at best, but its circular in logic, therefore not a proof. You introduce not only i, but rely on a very in-depth property of i, to prove that i exists...

[CogitoErgoCogitoSum]

The proof that e^(ix) = cos(x) + i sin(x) is not particularly difficult. Not obvious perhaps, but not hard either.

[CogitoErgoCogitoSum]

Professor? Right. If you were qualified to call yourself a teacher of any sort, you'd know the answer... especially a professor. PhD my asss. Are you a high school kid or something?

[staahl]

Thank you. Very informative. You might consider using a smaller font though..

[camgere]

Expressing a complex number in polar co-ordinates let's your rethink the meaning of square root. The magnitude of the square root of A is the square root of the magnitude of A. This exactly conforms to our experience with the real number line. The angle of the square root of A is half the angle of A. This makes us rethink what it means to be a square root. Since I neglected mathematical rigor I'm quite happy that anybody got anything out of this.