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)?
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.
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?
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...
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.
nice vid. the moving sinewave was cool, but i fell of at 2:44. i mean, i understand the calculations at the right side in an algebraic way, but it still doesnt make sense way the square root of minus one is equal to the square root of eulers number in the power of pi times i - in a graphical way. so, well, i guess youre right, but the thing is that you kind of start of good with explaining with the plane and axis, but yeah... well. nice to see thoser numbers anyway :)
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.
all u did at 1:29 was show that e to the io power equals itself
blizzard477 9 months ago
@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.
camgere 9 months ago
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)?
blower05 4 years ago
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 4 years ago
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?
CogitoErgoCogitoSum 2 years ago
The proof that e^(ix) = cos(x) + i sin(x) is not particularly difficult. Not obvious perhaps, but not hard either.
CogitoErgoCogitoSum 2 years ago
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 2 years ago
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.
camgere 2 years ago
Thank you. Very informative. You might consider using a smaller font though..
staahl 4 years ago
nice vid. the moving sinewave was cool, but i fell of at 2:44. i mean, i understand the calculations at the right side in an algebraic way, but it still doesnt make sense way the square root of minus one is equal to the square root of eulers number in the power of pi times i - in a graphical way. so, well, i guess youre right, but the thing is that you kind of start of good with explaining with the plane and axis, but yeah... well. nice to see thoser numbers anyway :)
MIFFLINITY 4 years ago
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.
camgere 4 years ago