Can You Sketch This Graph?

[@dorol6375]

You can also look at the greatest common divisor (4 in this case) and plug in values in each interval of size 1/gcd

[@wesleydeng71]

Since ⌊x⌋ is an integer, it can be moved outside of other floor functions and also is ⌊2x⌋ which means, y = ⌊4x⌋ - ⌊2x⌋ - ⌊x⌋ which is easier to analyze.

[@DrBarker]

This is an interesting point - it would be fun to further explore/prove how we can manipulate expressions with the floor function, e.g. when does ⌊ab⌋ = ⌊⌊a⌋⌊b⌋⌋, or ⌊a + b⌋ = ⌊⌊a⌋ + ⌊b⌋⌋.

[@JuniorTheCuber]

I don't believe that holds for -ve x, say u have x = -1.5. Then floor(2x+floor(x)) = floor(-3-2)=-5 But floor(2x)+floor(x) = -4. I believe in general u can only split, for some integer n, floor(n+x) into n+floor(x) only for positive values of x

[@charlottedarroch]

@@JuniorTheCuber If x = -1.5, then floor(2x) = -3 and floor(x) = -2. So floor(2x)+floor(x) = -5. If you work with the piecewise definition of the floor function, you'll see that floor(x+n) = floor(x)+n for all integers n.

[@JuniorTheCuber]

@@charlottedarroch floor(1-0.5)

[@Cosmonaut_Sloth]

⁠​⁠@@JuniorTheCuber … gives 0, since 1+floor(-0.5) = 1-1 = 0 = floor(0.5) = floor(1-0.5). If your argument here is that n is -0.5, note that -0.5 is a rational, but not an integer (as Daniel pointed out that it must be), so this doesn’t contradict anything.

[@composerlmythomorphic2635]

Suppose x=a+r where a is the integral part. y=[4a+4r+[2a+2r+a]]=[4a+4r-[3a+2r]] For rr>1/2, y=[a+4r-1]. Then consider the case r

[@5ucur]

Listening to this Desmos makes an interesting tune: rising, falling, rising again but longer, and so on! Especially when zoomed in a bit or the playback is slowed down a bit.

[@txikitofandango]

Very interesting and beautiful problem here. The reverse problem would also have been interesting, if you are given a graph with some strange but repeating pattern of steps, can you construct a rule for it using the floor function? Also, it would look cool as a polar graph, you could scale theta by a multiple of pi, or not

[@DrBarker]

The reverse problem sounds like it should be possible to solve, as long as our graph is made up of straight lines. It might be easiest to construct such a function as a sum of indicator functions, then work out a general way of converting from indicator functions to floor functions.

[@hylianpikachu37]

I found it easier to simplify by noticing that ⌊4x - ⌊2x + ⌊x⌋⌋⌋ = ⌊4x - (⌊2x⌋ + ⌊x⌋)⌋ = ⌊4x⌋ - ⌊2x⌋ - ⌊x⌋ since the integer values can be "pulled out" of the floor functions. From there, it is easier to analyze this function on the four intervals [n, n + 1/4), [n + 1/4, n + 1/2), [n + 1/2, n + 3/4), and [n + 3/4, n + 1) to get the behaviour.

[@MrRyanroberson1]

4:36 i have somewhat of a simpler method. notice the floor function is blind to integer steps: floor(x+1) = floor(x)+1 always. notice also the floor function always returns integers... so in floor(4x-floor(2x+floor(x))) floor(2x+floor(x)) = floor(2x) + floor(x) and floor(4x - floor(2x) - floor(x)) = floor(4x) - floor(2x) - floor(x) the overall slope should be approximately 1 since 4x-2x-x = x, and because the common period is also 1 that means you only need to look at the range [0,1) to know how this all behaves

[@lexinwonderland5741]

Would love to see somebody make a graphing calculator (desmos/geogebra) app for this problem with various parameters!!

[@kummer45]

Outstanding video. I need a lot of training with floor and ceiling functions.

[@MathOrient]

A very distinctive problem 👍

[@jamiewalker329]

I would have just written x = floor(x) + x* where x* is the fractional part. You then get y = floor(4floor(x) + 4x* - floor(3floor(x) + 2x*)) Then this will give one case where 0

[@osneket5713]

Since I couldn't be bothered to get a piece of paper I tried to solve it entirely in my head. Quite a fun challange actually. I belive the graph can be simplified to: y = [x - 0.25] (The square brackets are floor brackets). My math might be completely off though. I haven't seen the solution as of writing this.

[@osneket5713]

Turns out my math was wrong. But at least my answer matches the correct one for 75% of all x values

[@mrigayu]

What a fun task! I wonder what patterns analogous to the one in your video would look like with logarithms, sinusoids, or even polynomial functions of higher order.

[@AndrejPanjkov]

square wave: y = 1- [x - [x/2 + [x/2]]]

[@rob876]

I really enjoy your videos. I never miss one.

[@DrBarker]

I'm glad you're enjoying them!

[@alexandermorozov2248]

Great, interesting! Good work! 👍

[@NoNameAtAll2]

axis scaling slightly confused me, as I expected roughly y=1*x growth it is indeed so, just not as straightforward as 45 degrees :)

[@kianushmaleki]

Lovely

[@joefarrow1599]

Is there an algorithmic way to take functions of this form (kind of roughly 'floor linear') and return your plotting algorithm of, "go to the right 1/4 go up one, go to the right 1/2 go down one etc."?

[@maxvangulik1988]

Is there a graph of the limit case for this? (That is to say, would the sum of floor((-2^n)x) converge to a continuous function of x?)

[@maxvangulik1988]

The next partial sum would be floor(8x+floor(4x-floor(2x+floor(x)))), then floor(16x-floor(8x+floor(4x-floor(2x+floor(x))))), etc

[@maxvangulik1988]

I have a gut feeling that approaches a line

[@DrBarker]

I'd need to think quite carefully about this. Given that the graph jumps up and down, I wouldn't be surprised if we end up with quite a complicated fractal graph.

[@TaladrisKpop]

The problem becomes much simpler when remarking that f(x+1)=f(x)+1

[@ilafya]

It will be very easyer if you use the periodicity of the fonction wwuch is equal to one

[@ngc-fo5te]

Yes.

[@General12th]

Hi Dr. Barker! Is there anything the _ceiling_ function could add here? Or is taking the ceiling function usually the same as taking the floor function, but everything is shifted vertically?

[@DrBarker]

They're very similar, the only big difference is that the discontinuities are the opposite way around. So the ceiling function could be useful if we want our function to be left-continuous, rather than right-continuous.

[@kappascopezz5122]

An easy identity to relate them is ceil(x) = -floor(-x), so they're essentially both doing the same thing. Though it's easy to make the mistake of thinking that ceil(x) = floor(x) + 1, even though as Dr Barker pointed out, doing that would shift each integer to be mapped to one greater than itself, rather than keeping all the integers the same.

[@nathanisbored]

i like the haircut

[@padraiggluck2980]

I have Desmos 🙂

[@hypebeastuchiha9229]

2:18 THE ONE PIECE IS REAL

[@tomctutor]

I notice a couple of things: There is periodicity associated with these mixed linear-floor combinations (n:1 hence no inverse function); a clearer example is simply x-⇂x⇃which is a sawtooth waveform sitting along the x-axis of magnitude unity... ... ╱╱╱╱ ... Then ax - ⇂x⇃ = (a-1)x+( x-⇂x⇃) which is simply a sloped y=mx line plus the sawtooth earlier. Finally cx - ⇂bx⇃ ≡ bax - ⇂bx⇃ = b(ax - ⇂x⇃) which is just more vertical scaling. I do admit, once you start nesting mixed functions like these, it can get very awkward to visualize. For DrB's example just copy the command, plot floor(4x-floor(2x+floor(x))) into WolframAlpha. 😗

[@BraileySims-qr1yx]

Denote 'floor', ⌊x⌋, by Fx ≡ F(x), and let {x} := x -⌊x⌋ denote the fractional part of x. So, x = Fx + {x} u := F(2x + Fx) = F(2(Fx + {x}) + Fx) = F(3Fx + 2{x}) = 3Fx, if {x} < 1/2, = 3Fx + 1 otherwise So, v := 4x - u = 4(Fx + {x}) - 3Fx = Fx + 4{x}, if {x} < 1/2, = Fx - 1 + 4{x}, otherwise. And, finally, y = F(4x - F(2x + Fx)) = F(v) = Fx, if {x} < 1/4, = Fx + 1, if 1/4 =< {x} < 1/2, = Fx + 1, if 1/2 =< {x} < 3/4, = Fx + 2, if 3/4 =< {x} < 1.