34,226 reputation
9106243
bio website blog.plover.com
location Philadelphia, USA
age
visits member for 2 years, 9 months
seen 50 mins ago

I am Mark Dominus, an amateur.

I have a blog that often carries articles about mathematics.

You can email me at mjd@plover.com.


2h
comment Prove that the set of points that make up the unit circle are uncountable
At your stage¸‘exhibiting’ means constructing an explicit one-to-one and onto function, with as much explanatory detail as is necessary to convince the grader that you understand exactly what the function is.
2h
comment Why can't a direct proof be made backwards?
I thought your idea was to show that $5x-2$ is odd (“$A$”) if and only if $x$ is odd (“$X$”), and $3x+1$ is even (“$B$”) if and only if $x$ is odd (“$X$”). Then you want to show $A\iff B$, and instead you show $A\iff X$ and $X\iff B$. Then $A\implies X$ (because $A\iff X$) and $X\implies B$ (bceause $X\iff B$), so $A\implies B$. And $B\implies X$ and $X\implies A$, so $B \implies A$.
2h
comment Why can't a direct proof be made backwards?
The phrase “$A$ if and only if $B$” is short for “if $A$ then $B$, and if $B$ then $A$”, so you don't have to worry about getting it backwards. “If and only if” works in both directions at once.
4h
comment How to select the right books?
I don't see how you could possibly find out what book you like without looking at a bunch of books and picking the one you like.
5h
comment How many pairs of two distinct integers chosen from the set {1, 2, 3, …, 101} have a sum that is even?
Sure, I'm glad I could help.
6h
comment Why can't a direct proof be made backwards?
Yes, exactly so.
6h
comment Why can't a direct proof be made backwards?
@Luke I'm glad I could help. You should also observe that the last statement in your proof is “$5x−7$ is even when $x$ is odd” but you were asked to prove “If $5x-7$ is even then $x$ is odd” which does not say the same thing. Your proof of “$5x−7$ is even when $x$ is odd” is flawless, but it wasn't what you were asked to prove. “If… then” and “when” are not the same.
16h
comment What is the “fastest” increasing function that's useful in some area of math?
Here is another related blog article I enjoyed, "Who Can Name the Bigger Number?"
16h
comment What is the “fastest” increasing function that's useful in some area of math?
I looked into Graham's number recently, and it turns out not to have ever been used in any proof, and in particular it does not appear in the paper that it is claimed to have appeared in. The bound given there is vastly smaller, and also Ackermann-like.
16h
comment What is the “fastest” increasing function that's useful in some area of math?
Incidentally, Ackermann's function does appear naturally as the running time of certain algorithms; the running time of the important union-find algorithm is the inverse of the Ackermann function. Ackermann-like functions also appear naturally in the proofs of several theorems of Ramsey theory; van der Waerden's theorem is a well-known example.
18h
comment $\mathbb{N}$ is a Compact Space with the Co-finite Topology?
You didn't realize it, but you knew that already. Consider the usual topology for $\Bbb R$. Every open set (except $\varnothing$) is infinite. So you could have encountered the same puzzle by asking how there could be any compact subsets of $\Bbb R$.
21h
comment $\mathbb{N}$ is a Compact Space with the Co-finite Topology?
A finite cover means a finite family of neighborhoods, not that the neighborhoods themselves are finite sets.
1d
comment How do I correctly measure the circumference of a circle
You don't have to measure the circumference by measuring the diameter and multiplying by $\pi$. You can get a tape measure and measure the circumference directly. If you're measuring something like a column or a tree trunk, this is much the easiest way to do it.
1d
comment If two non-disjoint subsets are connected, why does their union have to be connected?
Are $X$ and $Y$ sets in any topological space, or are they subsets of $R^n$?
1d
comment Real roots of an nth order polynomial
I believe you are mistaken. See the section on page 5 that begins "concerning the number of iterations...".
1d
comment Is Cantor's diagonal argument dependent on the base used?
Let $d(x, i)$ mean the $i$th binary digit of the base-2 expansion of $x$. Define $s_P$ so that $d(s_P, P(i)) = 1- d(s_i, P(i))$, where $P$ is any permutation of the positive integers. Then $s_P$ is different from every $s_i$ because for each $i$, $s_P$ is different from $s_i$ in the $P(i)$th position. This constructs a large family of numbers not on the list.
1d
comment Real roots of an nth order polynomial
Mark McClure's answer here deserves some more upvotes. He beautifully summarizes a recent paper, "How to find all roots of complex polynomials by Newton's method", by Hubbard, Schliecher, and Sutherland, that does just what you want: "Not only is there a method but, due to the stability of the fixed points under iteration of the Newton's method function, there is a very good method."
2d
comment Is there any solution to this problem?
There is a solution.
2d
comment Is the square root of 2 really an irrational number?
In Mathematics and its History, Stillwell suggests that this crisis was not resolved until the invention of the real numbers starting in the 1600s.
Dec
14
comment Why is [b,a] the multiplicative inverse in the field of fraction of an integral domain?
I don't understand what your question is. The title says you want to know why $[a,b]$ and $[b,a]$ are inverses, but you seem to have shown that $[a,b][b,a]\sim [1,1]$, so it's not clear what you want to know.