Solving a Floor within Floor Equation

This video is about solving a Floor Equation
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[@leickrobinson5186]

Very nice! I think bprp and Michael Penn have a new contender! :-D

[@SyberMath]

Thank you for the kind words! 😊

[@givrally7634]

I solved it a different way, using some calculus. We know x*floor(x) has to be between 10 and 11. The derivative of x*floor(x) is floor(x) (since the derivative of floor(x) is zero everywhere where it is defined), so x*floor(x) is actually the integral from 0 to x of floor(t) dt. Since floor is positive, we know that its integral is increasing, so we just need to find out where it crosses 10 and when it crosses 11. x*floor(x) = 10 implies that x = 10/floor(x). Since floor(x) can only take a few possible values, we can check and find that floor(x) = 3, so x = 10/3. We know that the integral is increasing, so we just need to find how much we have to "push" to add 1 to the integral. Since the x value that gives 10 is 10/3, we know that for the next 2/3, it will increase linearly, since floor(x) = 3 on that whole range. So we just need to find the solution to 3h=1, so 1/3, and that is how much we have to add to the previously found 10/3. To conclude, the solution to the original equation is 10/3

[@SyberMath]

That's interesting. I never thought about differentiating the floor function

[@BiscuitZombies]

I solved it by saying that the graph of x\x/ is a sequence of straight lines with width one and is essentially a mapping from [n,n+1] to [n^2, n^2+n] (this is as it will start at n*n=n^2, and there is a gradient of n, so after 1 step, it is n^2+n). This boundary must contain 10. Hence n=3. So the n boundary is (3,4) and the output is (9,12). We have two points of this line (3,9), (9,12). This is a line y=3x. So x must be 10/3 for y to equal 10.

[@kennnnn]

This is pretty easy to do in your head: x must atleast be 10/3 because (10/3)*floor(10/3) = 10 and x is less than 11/3 because (11/3)*floor(11/3) = 11 if you take the floor of any x where 10/3

[@SyberMath]

Good job!

[@georget8008]

The floor and ceiling function equations were not taught back in the 80s when I was in high school.

[@danielbodovs1195]

I wasn't taught them in high school, too, even though I am quite a lot younger. I've only seen them in videos like these and in competition problems.

[@SyberMath]

@@danielbodovs1195 This is a really cool topic. I am curious as to why they would not teach this. Number theory is also fun and they don't teach very much of it either

[@jofx4051]

I don't think high schools teach this; this would be math competition problem

[@aashsyed1277]

@@SyberMath they will eventually ad this topic!

[@goodplacetostop2973]

8:09 Almost 5K subs, nice!

[@SyberMath]

Thank you!

[@marklevin3236]

Much simpler than that. If x is positive the floor value of x must be 3. If it is 4 or more the value function will be at least 16. If floor is 2 or less then value of the function is less than 9. That means floor of 3x is 10. 10

[@SyberMath]

Very nice! Did you know that I'm good at complicating things? 😁

[@marklevin3236]

@@SyberMath And I am even better than you... Once upon a time when I was getting a lunch at college cafeteria I needed to calculate its price.Calcolations involved graphs dimensional integrals of food moving through my digestive organs...Then I came up with formuls for amount of weight weight my walled need to lose.... Solving many differential and integral equations produced the correct price.

[@СтепанПрошин-и7ю]

very good equation, thank you)

[@SyberMath]

Glad you liked it!

[@sonalichakraborty6830]

What about the negative solutions

[@seany.1216]

For the case of n < 0, we can apply the same method as shown in the video, bearing in mind to change in direction of the inequalities as needed. However, doing so, we will arrive at the strict inequality -4 < n < -3. Hence, there are no negative solutions.

[@sonalichakraborty6830]

@@seany.1216 How do u conclude there's no negative soln?

[@SyberMath]

Thanks for bringing that up. I forgot to mention the case for n

[@sonalichakraborty6830]

@@SyberMath ah ok thanks

[@SyberMath]

@@sonalichakraborty6830 Absolutely!

[@MizardXYT]

The solution seems to jump between positive and negative as you change the RHS. There are never both positive and negative solutions, except when RHS is a perfect square. For example, ⌊x ⌊x⌋⌋ = 10 have only a range of positive solutions [10/3,11/3) ⌊x ⌊x⌋⌋ = 12 have only a range of negative solutions (-13/4, -3] ⌊x ⌊x⌋⌋ = 16 = 4² have a single negative solution, and a range of positive solutions. [-4] ∪ [4,17/4)

[@SyberMath]

That's right! Good find! If you graph it, you'll notice that.

[@mauricemaths]

@@SyberMath But you should have tested that!! It does not say that x must be positive...

[@Samiul_007]

Hello Sybermath, Can you make a video of Matrices? or Vectors or Statistics & probability

[@charlottedarroch]

The problem can be solved without resorting to solving quadratics. Let y = Floor(x) and z = Floor(xy). If x < -3, then y 12. If -3

[@Velstadt_The_Royal_Aegis]

Very unique and interesting topic! Keep up the good work, your videos are very entertaining. ✌

[@SyberMath]

Thank you!

[@АзизПиров-ы3й]

Beatiful! Can you solve more floor equation? Please!! 🤗🤗🙏🙏

[@SyberMath]

Thank you! ☺️

[@U9191-e6s]

the negative solutions -10/4 >= x > -11/4

[@tonyhaddad1394]

Nice problem !! Good job

[@SyberMath]

Thanks!

[@setokousuke6548]

Congratulations 🎊 you got 5k subscriber

[@SyberMath]

Thank you so much 😀

[@skilfuldude7312]

What's the difference between floor and GIF ?

[@soumyanamdeo2347]

Both are same function.

[@jofx4051]

wait what 😂

[@elias69420]

Ah yes. The floor here is put inside floor.

[@ramaprasadghosh717]

Writing floor (x)= n and fractional part (x) = f, one gets floor ((n +f)n) = 10 or n*n + floor(nf) = 10 Here 0

[@tamarpeer261]

10xfloor(x) ->floor(x)=10>6=3*2>(2+a)floor(2+a) for a2, which makes floor(x)=3 floor(3x)=10 10

[@Germankacyhay]

👍👍👍👍👍👍👍👍

[@barakathaider6333]

👍

[@CookieGod24]

i am floored

[@SyberMath]

😁😂🤣

[@ganda3454]

x^2=10 x=√10

[@leonhardeuler5211]

Old but gold!

[@coefficient1359]

Nice one

[@SyberMath]

Thank you!

[@זאבגלברד]

Very beautiful. Thanks.

[@SyberMath]

Thank you too!

[@mariomestre7490]

Que pasa si n

[@SyberMath]

It cannot be

[@ciaopeople9664]

​@@SyberMath Are you kidding ? If x n xn>0 !!!

[@adheensheikh6896]

Beautiful stuff

[@SyberMath]

Thank you!

[@deepjyoti5610]

Nyc vedioo

[@SyberMath]

Thank you so much 😀

[@shekhar9750]

Nice solution but little lengthy.