This Clickbait Harvard Problem Has No Real Solution

Added: 2026-05-11

[00:00:00]Okay, this looks like another clickbait Harvard entrance exam question that was suggested by one of my viewers. So, what is the value or values of this negative one to the power of square root of the pi?

[00:00:11]This looks like a simple calculator question, but this is a complex number trap.

[00:00:15]For example, if you consider real number system, then this negative one to the power of square root of the pi is not defined as a real number.

[00:00:23]But, if you consider complex numbers, something even more surprising happens.

[00:00:27]So, we need to rewrite this negative one as complex exponential form.

[00:00:31]And like I said, if you have integer exponent, then there would be not any issue, right? Negative one squared is positive one. Negative one to the power of three, negative one cubed is negative one, and so on.

[00:00:42]But then again, we have this square root of the pi.

[00:00:47]This is irrational number.

[00:00:52]So, that's why we cannot treat this just like an ordinary integer power, right?

[00:00:57]And we will be rewriting this negative one as complex exponential form.

[00:01:01]So, let me just draw this complex plane.

[00:01:07]Real axis and imaginary axis. And negative one lies on the real axis.

[00:01:15]And the angle has to be just a pi.

[00:01:21]So, negative one could be written as standard form of a complex number.

[00:01:24]Negative one is the same as negative [applause] one plus zero times i.

[00:01:29]So, that is why one form of negative one is going to be e to the power of i pi.

[00:01:36]Pi as an angle, right?

[00:01:38]But then again, angle repeats every two pi. So, same value will be gotten by adding the multiple of two pis or subtracting the multiple of two pis from the angle. Meaning, negative one could be written as also the same as e to the power of i times pi plus two pi.

[00:01:56]Or this is the same as e to the power of i times pi minus 2 pi. Also the same as e to the power of i times pi plus 2 pi plus another 2 pi and so on.

[00:02:09]Like I said, we can add the multiple of 2 pi's or subtract a multiple of 2 pi's to or from the angle, right?

[00:02:17]So that is why -1 in general could be written as e to the power of i times 2 times an integer k + 1 >> [applause] >> times pi. Of course, k is an integer.

[00:02:34]That means -1 to the power of square root of pi.

[00:02:39]>> [applause] >> This is the same as e to the power of i times 2k + [applause] 1 times pi times square root of the pi. Of course, k as an integer.

[00:02:51]So looks like we have infinitely many values of -1 to the power of square root of pi in the complex number system.

[00:03:00]And at the same time, we can use Euler's formula to represent -1 to the power of square root of the pi using tricks.

[00:03:07]So let's use Euler's formula.

[00:03:18]So the Euler's formula is now e to the power of i theta. This is the same as cosine theta plus i times sine theta.

[00:03:31]Okay, then our theta is going to be now that part, right?

[00:03:37]So in our case, the theta was 2k >> [applause] >> + 1 times pi times square root of the pi.

[00:03:47]Okay, so that is why using this, this e to the power of i * 2k + 1 * pi * square root of the pi is going to be then the same as cosine of 2k + 1 * pi * square root of the pi plus sin i times >> [applause] >> sin of the same thing 2k + 1 pi * square root of the pi.

[00:04:14]And of course >> [applause] >> k has to be an integer.

[00:04:21]Okay, so this is going to be the same expression for -1 to the power of square root of the pi.

[00:04:27]>> [applause] >> Maybe we can talk about the principal value, right?

[00:04:37]So the principal value of -1 to the power of square root of the pi.

[00:04:41]So the principal value of -1 FIRST PRINCIPAL VALUE OF -1 IS just uh pi, right?

[00:04:56]So principal value of -1 is pi and then we can just plug it in then 0 to the k.

[00:05:04]So if k is equal to now 0, then we can just plug it in.

[00:05:07]Then e to the power of only i * pi * square root of the pi is going to be the principal value of -1 to the power of square root of the pi.

[00:05:19]This is the same as then cosine of pi * square root of the pi plus i * the sin of pi * square root of the pi.

[00:05:31]And about pi * square root of the pi, right? So if you estimate the value of pi * square root of the pi, it is going to be 5.

[00:05:41]568 and so on.

[00:05:44]So looks like this is the fourth quadrant, right?

[00:05:48]So, in the fourth quadrant cosine is going to be now positive, but then again, sine is negative.

[00:05:59]So, that is why the principal value could be represented as now then, which is e to the power of i times pi square root of the pi. Okay, this is then going to be around 0.755.

[00:06:15]>> [applause] >> minus uh 0.6 56 times i.

[00:06:24]This would be the estimated the principal value of -1 to the power of square root of the pi, right?

[00:06:34]Okay, so like I said, if you consider only the real system, the -1 to the power of square root of the pi is not defined as a real number, but if you consider complex numbers, we will have infinitely many values of -1 to the power of square root of the pi. How amazing.