The Clock Hidden Inside the Number One
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- Опубліковано 20 гру 2022
- Let me show you about the “Roots of Unity” and their amazing clock-like patterns. Special thanks to the viewers supporting the Patreon I started this month (the top tier supporters are listed in this video’s credits, and all supporters have their names in this description) / comboclass
Special thanks to: Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Max, and Joost Doesberg.
more Combo Class links:
Bonus channel (which just hit 80k subscribers!): / @domotro
Reddit: / comboclass
Discord: / discord
Topics mentioned in this video include: square roots, cube roots, more roots (particularly "Roots of Unity", also known as "De Moivre numbers"), unit circles, imaginary numbers, the complex plane, homemade clocks, and more!
Disclaimer: Do not copy any physical actions you see in Combo Class episodes, including any use of fire, tools, or other science experiments.
If you read this, leave a comment containing the word "clock" :)
Thanks for watching! Consider checking out the Combo Class Patreon I started this month: www.patreon.com/comboclass
How about we rename complex numbers to composite number? Damn it, that name is already taken...
Absolutely entranced by this chaotic punk rock approach to mathematics.
It's like unhinged in a totally logical way
Mad Scientist of Math.
Geez, all of this clock geometry hidden within relatively simple math just makes me wonder how much crazy stuff has been under our noses this whole time.
Also, the camerawork is seriously underappreciated. Thank you, Carlo!
There’s always a ton of surprisingly good stuff in seemingly simple things.
I'm a new math teacher who sometimes misses learning about new math, and your videos have been scratching that itch for me, and helping me hold onto the passion I have for math. Keep up what you're doing!!
I'm a retired math teacher. I still keep learning new to me math. Good luck with those students.
What a twist of fate, all numbers are mental constructs that help us to understand such properties as quantity, but only the roots of negative numbers are called out for being imaginary creations of the mind.
Yes. Yes. Yes. I'm particularly interested in the properties of the labels given to numbers (eg 1 2 3) rather than the properties of the numbers themselves.
For example how to represent a 2s compliment 8 bit binary number on the number line. How do you tell an unsigned number from the above - given only radix 2 and no other information ?
What do you call a number line where the numerical value of the "next"point is some function of some or all of the previous LABELS (not VALUES) of the points?
I think this is a whole field of maths that no-one addresses. Am I missing something?
With the whole omega thing, I feel interestingly reminded of the golden ratio and some of its properties...could the two somehow be mathematically related? Sounds like a cool video concept if so.
The golden ratio is the (positive) solution to x^2-x-1, these are the solutions to x^2+x+1, it makes sense they're parallel
You are getting to the "metallic" ratio of the "means".
Look here,I have a clue.
Not in the way that’s shown in the video. They are both roots of polynomials over the complex numbers, but one root forms a cyclic subgroup of the complex numbers:
(I.e if you multiply two roots of unity, you get another root of unity, the inverse of a root of unity is also a root of unity)
e^(i*2pi/3) = omega
15:25 i’m surprised you didn’t mention the fact that in your 12th root clock, the coordinates for the real parts are just the cosines of 0°, 30°, 60°, 90° and so on, and the imaginary parts are just the sines of those same angles… perhaps you considered it to be too complex (pun intended) for an already complicated video, hahah, still, i think you should make a followup video where you point this out… anyways, great work as always and thank you Domotro for this amazing channel :)
Yeah I saved the clock to include more details about those notable coordinates over time :)
@@ComboClass🥳🎉
I haven't watched the video, so I'm not sure how much of this was already talked about, and I might be repeating things here.
The complex numbers are represented as points on the cartesian plane (also called R2). The number a + bi would land on the point (a, b) and as such the x-axis represents the real part of this complex number, and y-axis represents the imaginary part.
Now let's draw a line from the origin to this point, we can see that the numbers we were just representing using real and imaginary parts can also be represented using the length of the line and the angle the line makes with the x-axis (taken in a counter clockwise direction). We usually call the angle θ (theta) and the length r.
If we connect the tip of the line, the point (a, b), with the x-axis using a perpendicular line we have a right triangle. It should be obvious now that trigonometry is involved when we want to find a + bi in terms of an angle and radius and vice versa. r is easily found using Pythagoras to be sqrt(a^2 + b^2) (NOTE: m^n represents m raised to the power of n, sqrt(m) represents the square root of m).
a and b can be written in terms of θ and r as
a = rcos(θ)
b = rsin(θ)
which means the number originally had a + bi is equal to rcos(θ) + rsin(θ)i = r(cos(θ) + sin(θ)i).
If you want to look into the topic further, a + bi can also be written as re^(iθ) where e is the number e (approx. 2.71828). You can look up Euler's Formula which should explain where this comes from. Also look up "polar form of complex numbers" if what I explained didn't really settle with you.
I'm a first-year CS student and this video actually answered one of my questions: why can real polynomials even have complex roots? I really thought this was just a silly HowToBasic-esque math channel, but I actually learned something useful for university.
The fact that you can keep squaring ω and it'll take you 1/3 around the circle, then 2/3, then 1/3 again, is so mindblowing to me
Yeah it’s interesting to play around with
ω^1
ω^2
ω^4 = ω^1 (4/3 of the way around is the same position as 1/3 of the way around)
ω^8 = ω^2 (same thing with 8/3 and 2/3)
ω^16 = ω^1
ω^32 = ω^2
and so on :)
how do you type the w-ish thing
@@kdr2 Just search for “unicode omega”
If you have an Apple device you can go to Settings > General > Keyboard > Text Replacement and save it there. I entered “ω” as the phrase and “/omega” as the shortcut
Thank you for making these videos, they really allow me to gain a greater love for math.
meth
i would also like to add to my previous comment, that for any given root, say root N, all that needs to be done is calculate 2π / N, or equally 360° / N, and let’s call the result of this K, so that K = 2π / N = 360° / N, then calculate all multiples of K from 0 to N-1, naming them A, and finally compute the cosines and sines for those values, and assign the cosines to the real parts of complex values and the sines to the imaginary parts of complex values, and this gives you the coordinates of your roots
it may seem like a lot, but it’s actually pretty easy, i’ll do the first 3 examples
for N=1, therefore (X^1) - 1 = 0; K = 2π = 360°, and N-1 = 0, so 0 will be the only multiple of K, and since A0 = K*0 = 0, now we compute cos(0) and sin(0), which are 1 and 0, respectively, and assigning 1 as real and 0 as imaginary, we get the complex number 1+0i, which is just 1, in conclusion, 1 is the solution when N=1
for N=2, therefore (X^2) - 1 = 0; K = π = 180°, and N-1 = 1, so 0 and 1 will be multiples of K, and so A0 = K*0 = 0 and A1 = K*1 = π = 180°, now we compute cos(0), sin(0), cos(π) and sin(π), which are 1, 0, -1 and 0, respectively, and assigning 1 and -1 as real and both 0’s as imaginary, we get the complex numbers 1+0i and -1+0i, which are just 1 and -1, in conclusion 1 and -1 are the solutions when N=2
for N=3, therefore (X^3) - 1 = 0; K = 2π/3 = 120°, and N-1 = 2, so 0, 1 and 2 will be multiples of K, and so A0 = K*0 = 0, A1 = K*1 = 2π/3 = 120° and A2 = K*2 = 4π/3 = 240°, now we compute cos(0), sin(0), cos(2π/3), sin(2π/3), cos(4π/3) and sin(4π/3), which are 1, 0, -1/2, √3/2, -1/2 and -√3/2, respectively, and assigning 1 and both -1/2’s as real and 0, √3/2, -√3/2 as imaginary, we get the complex numbers 1+0i which is just 1, (-1/2)+(√3/2)i and (-1/2)+(-√3/2)i, in conclusion these are the three results when N=3
as you can see, you can do this same process for any positive integer N (from 1 to infinity) and get the results for any root N that is equal to 1
thank you for coming to my TED talk :)
p.s.: Domotro, you should totally make a video about this, please, i know you have power to make it so much more interesting and entertaining than i just did :P
Is there a name for this rule?
Would have killed to have a math teacher like you growing up. Love all the topics you explore and effort you put in to making it fun!
Yeah, if you had a math teacher that gave you a 20 minute lesson once a week, I'm sure you'd turn out fine.
@@B3Band I would love to have a class like this every week, a chance to explore new and interesting topics in math with my teacher. Probably would still have to go through all the mundane memorization stuff some of the other days of the week though.
I love how it's becoming tradition for something to fall over at the very beginning of every video
Finally math that both ignites my interest towards it and doesn't make my stomach ache. Awesome channel, I love the clunky style.
Nice video, just wanted to add something to it...
If I'm not mistaken, complex numbers can be written as r(cos(a)+sin(a)i), with a as the angle and r as the distance (normally you'd use theta instead a, but I'm on my phone rn).
In the end of the video you covered all notable angles except 45° (and symmetries). If my formula is correct, then it should look like (√2+i√2)÷2! If you were to do x²⁴-1=0 you would get the ones you showed and 45° (along with his symmetries). Also, I think you could shorten it to (1+i)√2÷2?
The exclammation mark is just ponctuation, not the factorial sign.
Edit: I'd also like to add that these are values of e^(ix) where x is the notable angles (in radians). Example: e^(iπ0.25) = (1+i)√2÷2 (notice how π0.25 radians is 45º).
Well yes. You’re just describing points on a trigonometric circle :) pi/4 radians (45°) is situated at point ( sqrt(2)/2, sqrt(2)/2 ). It only makes sense to add the i to the y coordinate since we’re talking about an imaginary y plane.
I didn't remember or know this, but relating this stuff to trig, pi, etc. would definitely be something I'm curious to see more about.
I wish my students had this sort of enthusiasm. I wish I had students. Or that I was a teacher. Or understood maths. Mostly I just wish I had more clocks.
I would love to see a video on Euler's identity, or e in general :) I have a physics degree, and your videos are bringing back the joy and beauty of math that I lost touch with in the (very difficult) process of getting it.
not sure if this is obvious or not but omega can be derived using the quadratic formula.
x^3 - 1 = 0 factors to (x - 1)(x^2 + x + 1) = 0, which has the trivial solution 1, and (-1 ± sqrt(1^2 - 4 * 1 * 1) / 2 * 1) or (-1 ± i * sqrt(3)) / 2, aka omega, which also explains why omega shows up twice in its two "forms" on the clock!
I love to watch your videos and i’m always blown away by the amount of stuff that I just don’t know sometimes. It was so funny to click on this video and actually understand what you were talking about. You explained this concept better than my teacher- this video is going in my study playlist. Thanks a ton for such a concise, logical AND entertaining explanation!!
You have a great personality for teaching math! i cant wait for your channel to blow up!
Ok, i learned more in half hour than in 6 months course of lineal algebra
Sensei Domotro never disappoints!
bro you cant sleep on the linear algebra classes 💀
I really love you chaotic videos!
I need one clock like this!
What a great way to approach and talk about the roots of one and the complex plane. I totally agree with you that the "imaginary numbers" should be called something more apt such as the "perpendicular numbers" or "orthogonal numbers." The silly "imaginary" name for them has made so many people be turned off and discount such a fundamentally important concept as orthogonality and how to numerically parameterize it!
and if you take the 12th root of 2, instead, you end up with the western well tempered music scale, where repeats (full rotations) occur at the octaves, where you double the frequency you started with.
Loved this vid! Thanks for teaching :)
I just tried using the quartic formula for
x^5 = 1 --> (x - 1)(x^4 + x^3 + x^2 + x + 1). It worked, but gave me a much longer and more complex number than the numbers shown here
There's a trick: x^5 = 1 iff x^5 - 1 = (x-1)(x^4 + x^3 + x^2 + x + 1) = 0, so the solutions are 1 and the roots of the second factor -- looking at the second factor and dividing by x^2 we have x^2 + x + 1 + 1/x + 1/x^2 = 0. The trick is to write this in terms of x + 1/x and (x + 1/x)^2 = x^2 + 2 + 1/x^2 -- so we have (x + 1/x)^2 + (x + 1/x) - 1 = 0, and you can use the quadratic formula to solve for the two values of x + 1/x and then finally use the quadratic formula again (twice) to find the four values of x :)
Lovin' the content. Keep it up.
So glad I 'clocked' this video! Another great episode from my new favourite UA-camr!
This is fantastic! Amazing math, teaching, and manner. Comforting, fun and enlightening. Thank you!
Thanks Domotro, always great to watch your videos! Love the roots of unity clock, gotta make one for my classroom at school 😎👍
Fantastic! Thank you for the wonderful and informative lesson that was full of entertainment.
It's funny how omega gets used for 2 somewhat popular things (Ordinals and Roots of Unity)
Would be interesting to see a video about ordinal numbers though
Wonderful as is expected…
We are certainly an apt pupil…
🦈👁
I'm pretty sure that this man loves the mathematical concept of a clock, but hates the physical representation of it.
I was watching a video with Carl Bender describing how he things that we live in the "real" plane that is described by real numbers while some quantum mechanical properties can take place, travel, and manifest within the complex plane inwhich real numbers no longer describe their behavior.
For example, quantum tunneling. For the fraction of a second that a particle is "tunneling" is exists somewhere other than the real plane. We do not have the language (math is a language) to describe where or in what state a particle is in where its actively tunneling. Carl Bender think it doesnt disappear, but instead moves into the complex plane descrived by a language we havent mapped yet
Loved how the beat dropped when i came into the room (6:20)
15:18
cameraman foot reveal
Killer math content Domotro!
Another great video, Dom0trO, thanks. I love your nicknames for things like the hyperelevens and hollowelevens, etc. Iff you're so miffed by the naming of imaginary and complex numbers, why don't you coin some new terms for them?
Oh hey, I was learning about roots of unity in class a few weeks back, this is like nice revision
Wow! I'm fairly certain that I still don't completely understand, but this is absolutely fascinating! Thanks for explaining!
Multi-dimensional numbers, might be most descriptive? It also allows for the intuition of adding dimensions past a second.
@comboclass Maybe taking a page from StarTrek would work? They would refer to any dimension that presented this behaviour as "multi-phasic".
Awesome!
Another good one, Domotro!
Beautiful. The unit circle is like a great work of art, the more you look at it the more you appreciate it.
Feel free to use some of the Patreon money for laundry : )
I love these vids because they make me feel smart and knowledgeable even though I have no clue what's going on half the time
I like complex plane concept to visualize the solutions of equation. Thxs for your easy explanation.
Also, if you take 2 pi and divide it by x’s exponent, you get the amount of radians in between each point. Taking the cosine and sine of these angles will result in the real and imaginary numbers of the coordinates respectively (just multiply the angle by i after taking the sin of it)
Simple and beautiful.
You would be the best teacher on time mechanics potentially create time travel
9:50 oh i get it now. When i learned this in Complex Analysis class, i always thought the name was excessively philosophical...but now i see, unity means "in regards to the unit circle" not some vague philosophical mathematical harmony with the universe....(necessarily 😏)
Also... I'm really glad you explained the origin of complex for the C.
I usually just call them Lateral numbers, as they are often modeled perpendicular to the numbers as another atribute of the same number. However since i imagine them as the flipside of a coin, i might even accept Complementary Numbers or Swing Numbers. At least with Complementary, you dont need to change the C for their symbol.
a math video I can actually understand and enjoy
i love this channel
In a way they're kind of like finite kernels of endomorphisms of the unit circle on the complex plane under multiplication.
This episode was a great time.
Any point of the movable circle will move an equivalent distance to it’s circumference. A point on a circle with circumference of 1 rotating 360° around a circle of circumference of 2 will travel a total of 4 units, and a point on a circle with circumference 1 rotating around a circle of radius 1 will travel 2 units. If it only rotates 180° then it has successfully traveled 1 unit and remained upright.
Imagine if you didn’t hold one quarter still and you rotated them 180° like cogs, their midpoints would both stay stationary and any given point on their edges would move only 1/2 rotation. By allowing one of the circles to move around the other, any given point on its surface can move twice as far at the expense of the other circles points remaining perfectly stationary. If you rotate both quarters at once like gears until the heads are upside down, they will also “swap places”, and when viewed from the opposite side the quarter that was previously heads up on the left, will now be heads up on the right.
who's here from the stream?
@@BTCRATES Domotro streams pretty often on the Combo Class Bonus channel. You can also find full vods of all previous streams with a synced chat replay there!
The Roots of Unity is a sweet band name.
It's omega o'clock I am going to bed guys.
In the moment I did see the roots of unity I knew this was gonna go imaginary
TYFYS
Thank you for a very special presentation. I love your videos!
Although I internally usually call your channel "the homeless mathemagician" :-).
This is awesome learn more math on yt then all of my public education
Combo Class.. where all clocks go to perish.
More good shit from Domotro
what i figured out before the video is that from x^n-1=0 the solutions can be found at the complex plane by starting at 1 and rotating (360/n)° counterclockwise with trigonometry daamnn thats fucking interesting
Cool video. This makes me think, it would be really cool if you could just do a video on i. I've heard other mathematicians talk about how calling them imaginary numbers is sort of a misnomer. But it does seem like just a made up number from a layman's perspective? 🤔
Looks to me like the 6th root of unity (1+i√3/2) Is where we can do degenerate things because it would tile the complex plane in hexagon.
If i'm not mistaken, that would mean that -2*(1+i√3/2)^3 = (1+i√3/2)+2-(1+i√3/2). Ok, I know it sounds lame because that's just 2=2... but trust me in my madness it's pretty rad!
happy hannukah
Nice clock
I’m interested to know why root 3 and 1/2 have such a large role in these roots of unity. They appear a lot, especially looking at the 12 roots of unity. Why?
wow 😳
I need to buy this clock to troll my students...
Do quaternions fit into this picture at all? Great video as always!
Quaternions don't follow the fundamental theorem of algebra (about factoring) because i² = j² = -1 so x² + 1 = 0 has 2 ways of factoring and 2 roots.
@@tupoiu k² = -1 too
Geometrically, just as the imaginary part of a complex number can represent an angle, quaternions can represent three angles (and the real part is the radius). This can be a hypersphere, or a regular 3D sphere with the extra coordinate representing the orientation of the point on the surface location described by the other three points.
Rotations in three dimensional space are kinda funky, and there are many possible combinations of turns which will bring you to the same result. (So, there's not an inverse to multiplication.) And this is why a combination of two turns on two different orthogonal axis, can be the same as a single turn on the third axis.
i think, i found what causes your clocks to fall. You tie your leg with them.
I got a vsauce ad before this. kinda funny since you guys are similar
You should make a video on the game 5D chess with multiverse time travel
filming in below 5 degrees c
I love how he and the camera operator seem to get more and more distractible/excitable each video
What video talks about prime numbers and the fundamental theorem of arithmetic? I tried searching your channel for it and could not find it. Thanks
One of the first ones called “which ancient questions about prime numbers can we answer”
On the 12th root of unity my true love said to me....
You can make a song out of this video, I wouldn't want to watch it but it's doable
So could x^n = 1 define the equation of a circle in the complex plane? If so, is there any restriction on the types of numbers that go into n? I can see how it could work with positive integers, but struggle to imagine how it would look if using negatives, irrationals or even complex numbers, so wonder if they would cause problems.
So is every point on the unit circle an n-th root of one? If not, what do we know about the ones that aren’t?
Brilliant! Who makes these clocks?
Most of the background clocks are old/broken ones I got for free or cheap so I don’t know their full history. I made the “roots of unity” clock in this episode though (using some broken clock parts and art supplies)
@@ComboClass To make a dial out of the unit circle is beautiful and a great idea for a watchmaker.
Bro image on left of thumbnail made me realize this immediately lol
Pointed to x-1 when x=-1 and vice versa
*same for x+1
Eweu.. a spider on me...
Good one...!
how many stopped clocks do you have
⌚️
11:58 haha I can see the camera persn
Never knew you had a cat! Hopefully he isn't injured by your chaotic nature... anyways, another excellent video!!!
3 cats haha and yeah I keep them safe :)
@@ComboClass Oh cool, I have 3 cats too!
bro i read all this when i was in class 11
. Omega stuff reminds me of the song, I'm my own grandpa.
What do you think of calling 'imaginary' numbers 'orthogonal' numbers?
Oh hell no. Descartes was probably one of histories best logicians. Their contribution to what we have today shouldn't merely be taken for granted. And, yes. Descartes considered the idea of an imaginary number ridiculous. But, Schrodinger felt the same way about quantum mechanics. The whole Schrodinger's cat bit is him trying to highlight the absurdity of the conclusions. I'm just saying, cut Descartes some slack. The imaginary line isn't _directly_ observable like the natural line is. They merely neglected the effort to imagine further the consequences of "imaginary" values. Which, I'd say is forgivable. Since, putting a lot of time and effort into something you _might_ _not_ _be_ _able_ _to_ _demonstrate_ _exists_ is usually called, "crazy".
i too am scared of imaginary numbers
People hate i?? When I first learnt it existed I thought it was the coolest shit ever. Although, I was introduced to it by Kjartan Poskitt's 'Murderous Maths' kids book series, not like... School.
At least "complex numbers" weren't called "complicated numbers" :D
As a philosophy student, the hate for Descartes is so damn vindicating
If you won't call it an imaginary number or a complex number, what would you rather it be called?
Whataabbaout if I doun't accedpt negativ numbers? or complex numbers? The only numbers are whole numbers. I don't think there are alot 'o numbers that represent alota numberers that are not based on numbers.