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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
Oct
17
comment Limit of a 0/0 function
@NikolajK that depends on what 'is' is.
Oct
17
comment Why do we stop at exponentiation stage in arithmetic of natural numbers?
See "my number is bigger" games among mathematicians. But part of the problem is that exponentiation already grows really fast: significantly faster growing functions quickly stop talking about physically realizable numbers. Which makes applications limited.
Sep
17
revised How many ways are there to write $675$ as a difference of two squares?
added 313 characters in body
Sep
17
awarded  Teacher
Sep
17
answered How many ways are there to write $675$ as a difference of two squares?
Sep
15
comment Why exactly does the distributive property work?
@Nick the distributive property of rationals can be derived from integers, and then reals from rationals, partly due to how they are constructed. In a sense, rationals/reals are certain extensions of the natural numbers (parts of them act like natural numbers) that obey nice properties, like distributivity, because we want them to. "Because we made it so" is, however, less interesting. ;) (The naturals, naturally, are not "made" but "found", if you believe that pithy quote)
Aug
18
comment Why doesn't this converge?
@Mathmo123 Not all techniques are bothered by that: the difficulty for a particular technique to be used shouldn't be treated like a fundamental problem. It really is not the fundamental problem. The fundamental problem lies elsewhere.
Aug
18
comment Why doesn't this converge?
A function being undefined on a set of measure zero doesn't seem to be much of a barrier to integrating. As an example, integrate $1/(sgn(x) * |x^{(1/2)}|)$ from -1 to 1 -- it is undefined at 0 as well.
Aug
5
answered What are differences between affine space and vector space?