What is i?
98,456
Published 2021-02-07
0:09 i=sqrt(-1)
0:31 solving for i from the Euler's formula
5:42 solving for i involving Lambert W function
Lambert W function explained: • Lambert W Function (domain, range, ap...
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All Comments (21)
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Can u solve for z?
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Imagine some kid searching for alphabet 'i' and gets this.
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5:15 cos^2 z -1 is always negative and we have sqrt negative which involves z. So "i" is still on both sides, just in disguise.
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5:46 the 3rd way
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Math in the 20th century: Galileo and Newton are wrong. Math in 2021: .._.. prank your friend 5:10
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I bet Euler did this kind of tinkering with math for 12 hours every day. 😂
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2nd answer is wrong, you got a nested i in the radical
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Hahahah intro is too good 👍👍👍
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alright, alright the next philosophical question is: Who am √-1 ?
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This actually doesn't make any sense. Also in the 2nd answer you would almost always have a negative under the sqrt. Or a zero (in the case that z = pi*n). This means you still have an 'i' on both sides. What a nice property of 'i' -- no matter what you do, it won't go :)
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What is i for u?
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For "second" form, you can simplify cos²Z-1 = -sin²Z, which of course brings out another √-1.
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3blue1brown’s lockdown series was I think a great introduction to i- one of the ways of thinking of i that stuck with me is a 90 degree rotation in the complex plane
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The fact that you could have simply written e^iz=cosz+i(1-cos²z)½ =cosz+(cos²z-1)½ i=1/z ln(cosz+(cos²z-1)½) Without actually solving the quadratic is pretty amazing!
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Okay I just noticed that you could just take the euler's formula change isin(z) to √(-1)*√(1-cos²z) which is √(cos²z-1) and its a tremendous shortcut
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For that derivation could you not just start with Euler's identity, take the log of both sides, rearrange to isolate i and then use the Pythagorean identity to rewrite sin z?
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It’s an old joke, but Here goes: i don’t want to be on the bottom
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With the first formula, cos^2 (z) will always be less than or equal to one. When it's less than 1, we have a a square root of a negative number (aka i on both sides again). When it's equal to one, you get 1/z * ln(0), which also doesn't work since ln(0) is undefined 😭
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As a physics man, I hate the W function. Non-analytical functions disturb me at a deep level.
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I thought this was going to be about explaining why we should say i^2=-1 instead of using the square root in the definition. Square roots are not nice because (-i)^2 is also -1.
