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 Yearling
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Jun
12
answered What are “instantaneous” rates of change, really?
Jun
12
comment What are “instantaneous” rates of change, really?
This answer seems to neglect the fact you can use infinitesimals to build a rigorous definition of calculus.
May
26
answered Understanding derivatives
May
14
awarded  Yearling
May
11
answered Why is $1/i$ equal to $-i$?
May
7
answered Is there an intuitive, not-too-mathematical way of thinking about limit points?
May
5
comment Universal quadratic formula?
@Mehrdad Then if b and c is zero, the above returns an indeterminate value for x2 -- 0/0. If b and a are zero, then the above returns an indeterminate value for x1 (again 0/0). Even in the extended reals, those don't work.
May
5
comment Universal quadratic formula?
This finds solutions in the extended real number line. What does sgn return if b is zero?
May
4
awarded  Critic
Apr
13
answered Limit of f(x) knowing limit of f(x)/x
Mar
17
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
I might expect a high school student to prove division by $(r+a)$ is valid (neither are $\leq 0$). And $perimeter of square = 32/5 r = 6.4 r$ explicitly might make people happy. The ratio math to show that $6.4 > 2 \pi$ while cute might even be omitted at that point.
Mar
16
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
If you set radius to r, half the side length to b, and a such that a+r=2b (distance from center of circle, to closest midpoint of square), then solve for a in terms of r and simplify fractions along the way, the 3,4,5 triangle falls out. No coordinate system needed, not assumption that something is unit distance.
Jan
13
comment Is there a law that you can add or multiply to both sides of an equation?
@DanChristensen or the observation that $f(x) = x+b$ is injective
Jan
7
awarded  Autobiographer
Dec
15
answered Can a matrix have the same range and nullspace?
Nov
12
answered Disproving existence of real root in some interval for a quintic equation
Oct
30
comment Given any computable number, is there any algorithm to decide whether it is transcendental?
@TonyK Let $X$ be the TM that takes as input $n$. It then starts enumerating all syntactically valid proofs (easy in most formal systems). For each proof, it checks if it is a proof that TM $n$ halts (easy). If so, it outputs 1. It then checks if it is a proof that TM $n$ does not halt. If so it outputs 0. Otherwise it continues running. $X$ halts on input $n$ if and only if there is a proof that says if TM $n$ halts. If for every $n$ there is a proof if TM $n$ halts, then $X$ solves Halt. As $X$ does not solve Halt, the result follows. Flaw?
Oct
29
comment Given any computable number, is there any algorithm to decide whether it is transcendental?
@TonyK Why? I can reduce my statement to yours (simply make a proof writing & checking bot for each $n$ -- if no $n$ exists for which there is no proof (it halts/does not), then my bot solves Halt. Thus there exists an $n$ for which there is no proof). I guess I do have to qualify it with a reasonable (you can mechanically verify proofs for validity, and write them) and fixed system to do the proof in.
Oct
29
comment Given any computable number, is there any algorithm to decide whether it is transcendental?
To clarify, as it took me a second: $a_n$ is transcendental iff Turing Machine $n$ halts, and $a_n$ is computable. As the halting problem cannot be solved, there exists an $n$ for which we cannot prove that Turing Machine halts, and hence an $a_n$ for which we cannot prove it is transcendental, but that $a_n$ is still computable.
Oct
17
awarded  Commentator