11,159 reputation
11230
bio website math.brown.edu/~lubinj
location Minnesota
age 77
visits member for 2 years, 6 months
seen 4 hours ago

Aged mathematician


Apr
9
answered Do spherical triangles with the same base and altitude have the same height?
Apr
9
comment Do spherical triangles with the same base and altitude have the same height?
Are you aware that the area of a triangle is proportional to the sum of the angles, minus $\pi$ (or $180^\circ$ if you’re using degrees)? I don’t know what you mean by “skewing a triangle will no longer have...”, since the sides of a triangle are presumed always to be arcs of great circles.
Apr
9
awarded  Nice Answer
Apr
9
comment Regarding a Basis for Infinite Dimensional Vector Spaces
By definition, linear combinations are finite. When one says $B$ is a basis for the vector space $V$, one means that every element of $V$ may be written as a finite linear combination of elements of $B$, and in only one way.
Apr
7
comment Given a function $f(x,y)$, can two different level curves of $f$ intersect? Why or why not?
This answer explains very nicely why OP’s “example” might lead to confusion.
Apr
7
comment Prove that the set of all roots of a continuous function is closed
I would have said that the inverse image of a closed set is closed, but this answer still is best.
Apr
7
comment $F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.
@DilipSarwate, right you are! I was perhaps too concentrated on my own method.
Apr
7
answered $F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.
Apr
7
comment $F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.
It’s a result in basic group theory that in a finite group, the order of an element divides the order of the group. Since $2^m-1$ is odd, it is not divisible by $2$. So there are no elements of order $2$.
Apr
6
comment An isosceles triangle with a measure of the sides abc of $5,5$ and $4$.
You should say how much trigonometry you already know, and what you’ve tried. Nobody wants to hand you the answer without knowing how far you’ve gotten so far.
Apr
6
comment Expand a function in Maclaurin's series
There are many ways of doing this, but your method is the most efficient, I suspect. You set $\alpha=-1/3$ and replaced $x$ by $x^2$, right? And then increased each $x$-exponent by $1$?
Apr
5
comment Connections between prime numbers and geometry
I think this is the best example of a purely geometric question that’s answered with arithmetic information.
Apr
5
answered Connections between prime numbers and geometry
Apr
5
comment Continuous bijective function between $\Bbb{R}$ and $[0,1]$?
To be clear, you’re not asking for the inverse function to be continuous, right?
Apr
5
comment Is (x,0) the only solution?
Sorry, I was on a plane for a few hours. As you have stated the problem, the “square” issue is beside the point. Clear your mind of cases when $n$ is a square and answer the question fully: which numbers are the difference of two squares? If you’re having difficulty with this, write down a list of the ten first squares, and see which numbers are the difference of two things on your list. I keep harping on this, I know, but knowledge proceeds from examples. Your problem is that you haven’t been thinking of examples. (AND MAKE SURE YOUR LIST OF SQUARES INCLUDES ZERO.)
Apr
4
comment Is (x,0) the only solution?
But the problem itself is asking which numbers are difference of two squares, right?
Apr
4
comment Is (x,0) the only solution?
Nothing in the problem seems to ask for $n$ to be a square.
Apr
4
comment Invalid subtraction when solving system of equations?
Check them all! The job is not done until you do. Sometimes square-rooting will give you false solutions...
Apr
4
comment Invalid subtraction when solving system of equations?
Here in the US, we don’t make it clear to students what they’re doing when they “solve” an equation. That’s really the uniqueness part of the story: here, if there is a solution, it’s either $(1/2,1/2))$ or $(0,0)$. But the existence part is what we call “checking”: to see whether the possibilities you found really do satisfy the equation(s). Here, OP has done the checking and found that one of the apparent possibilities is in fact not a solution. In teaching, we haven’t been stressing strongly enough that checking is an essential part of the process.
Apr
3
comment Quotient field of a localization
If by “quotient field” you mean the total ring of fractions, it is true, and you show it simply by writing down what the elements of each turn out to be.