Newton: _has to calculate the factors of 51._ My brain: "So Newton does not play darts." More on topic, I have enjoyed this dive into how to work the floor function. I await to see how deep you take this.
@@lukaskamin755 A standard dart board divides the area around the bull's-eye into 20 areas, each with a score ranging 1-20. There are also rings dividing the whole board into single, double (D), and treble (T) scores. The average darts player knows the scores for each ring by heart, including T17 being worth 51 points.
In your approach (I will rather write x=k+a): Solve for a (consider k as a constant and solve the second degree equation) You should get something like a= -k ± (√(40k-51))/2, so k+a=x= ± (√(40k-51))/2, From this, the part inside the square root must be nonnegative (so k≥2, and actually only the positive root is valid and you get a formula for x given k). Now using that 0≤a
For completeness, we should mention that the inequality at 4:15 is only valid if x is positive. If x was negative, then the inequality would be reversed. However, it is clear that x must be positive since negative x values make the original equation never equal to 0.
You are right. Indeed the case when k is negative would need to be done appart (reversing both inequalities) but that will eventually lead to an empty feasible solution for k (obtaining the complements of the regions obtained in the video)
Recently discovered Prime Newtons' channel and it's fantastic! Really love how he not only puts the equation in the video title but takes the time to get the proper math symbols in there :D
That's a beautiful question! Most people would generally stop at finding the values of x for which the original equation becomes negative & hence would wrongly give answer as 2 to 8. Thanks for solving it
9:00 we can use the quadratic criteria for sign : the sign outwards the critical points matches the sign of the a number in ax²+bx+c :) But indeed a table is way more intuitive and easy to understand !
For me: Always re-evaluate the results! After having found the zero crossing at the lower end, showing no additional crossing in the neighboring "cells", i thought to have finished after finding two of the zero crossings at the upper end. Seeing an additional result here, i could smell at first glance from a glimpse at my so far resulting coordinates that and where the fourth solution is to be expected. But: Slip error is my second name.
it's like trying to draw a number line with integers labelled on it, find the range the number is in, and then output the number on the left side of the range so yeah
I have a video idea. Actually it is a series. You do like the national math contest quizes from every country and make it into a video series. I think i would be excited to see my country's math contest featured and how hard it is for you to solve them
Crazy thing, l need to admit I never used to solve equations with floor or ceiling functions, and as you broke it down it looks quite doable, but still daunting. I also admit I have a gap in solving tasks where you have kinda check a bunch of values to find the ones, that fit. I find them exhausting even as I watch them . Btw fun fact in both Ukrainian and Russian only floor function is reviewed, at least at schools, and it's actually called the integer part of x, I'm not sure if there's a term for ceiling function. I used to encounter the durect translations of these words( by their initial meaning as parts of the room) but that was mostky in context of translation from/ into English, so I'm not sure if it is used in Maths 😊
Shouldn't we check that the obtained irrationals have the integer part equal to the corresponding k at which they were calculated? I mean when you see the graph, you can obviously check it visually, but if it was an Olympiad and moreover in 1999, I don't think it implies using computer or even calculator.
idk but 12:48 if (2k-11)(2k-5) > 0 then we look for outcomes that are greater than 0 so (-) times (-) and + times +, we skip - times - because of the first inequality so that leaves us with + times +, then we see that when k > 5,5 both of the multipliers are positive so that makes the inequality true. Does that mean that 2 is out of the question and is not an anwser? Love your videos btw, I watch them everyday lol
Habrá un correo electrónico para mandar mi resolución del ejercicio?? Buenas, gracias de antemano y saludos cordiales desde Perú - Lima - Villa El Salvador.
those type of questions should be done on computer, or even better plot the graph with real graph paper if you're asians students. visualize the curve not only help to find the K values, also very easy to understand how the floor function acting on a normal quadratic equation.
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Newton: _has to calculate the factors of 51._
My brain: "So Newton does not play darts."
More on topic, I have enjoyed this dive into how to work the floor function. I await to see how deep you take this.
Neither do I, what it has to do with darts?
@@lukaskamin755 A standard dart board divides the area around the bull's-eye into 20 areas, each with a score ranging 1-20. There are also rings dividing the whole board into single, double (D), and treble (T) scores. The average darts player knows the scores for each ring by heart, including T17 being worth 51 points.
@@Mycroft616 ok, I see, interesting to know
@@Mycroft616 interesting to know. I have never played darts. At least not when points count.
Instead of k55 (That is the empty set.) It must be written k55.😎
Correct. Sometimes, I just write what I'm thinking. Twisted mind!
Great stuff. The way I thought about it was to rewrite x as (x+a), where x is an integer and 0
In your approach (I will rather write x=k+a):
Solve for a (consider k as a constant and solve the second degree equation)
You should get something like
a= -k ± (√(40k-51))/2, so
k+a=x= ± (√(40k-51))/2,
From this, the part inside the square root must be nonnegative (so k≥2, and actually only the positive root is valid and you get a formula for x given k).
Now using that 0≤a
used x = k + c, got k² - 10k + 51/4 = -2kc -c², then used the min and max values of 2kc-c²
For completeness, we should mention that the inequality at 4:15 is only valid if x is positive. If x was negative, then the inequality would be reversed. However, it is clear that x must be positive since negative x values make the original equation never equal to 0.
Isn't x^2 always positive so you don't need to worry about that?
@@Monero_Monello We have k < x. If k=-3 and x=-2.9, for example, then we can’t say, for example that k^2
You are right. Indeed the case when k is negative would need to be done appart (reversing both inequalities) but that will eventually lead to an empty feasible solution for k (obtaining the complements of the regions obtained in the video)
@@GreenMeansGOF pretty easy to show that k needs to be positive so that's problem solved here
4x^2 = 40k - 51
0 < 40k - 51
k > 51/40
if x0
Recently discovered Prime Newtons' channel and it's fantastic! Really love how he not only puts the equation in the video title but takes the time to get the proper math symbols in there :D
The floor function is really cool :)
That's a beautiful question! Most people would generally stop at finding the values of x for which the original equation becomes negative & hence would wrongly give answer as 2 to 8. Thanks for solving it
I never stop learning, but it always seems to be after midnight for me. :)
A bit loss at 3:29, you say that because K
Good observation. You can show that with x0, therefore x must be >0. Though indeed it is something Mr. Newton forgot to address.
9:00 we can use the quadratic criteria for sign : the sign outwards the critical points matches the sign of the a number in ax²+bx+c :)
But indeed a table is way more intuitive and easy to understand !
For me: Always re-evaluate the results! After having found the zero crossing at the lower end, showing no additional crossing in the neighboring "cells", i thought to have finished after finding two of the zero crossings at the upper end. Seeing an additional result here, i could smell at first glance from a glimpse at my so far resulting coordinates that and where the fourth solution is to be expected. But: Slip error is my second name.
Is floor defined for negative numbers ? If so, floor of -9.8 is -10 ?
it's like trying to draw a number line with integers labelled on it, find the range the number is in, and then output the number on the left side of the range
so yeah
Yes. It is the greatest integer not above your number (or simply rounding down)
What's weird is the fractional part is +0.2 because it's defined as -9.8-(-10).
very interesting question and great use of simple but powerful mathematical techniques.
Stunned!
I have a video idea. Actually it is a series. You do like the national math contest quizes from every country and make it into a video series. I think i would be excited to see my country's math contest featured and how hard it is for you to solve them
Crazy thing, l need to admit I never used to solve equations with floor or ceiling functions, and as you broke it down it looks quite doable, but still daunting. I also admit I have a gap in solving tasks where you have kinda check a bunch of values to find the ones, that fit. I find them exhausting even as I watch them .
Btw fun fact in both Ukrainian and Russian only floor function is reviewed, at least at schools, and it's actually called the integer part of x, I'm not sure if there's a term for ceiling function. I used to encounter the durect translations of these words( by their initial meaning as parts of the room) but that was mostky in context of translation from/ into English, so I'm not sure if it is used in Maths 😊
Please do laplace transformation videos.
Thinking of it
@@PrimeNewtons 😍😍😍
You are a good teacher ❤
Shouldn't we check that the obtained irrationals have the integer part equal to the corresponding k at which they were calculated? I mean when you see the graph, you can obviously check it visually, but if it was an Olympiad and moreover in 1999, I don't think it implies using computer or even calculator.
In principle yes. Luckily in this case, checking it gets you to exactly the same inequalities (for k) that were solved in the video.
Just a minor issue the product of the root is equal c/a and not a*c and as you can see when you multiply the root you get 51/4 and not 204.
I needed the product to equal 204 and not 51/4. I didn't say they were the roots of the equation. They were tools for factoring.
Me salió a la primera, saludos desde Perú, excelente contenido 🗿🍷👍.
idk but 12:48 if (2k-11)(2k-5) > 0 then we look for outcomes that are greater than 0 so (-) times (-) and + times +, we skip - times - because of the first inequality so that leaves us with + times +, then we see that when k > 5,5 both of the multipliers are positive so that makes the inequality true. Does that mean that 2 is out of the question and is not an anwser?
Love your videos btw, I watch them everyday lol
You shouldn't skip - times -.
Yeah i reattempted it and included - times - and it worked out, thank you very much for replying !
Habrá un correo electrónico para mandar mi resolución del ejercicio?? Buenas, gracias de antemano y saludos cordiales desde Perú - Lima - Villa El Salvador.
why did you choose to restrict to positive integers for k?
can you make more trigonometry videos? I would love that but great video man.
if k=floor)x) how cN K=2 and x=sqr(29)/2
Cool video!
Cool!!
Mathematic is better then every logic what I BS can be done in M W
why aren't second degree inequalities resolved as such? it would seem much quicker to me.
However the video is interesting.
Sir, can you make more videos on functional equations please
50 seconds ago nice
those type of questions should be done on computer, or even better plot the graph with real graph paper if you're asians students. visualize the curve not only help to find the K values, also very easy to understand how the floor function acting on a normal quadratic equation.