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If you connect the lattice points of $f(x) = \sqrt{x}$ together through secant lines, you create a function $ g(x) = \left\lfloor \sqrt{x} \right\rfloor + \frac{x - \left\lfloor \sqrt{x} \right\rfloor^2}{\left\lceil \sqrt{x} \right\rceil^2 - \left \lfloor \sqrt{x} \right \rfloor^2}$.

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I was attempting to find the area between f(x) and g(x) for different ranges given by the lattice points that they shared ((1,1), (4,2), (9,3), etc), and then using those as boundaries (upper boundary being $x^2$, lower boundary being $(x - 1)^2$), which be: $$\int_{(x - 1)^{2}}^{x^{2}} (f(x) - g(x))dx $$ Using desmos, I figured that this value is 1/6 for all natural numbers (it changes for other numbers, but they would not matter because x represents different ranges, and having a non-integer number wouldn't make sense there). Is that a coincidence, or is there more to it? Thanks in advance!

Jean Marie
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Math Man
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  • Your question is almost impossible to understand. What do you mean by "lattice points of $f(x)=\sqrt{x}$ together through secant lines" ? Besides, what is the origin of formula $g(x) = \left\lfloor \sqrt{x} \right\rfloor + \frac{x - \left\lfloor \sqrt{x} \right\rfloor^2}{\left\lceil \sqrt{x} \right\rceil^2 - \left \lfloor \sqrt{x} \right \rfloor^2}$ ? A graphical representation would be welcome... – Jean Marie Nov 06 '22 at 22:52
  • Hi! Sorry about that! Here's a graphical representation: https://www.desmos.com/calculator/yznae5cd2b By lattice points, I meant the integer points on f(x) - (0,0),(1,1),(4,2),(9,3) - and how we could construct a function g(x) that is essentially composed of lines between the lattice points - if you check out the link i put above, it shows this representation very well (i highlighted the first few intersection points as a demo). – Math Man Nov 07 '22 at 01:32
  • Thank you very much. I just included your graphical representation into your question. – Jean Marie Nov 07 '22 at 07:46
  • If we switch the $x$ and $y$-coordinates, I believe this follows from the facts that the function is quadratic and all secant-line segments have the same width. – Greg Martin Nov 07 '22 at 07:57
  • How have you obtained the surprizing (and clever) compact form of function $g$ ? – Jean Marie Nov 07 '22 at 12:22
  • Well I obtained it by thinking of a method of calculating square roots of non-perfect squares in a linear fashion - that is, creating a number line between the nearest perfect square less than our number and the nearest perfect square greater than our number. For instance, if you had the number 37, we would create a number line between 36 and 49, find the difference between the two (13), and then write our number as a mixed fraction - in this case, we have 37 - 36 = 1, so the answer is 6 1/13 - g(x) is basically just an algebraic way of viewing this way of calculating non-perfect square roots. – Math Man Nov 08 '22 at 00:43

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