Why does -2^x look so strange?

[00:00] You see, approximately six years ago a Reddit user wandered into the r/mathpics subreddit they revealed this picture of a maths trumpet.

[00:18] Sadly, they then got exactly zero community engagement ["Sat Trombone" riff plays] so there you go... Now, I forget how I came across this...

[00:34] let's not get judgy here! and that's not the only picture of The Maths Trumpet; We get different angles of The Maths Trumpet:

[00:48] (I don't care to know) I mean that's head-on towards -x. I'm like: "that's an amazing trumpet,

[01:02] but what on earth is it?" when it says the complex values of (-2)^x? (Multiple lines... something with, fractional values...) [Noir style trumpet plays]

[01:22] Right, so to recap: gives you a positive answer, still gives you a positive answer.

[01:38] from squaring something. Which means: which I'm going to write the fractional way around

[01:50] So it's just whatever the root is, and get back to 4: that answer is either -2 or 2,

[02:03] and the positive one is the Principal Main Root, And, sadly because neither of these are negative, if we had -4 to the power of 1/2:

[02:15] I'll write that down... nothing. because if we have a positive, times a positive, times a positive

[02:30] that's still positive. a negative, times a negative, times a negative I'm going to cheat this slightly,

[02:45] so I'm going to make it an 8 that if we have 8 to the power of 1/3,

[02:57] And if we have -8 to the power of 1/3, so we still got two solutions. But instead of both of them being... When it's a positive base,

[03:09] And this is the beginning of the unravelling fractional powers represent roots hence, multiple lines on The Mystery Trumpet.

[03:23] at this spot here. it's using -2! when we're taking a fractional power of a negative number,

[03:39] You may remember, from my previous video where I can move the complex number "z" around and it shows its square, z^2, move accordingly.

[03:52] if you take "z" all the way around the unit circle And that's because, for the same on the complex plane that will give, give it.

[04:07] also made by my friend Sam, and see both of its square roots. and you're seeing the squared move now.

[04:21] and they're always exactly opposite each other. not being solutions. of a complex number.

[04:36] And, nicely, if I bring "z" down to be four perfectly align with 2 and -2. "It's worth noting this is subtly different to The Trumpet:

[04:51] not different powers. you get two complex roots." cube roots, fourth power roots, etcetera?

[05:05] but for those we have to explore a fun concept I love The roots of unity are the roots of one we want when we've got -2

[05:19] Well this is just the case of: that's equivalent to -1 to whatever our weird fractional power is, multiplied by regular old 2, to the same real number;

[05:33] all of this is a perfectly normal, boring, value. This is the bit that's going to be complex However, we can just look at regular,

[05:50] I'm going to draw a dotted line, For it's not exactly the same but it will be in a moment... and we're taking some root

[06:02] some 1 / n representing the nth root of 1? if we've got 1 to the power of a 1/2

[06:14] well 1 x 1 is going to give us 1. one is always going to feature However, as we saw before,

[06:28] there's the cheeky, the  non principal root over here: -1. So, square root of unity: two roots. Well, if we skip ahead; however, to the fourth root.

[06:42] and -1 x -1 = 1, And actually you can put in a -i as well

[06:54] So the fourth root of unity has four roots, they make a perfectly regular square!

[07:11] and that's because fourth roots are just so nice they give you the square. there's one down here,

[07:27] they're symmetric on -1/2; and that there is (√3/2)i, and over here is (-√3/2)i.

[07:40] I've done a video all  about how much I love √3/2. sure enough, these three roots form an exact equilateral triangle,

[07:59] any nth root of 1, all the corners of which are a valid root of unity.

[08:12] We, however, want -2^x, if this was negative, it just flips the entire thing. you start at -1 and then fill in all the points.

[08:29] And as we saw before, by multiplying by this real, we can get to -2^x. So what this means  is our value of x, up here,

[08:41] it'll equal 1/2, it'll equal 1/3, it'll equal all of them. I shouldn't be just  drawing it as some list of values, and we can add that to the diagram,

[08:56] is now in 3D! and imagine that like coming out of the board towards you! as the roots, as x changes

[09:11] and so they will move and change If only we could visualise it in 3D... using this, The -2^x Explorer.

[09:28] and a look at that! Ahh, I'm already in love... Look at that 3D spiral. So if I just swing it around like that,

[09:42] And as you vary x, gradually up as the principal root moves around on the plane.

[09:55] ah, so the third dimension now, we get this fantastic 3D shape one spiralally thing like this

[10:12] we've now got two of them. And you can see it's a similar shape, slightly tighter coil; um, and the dots are because we're sampling it at discrete points.

[10:26] look at that. we're seeing all of the principal ones and the symmetric roots on on either side of that.

[10:38] because you can either go all the way around a circle and that's the way we've set this up. And as I turn on more and more of these different roots,

[10:50] you can see it fills in The Mystery Maths Trumpet. we have recreated it from scratch. rendering it for us,

[11:03] I'm going to go right down the trumpet, can I look back up the trumpet? Why do I keep saying that?

[11:16] we zoom out and that is The Maths Trumpet. You've got to download the code,

[11:29] I'll make that code available, if you want to check out Zach's incredible -2^x explorer it's great!

[11:42] or rather, it looks like an object at all, is all the lines for the different roots are kind of confined to the one surface And I want to see the actual surface, so I 3D printed one out: [Trumpet fanfare plays]

[12:00] and I've just put the single k=0 root on there, and it actually,

[12:12] Ready? [Jazz trumpet music stops] The question that occurred to me, once I printed this is: we have a continuous line representing an everchanging primary route...

[12:26] this video is brought to you by Bamboo Lab 3D Printers. but there's no time now - I don't know if you noticed; but before, when Regular Matt was doing all the maths on the whiteboard,

[12:41] behind it is the Bamboo X1 Printer. oops! So it's a Milestone Desktop 3D Printer,

[12:55] You can use it at home or at work, It's extremely stable, depending on how you put it down, Previous 3D printers, I didn't bother getting one

[13:09] This one, it just works, it's great! because: it is a top choice for studios like  professional printing,

[13:21] every time enough of you  buy a Bamboo Lab printer People send me in things to print

[13:33] a Babylonian tablet  from around three and a half thousand years ago; and on my phone I can see like the live view,

[13:46] I can see the live view of what it's doing in there. and it will tell me if something goes wrong, Although, I am, um, going to pause that for now

[14:01] Once it's done, someone called Chuck sent it in, I'll show it now if that print works. First ever evidence of humans calculating the square root of two.

[14:18] Uh, normally I make like an  interactive, weird maths thing that someone sent in; So everyone please do check out Bamboo Lab 3D Printers It's currently (up until the 15th of July) their 3rd anniversary celebration.

[14:35] And if you haven't got your own Chucks you can check out Maker World: here's just the maths ones,

[14:47] [Maths Matt] Thanks Business Matt, a worthwhile interruption. I'll link to this file also made by Zach below. is it's a continuous line going around the trumpet

[15:05] and that represents a constantly changing And we looked at things like the square root or the cube root;

[15:17] for all the infinitely many in between values? I pointed out x is not this discrete value,

[15:29] So it kind of makes sense that we get a continuous line slightly more subtle. [Jazz band music plays]

[15:43] so, in that case x = 2. When x equals,

[15:55] When x = 1/3, so the cube root, What if x is something, you know, how many solutions now?

[16:10] Same as 1.3. that is working out the values for -2^(17/10),

[16:26] and the denominator, 10, number of solutions. In fact: 3, 3, 2, 2 and we could rewrite that So you turn it into a fraction

[16:40] the-the bottom bit is the number of solutions. there was a single comment, I don't know if you noticed this before, on the Reddit post and that was from the  user who showed us the trumpet

[16:55] In it they point out they actually  set x equal to some fraction m/n So they are actually only doing discrete values,

[17:07] which is why their trumpet has dots everywhere. uh, we switched to using it as a continuous line, The comment does also talk about how you can do this, uh,

[17:22] And that's just why I've been kind of handwaving my way through, The issue now, however, is; but there's always changing numbers of solutions.

[17:39] Continuous lines and a changing number of values, [Noir style trumpet plays] Whenever there's a point with fewer roots

[17:51] so there's fewer spots  at that point in the cone. And if I zoom in near the middle somewhere, see there, see how there's like a massive gap

[18:07] That's the point that corresponds to 1. there's only one value there's no fractional part, Up here, 2, that's just squaring it,

[18:19] So you can see there there's like a  wedge missing from the trumpet and a single value where all the lines cross. it takes a little while to drag it around to line it up.

[18:33] So, just for fun we thought we would, uh, print them out. Now, we couldn't get the curved trumpet shape, But, you can still see all the lines

[18:46] that might be 1, so either 1 or -1, see there's no other dots at all,

[18:58] Now, here we're recreating the original trumpet Of course, Zach also made a continuous version

[19:10] So this is just, you know,  filling in the continuous line and again you can see all the points where they they bunch up. When there's a single root, or sometimes there'll be a root opposite each other.

[19:25] but you can tell on the original You can, almost it looks like it's not.

[19:38] and that's what gives us a very faint line. That's The Mystery of the Maths Trumpet! how could you come and see me live on stage?

[19:52] so many shows! It's going to be a lot of fun. come to one of the shows.

[20:04] that just means you can pay extra, show up early, I'll sign your calculators ahead of the show

[20:16] And the VIP money makes the tour more fun cuz we use that money to pay for like the hotels we stay in the whole crew has a nicer trip around the country.

[20:32] I will also do a meet and greet after the show, Oh! And I got one show coming up in New York, Uh, and that's it for me,

[20:45] to find the world's best maths party hat...