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1500 questions
94
votes
5 answers
Root Calculation by Hand
Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please? Well, when I use a Casio scientific calculator, I get this answer: $105^{1/5}\approx "…
Kerim Atasoy
- 859
94
votes
9 answers
Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?
Choose a random number between $0$ and $1$ and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds $1$. What's the expected value of the number of random numbers needed…
user25329
- 1,147
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94
votes
4 answers
How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?
How can one prove this identity? $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$ There is a formula for $\zeta$ values at even integers , but it involves Bernoulli numbers; simply plugging it in…
E.H.E
- 23,660
94
votes
27 answers
Easy example why complex numbers are cool
I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly important in a daily day sense. I.e. complex…
ano
- 1,794
93
votes
6 answers
Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$?
Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbb{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbb{Q}\}$$ $$\mathbb{Q}(\sqrt{2}+\sqrt{3}) = \lbrace a+b(\sqrt{2}+\sqrt{3}) \mid a,b \in…
Tashi
- 1,641
93
votes
8 answers
Difference between continuity and uniform continuity
I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book defines continuity as: Definition 4.3.1. A…
user124005
- 943
93
votes
12 answers
Do groups, rings and fields have practical applications in CS? If so, what are some?
This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be applied (directly or indirectly) or has given me Aha!…
PhD
- 2,881
93
votes
6 answers
Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix
The rotation matrix $$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation…
Alf
- 2,667
93
votes
3 answers
Meaning of “arg min”
Would someone be so kind to explain this to me: $$\pi_nk=\left\{\begin{array}{cl}1&\textrm{if }k=\arg\min_j\left\Vert\mathbf x_n-\mu_j\right\Vert^2\\0&\textrm{otherwise}\end{array}\right..$$ Especially the $\arg\min$ part. (It's from the $k$-means…
Olivier_s_j
- 1,103
93
votes
9 answers
Learning mathematics as if an absolute beginner?
I dread mathematics, and I believe it's because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the epitome of evil. She would take any opportunity to…
Kirva
- 1,039
93
votes
15 answers
What are the practical applications of the Taylor Series?
I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: What are the practical applications of the Taylor…
Rivasa
- 1,605
93
votes
6 answers
Contest problem: Show $\sum_{n = 1}^\infty \frac{n^2a_n}{(a_1+\cdots+a_n)^2}<\infty$ s.t., $a_i>0$, $\sum_{n = 1}^\infty \frac{1}{a_n}<\infty$
The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a sequence of positive real numbers and the series $\displaystyle\sum_{n = 1}^\infty \frac{1}{a_n}$ converges. Show that…
Potato
- 41,689
93
votes
11 answers
Why does Friedberg say that the role of the determinant is less central than in former times?
I am taking a proof-based introductory course to Linear Algebra as an undergrad student of Mathematics and Computer Science. The author of my textbook (Friedberg's Linear Algebra , 4th Edition) says in the introduction to Chapter 4: The determinant,…
dacabdi
- 1,228
93
votes
10 answers
Is $0! = 1$ because there is only one way to do nothing?
The proof for $0!=1$ was already asked at here . My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or just because of the way it's defined.
Ali Abbasinasab
- 1,793
93
votes
4 answers
Why is the Möbius strip not orientable?
I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ…
Richard G
- 4,043