A Floor & Ceiling Equation

We solve the equation ⌊x^2 - x⌋ = ⌈1 - x^2⌉.
00:00 Sketching x^2 - x and 1 - x^2
01:53 Key values
05:26 Sketching ⌈1 - x^2⌉
07:13 Sketching ⌊x^2 - x⌋

[@shauryagupta3644]

Very nice explanation! I was able to make the graph, but when it came to plotting the points I got lazy and started looking for an alternative method :P

[@quandarkumtanglehairs4743]

Haha! Right? I've had to go back, recently, and force myself to do the long way through many different calculations from my youth, finding I'd become lazy, and weak, from dependence on my shortcuts. We need both, both the long-way for detail and thoroughness and reminders of justifications, and also shortcuts for mid-calculation summarizing for handling encapsulated values and relations in a manageable way.

[@user-cd9dd1mx4n]

Fantastic solution! I really appreciate how systematically you approached the problem, making it easy to follow and understand. Your channel is truly a gem for anyone looking to improve their math skills, and your way makes learning enjoyable. Please keep uploading such excellent videos! Your dedication and clear explanations are incredibly valuable. Apart from the maths, I like your English accent, it is amazing. Like from Saudi Arabia 🇸🇦

[@DrBarker]

Thank you so much!

[@TikeMyson69]

Agreed 100%

[@decare696]

omg! Two of my fav irrational numbers in one video!

[@ccrypt949]

Great video, will work great with my lower sixth pupils as a bit of a side note to the regular A-Level course

[@zathrasyes1287]

Thx for another good friday-math.

[@francescocostanzo8225]

This is an incredible video. Where do you get your inspiration?

[@DrBarker]

Thank you! For a video like this one, I usually decide, for example, that I want to solve a weird equation involving the floor function. Then it's a matter of playing with different starting equations, and trying to find a nice method of solving, until I find something that I'm happy to record.

[@archangecamilien1879]

Interesting question, lol...have to write this algebraically, etc, in equations, but basically you have to make sure that they are within two integers, etc, like if you draw a line at each integers, forming a grid of horizontal lines, you have to make sure they are within two grids, one (x^2 -x or something) above the middle line (so it's floor is on the middle line), the other (1-x^2) below the middle line (so its ceiling is on the middle line), but above the bottom one (so it's ceiling isn't there)...should be easy to write in terms of algebra, but I can't do that in my head, lol, not very good with my head and I'm in a hurry...

[@TheLuckySpades]

Algebraically you can exclude the outside regions by showing that one side is >=2 while the other is

[@archangecamilien1879]

Ah, I see...I actually typed a comment some 20 mins after I wrote that with my approach to it, but I don't see it...something about regions where 1< (x²-x) - (1-x²)

[@brachypelmasmith]

can this be solved mathematically, without drawing the functions or is graph required

[@anthonyserafini]

It can be solved, but is less pretty. You basically have to manipulate inequalities to show that the floor and ceiling will have different values at certain intervals.

[@alaricsnellpym]

Hmmm, you say "smaller" for "closer to negative infinity"... I try to use larger/smaller for distance or proximity to 0, and higher/lower for the proximity to positive or negative infinity, as I felt that's less confusing. But you're a far better maths educator than me. Am I wrong? Should -1000000 be considered a smaller number than -1? 🤔

[@samueldeandrade8535]

Man ... don't make up rules. Or if you want to do that, make some good rules.

[@F_A_F123]

It doesn't really matter

[@alaricsnellpym]

"Don't make up rules"? "It doesn't really matter"? Have neither of you ever met a typical mathematician before? 🤣

[@TheLuckySpades]

The relation you describe is different from what is typically meant with

[@alaricsnellpym]

@@TheLuckySpades I'm treating signed numbers as a vector quantity, with a magnitude and a direction (albeit only in 1d) when I talk about "larger and smaller", I'm comparing magnitudes!

[@Jono4174]

1/2 - sqrt(5)/2 = 1 - ϕ 😡😜