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1500 questions
76
votes
8 answers
Why does this "miracle method" for matrix inversion work?
Recently, I answered this question about matrix invertibility using a solution technique I called a " miracle method ." The question and answer are reproduced below: Problem: Let $A$ be a matrix satisfying $A^3 = 2I$ . Show that $B = A^2 - 2A + 2I$…
David Zhang
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76
votes
10 answers
Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?
I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, could one say that $5+2i> 3$ because the real part of…
Juanma Eloy
- 1,465
76
votes
11 answers
Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)
Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$…
user116
76
votes
14 answers
How to stop forgetting proofs - for a first course in Real Analysis?
I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read the first line of the proof and then try to…
MAS
- 1,971
76
votes
1 answer
Properly discontinuous action: equivalent definitions
Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is…
Pedro
- 7,006
75
votes
7 answers
What’s the difference between analytical and numerical approaches to problems?
I don't have much (good) math education beyond some basic university-level calculus. What do "analytical" and "numerical" mean? How are they different?
jbrennan
- 869
75
votes
2 answers
Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}}$ be a non-negative bounded measurable…
user54992
75
votes
6 answers
Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$
I am having trouble proving the statement: Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon > 0$ , the intersection of $S$ and $(0, \epsilon)$ is nonempty.
user11135
- 863
75
votes
6 answers
Similar matrices and field extensions
Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$? Any help ...…
Melesia
- 751
75
votes
2 answers
What is the difference between projected gradient descent and ordinary gradient descent?
I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of projected gradient descent? What does that projection…
erogol
- 1,177
75
votes
4 answers
How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$
Consider the following expression: (a - b) mod N Which of the following is equivalent to the above expression? 1) ((a mod N) + (-b mod N)) mod N 2) ((a mod N) - (b mod N)) mod N Also, how is (-b mod N) calculated, i.e., how is the mod of a negative…
J.P.
- 915
75
votes
2 answers
coincidental (?) patterns in logs of repeating decimals, e.g. $\ln(2/3)$ vs. $\ln(0.6666666)$
I was playing with logarithms under an arbitrary-precision calculator, and got some odd results. I happened to have the precision set to 20 places, and these are the initial results I got: ln 0.6666666 = -0.40546520810816938198 ln 2/3 =…
Steve Summit
- 769
75
votes
4 answers
Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions
Let $f_n(x)$ be recursively defined as $$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$ i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$: $$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{x+1}},\ \ \…
Vladimir Reshetnikov
- 32,880
75
votes
3 answers
Geometry question about a six-pack of beer
On a hot summer day like today, I like to put a six-pack of beer in my cooler and enjoy some cold ones outdoors. My cooler is in the shape of a cylinder. When I place the six-pack in the cooler against the wall, with three beer cans touching the…
Dan
- 41,295
75
votes
8 answers
How can one prove that $e<\pi$?
This question is inspired by another one , asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (expectedly) downvoted, though in my opinion it…
Start wearing purple
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