What is iⁱ ? Imaginary or Real? 

Conclusion: if you take something imaginary and give it imaginary power, it becomes real.

One of the most difficult challenges for any teacher is to make the video worthwhile for a broad range of student backgrounds. You've got an uncanny knack for doing this well.

i^i=x Taking logarithms i lni = lnx lni = -i lnx (1) Taking Euler's famous formula e^(i pi)= -1 And taking the sqrt. e^(i pi/2) = i Substituting in eq 1 (i pi/2) = -i lnx You get rid of the i. And obtain: x=e^(-pi/2)

So nice! You, sir, got me hooked on math again, almost 30 years after one lousy teacher ruined ir for me. Cannot thank you enough.

Thank you so much. Really happy to learn something new today. 😄😃

Very simple Question : i^i=e^(ln(i)*i)=e^(ln(sqrt(-1)*i) e^(1/2*ln(-1)*i) We know : e^(pi*i)=-1 Placement; e^(1/2*ln(e^(pi*i))*i)=e^(1/2*i²*pi)= 1/e^(pi/2) that is a real number 😁

Very interesting video!! I like learning new things!! Thank you!!!!!❤❤❤❤

I think a good Step One would be to define f (w) = z^w in the first place, for a given z. [That would involve the logarithm, which in passing helps to grasp the multivalued nature of this function.]

GOOD EVENING SIR The expression is > 1÷5 when k=0 and < 1÷5 when k > or = 1 and ＞1÷5 as k is less than or equal to -1 Thank you

Today's question was pretty simple. Suggestion of a problem to be solved: sin(x) = 2. The solution exists (obviously, it is complex) and can be obtained analytically.

In quale riferimento ad esempio l ' imbattibilita' interplanetaria è determinabile? Che disastro !

-It' far, far after midnight. What are you doing? -I'm doing i to power i

What is e^h, where h is a quaternion(a + bi + cj + dk)?

Neatness is a key element to communicate ideas. As a teacher, explaining such an advanced topic, it would help if you had better penmanship skills. Learn to make an algebraic symbol x (two letter 'c's back to back), to distinguish it from the common multiplication sign. If my brain has to spend less time deciphering your untidy presentation, then I'll be better able to focus on your teaching.

Another method may be starting from e^iπ = -1 so that iπ = ln(-1) = ln(i^2) = 2ln(i) -> π/2 = ln(i)/I -> ln(i) = iπ/2. Now we can write x = i^I -> ln(x) = iln(i) = i^2 * π/2= -π/2 -> e^ln(x) = x = i^I = e^(-π/2)

great video. Can you explain Cauchy's theorems? I have never understood Complex analysis because it was never explained to me in College. We just memorized the results and used them in examples. I would love someone to explain it because I have no idea how Cauchy came up with any of it. The man must have been an abstract genius.

Equivalent reasoning: x = i^i => x^i = i^(-1) = 1/i = i/i² = -i If x = e^θ: x^i = e^(iθ) = cosθ+isinθ = -i => cosθ = 0 and sinθ = -1 => θ = 3π/2 + 2kπ => x = e^(3π/2 + 2kπ)

Very cool video!

I checked with a calculator and I got the same number, even more accurate.

Great !

(-i)^(-i) resulta no mesmo número, results the same number.

I've seen this answered as i^i = (e^i*pi/2)^i = e^-pi/2 ~= 0.2. What I don't understand is that following this reasoning, it is also (e^5*i*pi/2)^i = e^-5*pi/2.

If you take a real number and raise it to i, you get e to the quantity of i times the natural log of the real number, which is equal to the cosine of the natural log of the real number plus i times the sine of the natural log of the real number. This is how Euler’s formula works. It works for every complex number. Example four to the quantity of five plus three times i is equal to four to the five times e to the quantity of i times the natural log of four to the three. But here’s the question. If you take a complex number to a complex number, will the formula work? If it does work, here’s how you explain it with only math symbols: i^i = e^i ln i = cos(ln i) + i sin(ln i). But we can’t take the natural log of i. But oenr’s comment that is liked by the creator of this video says it’s real. And Josep Martí Carreras’s comment and SyberMath’s answer says it’s e^-pi/2. So the answer is absolutely e^pi/2.

so is this solution a result of bad math or this solution is ACTUALLY the solution and the graph of x^x would intersect the solution if that graph was to be drawn in complex plane too.? someone elaborate please

i would say. i too the power of i is just more i but not r.

Seems this is a required problem for all math sites

1:14 Why? I mean, why the number with e? Why don't we say any other number for example 5^(ix) = cosx + isinx ?

so its indeterminate if there's a k?

i = exp[(90° + n•360°)•i] i^i = 1/exp(90° + n•360°) is a real and infinity solutions

For k=1 i^i = e^(1.5*π) = 111.318 When dividing the length of the equator by 360 degrees, we get 111.319 km Whatever that means? 😄

Yes

I used the same method and got the same solution, i^i=e^(-pi/2). But when I tried (i^i)^4 = (i^4)^i = 1^i = 1 != e^(-2*pi), why?

Nice!

hallo auther. I wonder if (a*i)^i belongs to a Realnumber, provided that a belongs to R.

this only works for the principal angle right? sorry i didnt watch the whole video

This is unreal, man!

But is e^(-pi/2) transcendental?

This is the advanced class....

Что то я не очень понял, но очень интересно! 😆

your understanding of math should be measured by how far you can make it through this video before you give up trying to understand it and just watch to see what the answer is

This could have been solved in a few seconds! Simply replace base i by e^i*pi/2 , raise it to power i then multiply the exponents to get -pi/2 Why such an elaborate video?

e^ipi = -1 sqrt(e^ipi) = sqrt(-1) = i (e^ipi)^1/2 = i e^(i^2pi/2) = i^i

What is the point of imaginary numbers?

make a video why 1 to the power Infinity is not 1

Ничего не понимаю i^i также легко вычислить как например 4^4.

imaginairy stated as conciouss descision.

"I'm gonna go get the papers, get the papers."

The set of Real and imaginary numbers put together is a complex number set. In other words all real and imaginary numbers are called complex numbers.

Yes it is.

i^i = e^-2pi•n x e^-pi/2 There are literally an infinite number of answers.

cos(i)=??? Imaginary or Real?

are videos on the internet made by criminals who hide behind camera operators real?

When you show this to a teacher, that person will faint.

very interesting ! respect from china

You can't just multiple powers of i time i since they are complex numbers. That rule doesn't apply to complex numbers, even though you got lucky the answer came out correct. That rule only applies to real numbers. Please re-think is over and make the necessary correction to the steps. You are on the right track.

Imaging, imagining is quite real.

I have a challenge for you: (7^x) + (4^x) = (13^x)

What will be the answer of i^2i Anybody???

愛の愛情は実数

For once I did it right … possibly because this was easy …

exp(-pi/2)

🤍 amazing

glad i knew about eulers formula and could instantly arrive at the answer

i^i=roughly 1/5. Damn...

arg 1 = 0, arg -1 = pi, arg i = pi/2, arg -i = -pi/2

what is the moral of the story here? I watched the whole video but the guy never makes a point at the end as to why he did all of this!

#bro face reveal

So... i*ln i is -P/2?

(e^(i*pi/2))^i=e^(-pi/2)...real

My guess: (e^ipi)=-1 so (e^(ipi/2))=i so i^i= e^(i^2pi/2)=e^(-pi/2) so using a calculator .2078…

not real is not the same as imaginairy. as classification. for example. the number 10 is a part of the number 100 axioma's 1.real is not imaginairy 2.reality alowes imagination. therefor. hypothesis: 1.imagination is a part of reality bit not the whole reality, true or false ? so of 10 is a part of a bigger number. 100 things you dont imagine and are not part of reality can also excist. then negative 10 is not a 100 but 10 too the power of 10 is. so the question you ask is. does your imagination covers the whole reality ? or. do you have an accurate view on reality or do you intrepretate reality well, without observing it ? is that the line of your real question ? or are you missing something. like is the language too abstract too describe reality. (the answer on the last one is, yes, the language is merely a symbolicle representation of the real thing but sufficient enought to work with. ) or are you just saying hi...

((-1)^0.5)^(-1^0.5) = ±1^0.25 ...

i^i = 1/8th root of -1

0:49 He fails a little

How to prove i^i>1/5 without using calculator

i for an i make whole world blind - MATHma Gandhi

Why in the world did you bring Euler's Identity into it? Completely unnecessary. Why didn't you write i = exp((i*Pi)/2) and then use your rules of exponents. Would have yielded the same result in less than 1 line.

Imaginary indicates a Tachyon enigma, Thanks, so I can become Etc., however, how do You pls. set that 'e' be? Thanks!

real 🤣: i=exp(i*pi/2) so i^i = exp(-pi/2)

Be careful when presenting graphs of solutions without proper identification of your graphs. You went from presenting the solution of i=cosx+isinx on the complex plane to presenting a solution of i^i=e^(-pi/2) on the real axis. Teaching complex numbers can be a bear ...

Let i^i = r[cos(a)+i*sin(a)] ---- (1) (i^i)^2=r^2[cos(2a)+i*sina(2a)] i^(i^2)= i^-1= 1/i= i/(i*i)= i/(-1)= -i= Then by comparing the real part and imaginary part, we can have the r and a. Sub r and a into (1), then problem solved. answer. 😊😊😊😊😊😊

1^10=1

Wolfram say it's real: e^(-π/2)

Real ?

пишут так что бы сбить с толку! 2:20 экспоненциальная запись это е в степени i * (фи), а если вместо фи используется х то и на рисунке должно быть х, но на рисунке пишут не х, не фи... а большую Тета! Вот зачем такие запутки? Написал х и на рисунке пиши х! А если не х то изначально пиши Тета и не морочь голову! 2:50 А объяснение синуса и косинуса через r, а потом в итоге выписано классическое определение синуса и косинуса (отношение определенного катета к гипотенузе)..... Жесть! Все сделано через левое плечо и правое колено! Жесть! синус есть синус, а косинус есть косинус! На кой тут r применили.... (я знаю что такое r но тут оно не к месту!) 3.00 а зачем писать нормальную форму комплексного числа для i и еще уточнять что мы имеем r=1 ? Вот на кой это? 3:30 ну да-да-да, 30 сек рассуждаем как комплексное число i будет отображаться на комплексной плоскости. Простите но это очевидно! (исходя из нормальной формы записи числа i) 3:50 записали i в тригоном.форме... а на 2:20 почему это нельзя было сделать? короче... до сих пор это все теория.... ладно... пусть так! 4:20 это можно было написать тогда когда экспоненциальную форму описывали т.е. 2 минуты назад.... (ок не ворчу......) 4:45 ну да.... только это одно действие и есть решение, а все остальное это теория..... которую должны знать те кто решают задачи с комплексными числами! (пол ролика теории.... нуладно! если автор так видит...) 5:20 вспоминает про период в два Пи.... мда.... печально! Вообще то это изначальное определение! И опускание этого факта в самом начале - это ошибка! короче! Баба Яга против! (ну не кошерно это!)

e^i* pi/2 = i, So (e^i * pi/2)^i = e^(-pi/2) [ As i^2=-1] So i^i is Real and it is e^(-pi/2)

Eu pensei nisso na escola e todos disseram que sou idiota😮

🐶

答え　i^i= e ^-Πi

if i = -1 × real. if imaginairy is a negative real. there are a lot of "notreal" but they are not all the opposite of real. yhe words come from latin or greek. 1real 2.anti real 3.contra real 4.pseudo real 5.quasi real 6.beyond real 7.before real 8. areal 9.non real 10.not real 11.advanced real 12.not real yet 13.was real once but not know 14.is real for you but not for me 15.phantasy 16.imaginairy 17.like a dream 18.surreal (more then real) something like that. imaginairy is not that black and white or diabolic as presumpted. like if real state too imaginairy then if turning left state too going up... as ? instead of simple turning right. it is possible that imaginairy is more .. then reality.. although it is reality that makes imagination possible. if you ask me... everything is real. found out have a very efficient minimal perceptual view on the matter. senses are body specific. so pretty sure reality is more then can be perceived.

wow nice

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i to the i = iphone max pro gold miaheheh

If i=e^i.pi/2 then (i)^i= (e^i.pi/2)^i Here folowing you will making a logic syntax error: You get power i in processin into e^i.pi/2 and resulting i^i= e^[(i^2).pi/2] You can not go like that!