What is the graph of x^a when a is not an integer? An unusual look at familiar functions #some2

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• A.

  @a.arredondo · 3 years ago (edited) · pinned

  <b>F.A.Q:</b><br><br><b>Q. How did you do the 3D animations?</b><br><b>A.</b> I programmed them in Python, using the manim library: <a href="https://www.manim.community/">https://www.manim.community</a><br><br><b>Q. How can I play around with these types of graphs myself?</b><br><b>A.</b> Any software that can graph 3D parametric curves will do, but you do have to define the real and imaginary parts of the parametric function yourself. Here&apos;s an interactive example for x^a using GeoGebra 3D: <a href="https://www.geogebra.org/m/wtqwhswu">https://www.geogebra.org/m/wtqwhswu</a><br><br><b>Q. Isn&apos;t x^2.4 just the 5th-degree root of x^12? (and thus just a negative number for negative x?)</b><br><b>A.</b> That is one possible way to interpret it, yes. There are other just-as-valid interpretations. This is what I wanted you to think about with the suggested exercise at <a href="https://www.youtube.com/watch?v=_lb1AxwXLaM&amp;t=830">13:50</a>. When the exponent a in x^a can be written as a fraction with an odd base (which is the case for 2.4 = 12/5) then one of the many values of x^a will be a real number, even for negative x. Which one of the multiple values to pick from can be subject to debate, and you can see that in the showcase starting at <a href="https://www.youtube.com/watch?v=_lb1AxwXLaM&amp;t=23">00:23</a> only one of the three graphing tools that we tested (Desmos) chooses this interpretation of always assigning a real value to x^a when possible.<br><br><b>Q. So you are defining x^a by using e^something-else? Isn&apos;t that a circular definition?</b><br><b>A.</b> No. The function e^x or exp(x) is not understood to be an exponentiation, but rather a function defined by its series expansion (this expansion can be seen at <a href="https://www.youtube.com/watch?v=_lb1AxwXLaM&amp;t=427">07:07</a>).<br><br><b>Q. Where do all of these definitions come from anyway?</b><br><b>A.</b> That&apos;s a great question, hopefully for another video.

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• BA

  @barrettellman3781 · 1 year ago

  Bro dropped a 3B1B caliber video, it blew up, and then hasn&apos;t uploaded another one since? 6.7k subs? 28k likes? Crazy channel honestly. Hope he comes up in my feed soon with something new.

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• DY

  @DynestiGTI · 4 years ago

  Complex topics don&apos;t have to be dumbed down, all it takes is a good visualisation and it can be taught to anyone.

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• LU

  @LunaAlphaKretin · 4 years ago

  My jaw hit the floor when x^a started rotating around the real axis and suddenly I understood exactly why even a curves up and odd a curves down. Fantastic video, thanks for sharing these concepts!

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• PR

  @PretzelBS · 4 years ago

  I love how math is always self consistent. Literally spins around in the complex plane to get to where it needs to go

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• YU

  @YummaMurong · 1 year ago

  Bro just posted this gem two years ago and never posted again

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• JO

  @johnredberg · 4 years ago

  This is HANDS DOWN the best of the 10 or so <a href="https://www.youtube.com/hashtag/some2">#SoME2</a> videos I&apos;ve watched so far. 1/ It centers around a &quot;simple&quot; (middle/high school) topic: graphing some of the most basic functions. 2/ It reveals beautiful hidden aspects of the topic that are likely surprising to many people, even those with a higher degree in STEM (usually because the visualization of complex functions is HIGHLY underexplored for the sake of rushing to more &quot;interesting&quot; topics in complex analysis). 3/ It encourages the audience to experiment, think about different approaches, and explore exercises. 4/ It is well produced (great visualizations, great audio quality, great script).<br>For my taste, you could have spent a little more time on the general formula for any x in C* and the more technical parts involving algebraic manipulations and the different branches of the logarithm (see how carefully 3b1b does it ;-) ), but in general I think that shouldn&apos;t bar you from being one of the top contenders for the win. <br>Keep up the good work!

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• DR

  @drawing-ology · 4 years ago

  Looked like that&apos;s some high level mathematics easily explained to a 16 yr dumb like me . Thank you sir for such a interesting session

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• LL

  @lloydgush · 4 years ago

  The fun thing is how complex numbers have this tendency to create rotations.<br><br>Essentially when you have an irrational exponent you are taking a root.

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• IA

  @IAm18PercentCarbon · 10 months ago

  Bud, this is incredible. Why did this just get recommended to me? I suddenly understand things I wish I had known decades ago!

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• KH

  @kharybdis · 3 years ago

  The difference between dumbing down for the viewer and raising the viewer up to the desired level is such a shockingly pleasant one to experience. Brilliantly satisfying video, even a comforting one in its beauty.

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• LO

  @lockaltube · 1 year ago

  But why nobody tries to continue the idea of <a href="https://www.youtube.com/watch?v=_lb1AxwXLaM&amp;t=791">13:11</a> graph and draw y=x^x at all k? It will look like a surface (like Riemann surface), and probably is the most correct visualization

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