qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,360,694 | <p><span class="math-container">$U(n)$</span> is the collection of positive integers which are coprime to n forms a group under multiplication modulo n.</p>
<p>What is the order of the element 250 in <span class="math-container">$U(641)$</span>?</p>
<p>My attempt:
Here 641 is a prime number.
So <span class="math-cont... | Dietrich Burde | 83,966 | <p>By Lagrange the order of an element divides the order of the group. Since we have <span class="math-container">$640=2^7\cdot 5$</span>, I tried powers of <span class="math-container">$2$</span>. Then we see immediately that <span class="math-container">$250^{16}=1$</span> in <span class="math-container">$U(641)$</sp... |
3,360,694 | <p><span class="math-container">$U(n)$</span> is the collection of positive integers which are coprime to n forms a group under multiplication modulo n.</p>
<p>What is the order of the element 250 in <span class="math-container">$U(641)$</span>?</p>
<p>My attempt:
Here 641 is a prime number.
So <span class="math-cont... | Mark Bennet | 2,906 | <p>We have <span class="math-container">$641=625+16$</span> so that <span class="math-container">$5^4\equiv -2^4$</span> modulo <span class="math-container">$641$</span> </p>
<p>and also <span class="math-container">$5\times 128\equiv -1$</span> so that <span class="math-container">$5^4\equiv 5\times 2^{11}$</span> an... |
85,126 | <blockquote>
<p>Show that any sequence of positive numbers $(a_n)$ satisfying $$0< \frac{a_{n+1}}{a_n} \leq 1+ \frac{1}{n^2}$$
must converge.</p>
</blockquote>
<p>I have tried taking the limit of the inequality which yields that $0 \leq \lim \frac{a_{n+1}}{a_n} \leq 1$. If $\lim \frac{a_{n+1}}{a_n} \lt 1$, the... | GEdgar | 442 | <p>Why not reduce to $y^{b/a}+y=1$, then you have only a one-parameter family to solve. </p>
<p><img src="https://i.stack.imgur.com/8J2Dj.jpg" alt="graph"></p>
<p>The solution for $y^r+y=1$, expanded in a Taylor series near $r=1$ is<br>
$$
\frac{1}{2} + \frac{\operatorname{ln} (2)}{4}(r - 1) - \frac{\operatorname{ln}... |
3,416,600 | <p>Show that <span class="math-container">$|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$</span> where <span class="math-container">$a,b,c\in\mathbb{R}$</span></p>
<p>I'd like to get an hint on how to get started. What I thought to do so far is dividing to cases to get rid of the absolute value. <span class="math-containe... | David Diaz | 431,789 | <p>For all <span class="math-container">$a,b,c \in \mathbb{R}$</span>,
<span class="math-container">\begin{align}
0 &\leq (b-c)^2\\
2bc &\leq b^2 + c^2\\
2a^2bc &\leq a^2b^2 + a^2c^2\\
a^4 + 2a^2bc + b^2c^2 &\leq a^4 + a^2b^2 + a^2c^2 + b^2c^2\\
a^2 + bc&\leq \sqrt{a^4 + a^2b^2 + a^2c^2 + b^2c^2} \l... |
79,726 | <p>Let $R$ be a commutative ring with unity. Let $M$ be a free (unital) $R$-module.</p>
<p>Define a <em>basis</em> of $M$ as a generating, linearly independent set.</p>
<p>Define the <em>rank</em> of $M$ as the cardinality of a basis of $M$ (as we know commutative rings have IBN, so this is well defined).</p>
<p>A <... | Georges Elencwajg | 3,217 | <p>Yes, a generating set of minimal cardinality must have cardinality $r=rank_R(M)$.<br>
It suffices to show that for <em>any</em> generating set of $M$ with $s$ elements, we have $s\geq r$ . </p>
<p>Assume that $M=R^r$.<br>
Our generating set gives rise to a surjective $R$-module morphism $R^s\to R^r\to 0 \q... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Daniel Moskovich | 2,051 | <p>The <a href="http://en.wikipedia.org/wiki/Four_color_theorem">Four Colour Theorem</a> might perhaps be a canonical example of a very hard proof of a major result which has improved, but is still very hard- no human-comprehensible proof exists, as far as I know, and all known proofs require computer computations.</p>... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Victor Protsak | 5,740 | <p><a href="https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem">Kolmogorov-Arnold-Moser</a> (or <strong>KAM</strong>) theorem. </p>
<p><strong>KAM theory</strong> gives conditions for persistence of invariant tori under small perturbations of a Liouville-integrable Hamiltonian system. It is ... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | ntc2 | 36,970 | <p>Look for theorems that have been, or are currently, the subject of
major formalization efforts!</p>
<p>The two highest-rated answers as I write this [<a href="https://mathoverflow.net/a/152418/36970">1</a>,<a href="https://mathoverflow.net/a/152412/36970">2</a>] -- concerning
the Four-Color and Feit-Thompson theore... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | user160393 | 62,818 | <p><a href="https://link.springer.com/content/pdf/10.1007/BF01202354.pdf" rel="nofollow noreferrer">Hadwiger's conjecture for <span class="math-container">$K_6$</span>-free graphs</a>.</p>
<p>This paper shows the equivalence of Hadwiger's conjecture for graphs with no <span class="math-container">$K_6$</span> minor and... |
27,490 | <h2>Motivation</h2>
<p>The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nic... | Ronnie Brown | 19,949 | <p>Thanks Andrew for the nice comments! </p>
<p>In relation to the comment on G-spaces by Donu, I should point out that Chapter 11 of "Topology and groupoids" is on "Orbit spaces, orbit groupoids". But I doublt many topologists are aware of the latter concept! </p>
<p>My new jointly authored book `Nonabelian algebra... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | Vidyanshu Mishra | 363,566 | <p>Your reasoning is appreciable, but the problem is that it is wrong.</p>
<p>While tempering with sides and area, you forgot about the angles. In this case , it is just the matter of sines and cosines. Let's see how:</p>
<p>Suppose $\angle ADC=\theta$ and $\angle BCD=180-\theta$. </p>
<p>On using trigonometric for... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | hmakholm left over Monica | 14,366 | <p>If you know two sides and the area of a triangle, there will generally be two <em>different</em> lengths for the third side that gives you that area.</p>
<p>Consider, for example: If the two known sides are $3$ and $4$, then the third side is somewhere between $1$ and $7$. A third side of length $1$ gives area $0$,... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | G Cab | 317,234 | <p>The answer by <em>Narasimham</em> is fully right.<br>
To visualize it better, consider to flip $\triangle BCD$ around the bisector of the common segment $DC$. </p>
<p><a href="https://i.stack.imgur.com/Tn7ks.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Tn7ks.png" alt="Quadr_Tr_t"></a></p>
<p... |
1,674,676 | <p>Let $p$ be an odd prime number. I want to show that $\mathbb{F}_{p^2}$ has a primitive 8th root of unity $\zeta$. </p>
<ul>
<li>I know that $\zeta^8 = 1$. So my idea is to define $f = X^8 - 1$ such that $\zeta$ is a root of $f$. But this is for a field extension of degree 8 and $p^2$ is at least 9.</li>
</ul>
<p>a... | Matt B | 111,938 | <p>Consider the unit group $\mathbb{F}_{p^2}^{\times}$, which has order $p^2-1$. </p>
<p>Finding an eighth root of unity is equivalent to finding an element which has order $(p^2-1)/8$ in this group. Since the unit group of finite field is known to be cyclic, this happens if and only if $p^2-1$ is a multiple of $8$, i... |
1,325,563 | <p>Can someone show me:</p>
<blockquote>
<p>If $x$ is a real number, then $\cos^2(x)+\sin^2(x)= 1$.</p>
<p>Is it true that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable?</p>
</blockquote>
<p>Note :look [this ] in wolfram alpha showed that's true !!!!</p>
<p>Thank you for your help</p>
| milka | 539,462 | <p>by definition (In complex Analysis):</p>
<p>cosz= ((e^iz)+(e^-iz))/2</p>
<p>sinz= ((e^iz)-(e^-iz))/2</p>
<p>L.H.S=(cosz+isinz)(cosz-sinz)</p>
<p>substitute with definitions above and manipulate algebraically.
You should get this at the end:
(e^iz)(e^-iz)= e^0=1= R.H.S</p>
|
668,664 | <p>Solve $\dfrac{\partial u}{\partial t}+u\dfrac{\partial u}{\partial x}=x$ subject to the initial condition $u(x,0)=f(x)$.</p>
<p>I let $\dfrac{dt}{ds}=1$ , $\dfrac{dx}{ds}=u$ , $\dfrac{du}{ds}=x$ and the initial conditions become: $t=0$ , $x=\xi$ and $u=f(\xi)$ when $s=0$ .</p>
<p>I believe this leads to $t=s$ , bu... | Pragabhava | 19,532 | <p>Your PDE is quasilinear, meaning it migh not have a <em>classic</em> solution for all time $t$. That said, we know that the quasilinear equation
$$
a\big(x,t,u(x,t)\big) u_x(x,t) + b\big(x,t,u(x,t)\big)u_t(x,t) = c\big(x,t,u(x,t)\big)
$$
where $a,\,b,\,c \in C^1$ with data $\mathcal{C}(\xi) = \big(x(\xi), t(\xi), ... |
2,292,656 | <p>Let $L/K$ be a degree $n$ extension of fields, where $K$ has discrete valuation $v$, which can be prolonged to the discrete valuations $w_i$ on $L$. We can therefore define the completion of $K$ w.r.t. $v$ to be $\hat K$, and the completion of $L$ w.r.t. $w_i$ to be $\hat L_i$, then in Theorem II.3.1 of Serre's <em>... | sharding4 | 254,075 | <p>The basic idea is to take $L$ to be $K[\alpha]\cong K[x]/(f(x))$ where $f(x)$ is the minimum polynomial of $\alpha$. Then $L\otimes_K\hat K \cong K[x]/(f(x))\otimes_K\hat K \cong \hat K[x]/(f(x))$ and factor $f(x)$ in $\hat K[x]$.</p>
|
384,318 | <p>Let $X$ be a topological space and let $A,B\subseteq X$ be closed in $X$ such that $A\cap B$ and $A\cup B$ are connected (in subspace topology) show that $A,B$ are connected (in subspace topology).</p>
<p>I would appreciate a hint towards the solution :)</p>
| FiveLemon | 76,591 | <p>Suppose $A$ were disconnected. Then $A$ is the disjoint union of $A'$ and $A''$ non-empty closed subsets of $A$. </p>
<p>If $A' \cap B$ and $A'' \cap B$ are both non-empty then $A\cap B$ is disconnected -- a contradiction.</p>
<p>If $A' \cap B$ is empty then $A'$ and $A'' \cup B$ is a partition of $A \cup B$. S... |
1,079,995 | <p>I can't understand how: $$ \frac {2\times{^nC_2}}{5} $$</p>
<p>Equals:</p>
<p>$$ 2\times \frac {^nC_2}{5} $$</p>
<p>If we forget the combination and replace it with a $10$, the result is clearly different. $1$ in the first example and and $0.5$ in the second.</p>
| JEET TRIVEDI | 115,676 | <p>$$\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b \times d}$$
So, taking your example
$$\frac{2\times10}{5\times 1}=\dfrac{2}{5}\times\dfrac{10}{1}=\dfrac{2}{1}\times\frac{10}{5}=\dfrac{10}{5}\times 2=\dfrac{2}{5}\times 10$$
As multiplication is commutative.</p>
|
1,463,567 | <p>We have the following theorem </p>
<p>If |G| = 60 and G has more than one Sylow-5 subgroup, then G is simple.</p>
<p>Since order of the rigid motion of the dodecahedron group is 60, so all we have to do is to show that it has more than one sylow-5 subgroup, but I don't know how to do this as I don't know the eleme... | bof | 111,012 | <p>The "topologist's sine curve" is a <em>nice</em> example. Wouldn't you rather see a nonconstructive horror?</p>
<p>Assuming the axiom of choice, there is a <a href="https://en.wikipedia.org/wiki/Bernstein_set" rel="nofollow">"Bernstein decomposition"</a> of the plane, i.e., $\mathbb R^2$ is the union of two disjoin... |
4,646,715 | <blockquote>
<p>Let <span class="math-container">$A\in \operatorname{Mat}_{2\times 2}(\Bbb{R})$</span> with eigenvalues <span class="math-container">$\lambda\in (1,\infty)$</span> and <span class="math-container">$\mu\in (0,1)$</span>. Define <span class="math-container">$$T:S^1\rightarrow S^1;~~x\mapsto \frac{Ax}{\|Ax... | Parcly Taxel | 357,390 | <p>The other two fixed points are <span class="math-container">$-x$</span> and <span class="math-container">$-y$</span>, when <span class="math-container">$x$</span> and <span class="math-container">$y$</span> lie on <span class="math-container">$S^1$</span> (they can always be chosen to lie on <span class="math-contai... |
393,293 | <p>I need an upper bound for
$$\frac{ax}{x-2}$$
I know that $1\leq a< 2$ and $x\geq 0$.</p>
<p>This upper bound can include just $a$ and constant numbers not $x$.</p>
<p>thanks a lot.</p>
| Federica Maggioni | 49,358 | <p>$$\lim_{x\rightarrow 2^+}\frac{ax}{x-2}=+\infty$$</p>
|
393,293 | <p>I need an upper bound for
$$\frac{ax}{x-2}$$
I know that $1\leq a< 2$ and $x\geq 0$.</p>
<p>This upper bound can include just $a$ and constant numbers not $x$.</p>
<p>thanks a lot.</p>
| Inceptio | 63,477 | <p><strong>Hint:</strong></p>
<p>For $2>x \ge 0$, you get a negative value, and for $x>2$ you will see that the value gradually reduces. What happens when $x \to 2$? Consider LHL and RHL.</p>
|
3,001,700 | <p>I am trying to find an <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that solve the equation <span class="math-container">$15x - 16y = 10$</span>, usually in this type of question I would use Euclidean Algorithm to find an <span class="math-container">$x$</span> and <span class=... | Ethan Bolker | 72,858 | <p>In this case you don't really need the full power of the Euclidean algorithm. Since you know
<span class="math-container">$$
16 - 15 = 1
$$</span>
you can just multiply by <span class="math-container">$10$</span> to conclude that
<span class="math-container">$$
16 \times 10 + 15 \times(-10) = 10.
$$</span>
Now you h... |
202,742 | <p>Consider a <a href="http://en.wikipedia.org/wiki/Circular_layout" rel="noreferrer">circular drawing</a> of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The <em>crossing graph</em> for such a drawing is the simple graph whose nodes correspond to the edges ... | Tony Huynh | 2,233 | <p>The answer to 1 is <strong>no</strong>. To see this, note that every edge-crossing graph is a <a href="http://en.wikipedia.org/wiki/String_graph">string graph</a>. A <em>string graph</em> is a graph which is the intersection graph of arbitrary curves in the plane. However, there are graphs which are not even stri... |
202,742 | <p>Consider a <a href="http://en.wikipedia.org/wiki/Circular_layout" rel="noreferrer">circular drawing</a> of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The <em>crossing graph</em> for such a drawing is the simple graph whose nodes correspond to the edges ... | Brendan McKay | 9,025 | <p>Such graphs are called "circle graphs" and if you search on that phrase you will find some literature. For example, some is around page 56 in <a href="http://books.google.com.au/books?hl=en&lr=&id=bAGW1L84hRQC&oi=fnd&pg=PR9&dq=crossing%20chord%20graph&ots=R_jz3vps1w&sig=4AltkQYV2XfPqElucx... |
2,296,256 | <p>I need help how to mathematically interpret an ODE (Newton's second law). I used to the ODE in this form:
$$
m\ddot x(t)=F(t)\tag{1}
$$</p>
<p>However, in another book they wrote:
$$
m\ddot x=F(x,\dot x) \tag{2}
$$
where $F: \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}^n$.</p>
<p><strong>Questions:</stro... | edm | 356,114 | <p>For $x\lt z$, consider any $y$ in-between, i.e. $x\lt y\lt z$, so that $f(x)\lt g(y)\lt f(z)$, i.e. $f$ is strictly increasing. Similarly, $g$ is strictly increasing.</p>
<p>Consider a point $x_0$ at which $f$ is continuous. When a sequence $(x_n)_{n\in\Bbb N}$ of real numbers is decreasing to $x_0$, the sequence $... |
2,150,552 | <p>I'm following a YouTube linear algebra course. (<a href="https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14" rel="nofollow noreferrer">https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14</a>)<br>
In part 9 there's the ... | egreg | 62,967 | <p>Let $R$ be a total order relation on the set $X$ (for instance the usual ordering $\leq$ on $X=\mathbb{N}$), so antisymmetric. Let
$$
S=R^{\mathrm{op}}=\{(x,y):(y,x)\in R\}
$$
which is obviously antisymmetric as well.</p>
<p>Then $R\cup S=X\times X$ which is only antisymmetric if $X$ has at most one element.</p>
<... |
3,189,303 | <p>What is the point of constant symbols in a language?</p>
<p>For example we take the language of rings <span class="math-container">$(0,1,+,-,\cdot)$</span>.
What is so special about <span class="math-container">$0,1$</span> now? What is the difference between 0 and 1 besides some other element of the ring?</p>
<p>... | Clive Newstead | 19,542 | <p>An <span class="math-container">$L$</span>-structure is not just a set, it is a set <em>together with</em> interpretations of the constant symbols, function symbols and relation symbols in <span class="math-container">$L$</span>. You need to keep track of the interpretations as additional data so that you can do thi... |
3,219,635 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$\Bbb R$</span>, define <span class="math-container">$F(x)=\int_a^bf(x+t)\cos t\,dt,x\in [a,b]$</span>.</p>
<p>How to show <span class="math-container">$F(x)$</span> is differentiable on <span class="math-container">$[a,b]... | fGDu94 | 658,818 | <p>With <span class="math-container">$u=x+t$</span>, we have</p>
<p><span class="math-container">$F(x) = \int_{a+x}^{b+x}f(u)\cos(u-x)du$</span>.</p>
<p>Now use liebniz integral rule</p>
<p><span class="math-container">$F'(x) = f(b+x)\cos(b)-f(a+x)\cos(a)+\int_{a+x}^{b+x}f(u)\sin(u-x)du$</span></p>
|
474,048 | <p>I am stuck with the following problem from a book.</p>
<p>It asks whether or not $f_n \rightarrow f$ converges uniformly on $A$ if for every $[a,b], f_n\rightarrow f$ uniformly on $A\cap [a,b]$.</p>
<p>The statement seems false to me (i.e. not necessarily true) because of this intuition I had:</p>
<p>If $A$ is no... | user71352 | 71,352 | <p>Take $A=\mathbb{R}$ and consider the sequence</p>
<p>$f_{n}(x)=\chi_{(n,n+1)}(x)$.</p>
<p>Consider any $[a,b]$. Then for all but finitely many $n$ we have $f_{n}=0$ on $A\cap[a,b]$. So $f_{n}$ uniformly converges on $A\cap[a,b]$ to $0$. Since $[a,b]$ are arbitrary then this holds in general. But $f_{n}$ does not c... |
3,991,572 | <p>I need to solve the following problem:
<span class="math-container">$\lim_{x\to 3}(x-3) \cot{\pi x}$</span>.
Can anyone give me a hint? I have no idea.</p>
| Rishab Sharma | 864,616 | <p>In this this type of problems just do telescoping now make the second terms in your partial fraction in powers of n+2 that is just divide and multiply by 16 in second term then do telescoping and possible answers are 33/2</p>
|
208,744 | <p>I was asked to show that $\frac{d}{dx}\arccos(\cos{x}), x \in R$ is equal to $\frac{\sin{x}}{|\sin{x}|}$. </p>
<p>What I was able to show is the following:</p>
<p>$\frac{d}{dx}\arccos(\cos(x)) = \frac{\sin(x)}{\sqrt{1 - \cos^2{x}}}$</p>
<p>What justifies equating $\sqrt{1 - \cos^2{x}}$ to $|\sin{x}|$?</p>
<p>I ... | preferred_anon | 27,150 | <p>$\sqrt{1-\cos^{2}(x)}=\sqrt{\sin^{2}(x)}$,
which is $|\sin(x)|$ by definition.</p>
|
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | Carl Mummert | 5,042 | <p>I believe it can be easier, at first, to look at the formal definition of continuity in terms of function application commuting with limits, which says that $\lim_{x \to A} f(x)$ is the same as $f(A)$. The explanation is more clear if also you write $f(A)$ as $f(\lim_{x \to A} x)$. </p>
<p>Thus:</p>
<ul>
<li>$f(A)... |
3,671,223 | <p>First and foremost, I have already gone through the following posts:</p>
<p><a href="https://math.stackexchange.com/questions/2463561/prove-that-for-all-positive-integers-x-and-y-sqrt-xy-leq-fracx-y">Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$</a></p>
<p><a href="https://ma... | Community | -1 | <p>This proof is hinted by the presence of the square root, which one will tend to remove by squaring. As all numbers are positive</p>
<p><span class="math-container">$$\sqrt{xy}\le\frac{x+y}2$$</span> is rewritten</p>
<p><span class="math-container">$$xy\le\frac{x^2+2xy+y^2}4,$$</span></p>
<p>which is also </p>
<p... |
3,131,516 | <p>I would like to know if this differential equation can be transformed into the hypergeometric differential equation</p>
<p><span class="math-container">$ 4 (u-1) u \left((u-1) u \text{$\varphi $1}''(u)+(u-2) \text{$\varphi $1}'(u)\right)+\text{$\varphi $1}(u) \left((u-1) u \omega ^2-u (u+4)+8\right)=0$</span></p>
| MPW | 113,214 | <p>You can <em>always</em> use the quadratic equation to factor a quadratic polynomial <span class="math-container">$Q(x)$</span>.</p>
<p>If the solutions to <span class="math-container">$Q(x)=0$</span> are <span class="math-container">$x=r$</span> and <span class="math-container">$x=s$</span>, then the polynomial can... |
3,720,677 | <p>There are two vertical lines, <span class="math-container">$l_1$</span> from <span class="math-container">$(0,0)$</span> to <span class="math-container">$(0,n)$</span> and <span class="math-container">$l_2$</span> from <span class="math-container">$(m,0)$</span> to <span class="math-container">$(m, n)$</span>.</p>
<... | user | 293,846 | <p>There are
<span class="math-container">$$
\binom{m+j-i}{m}
$$</span>
ways to start at point <span class="math-container">$(0,i)$</span> and finish at point <span class="math-container">$(m,j)$</span>, so that the total number of ways is
<span class="math-container">$$
\sum_{0\le i\le j\le n}\binom{m+j-i}{m}=\sum_{k=... |
3,720,677 | <p>There are two vertical lines, <span class="math-container">$l_1$</span> from <span class="math-container">$(0,0)$</span> to <span class="math-container">$(0,n)$</span> and <span class="math-container">$l_2$</span> from <span class="math-container">$(m,0)$</span> to <span class="math-container">$(m, n)$</span>.</p>
<... | Brian M. Scott | 12,042 | <p>Let <span class="math-container">$P$</span> be the set of lattice paths from <span class="math-container">$\ell_1$</span> to <span class="math-container">$\ell_2$</span> using only steps to the north and steps to the east, and let <span class="math-container">$Q$</span> be the set of such paths from <span class="mat... |
2,397,874 | <p>I am new to modulus and inequalities , I came across this problem:</p>
<p>$ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $</p>
<p>How to find $ x $ ?</p>
| Raffaele | 83,382 | <p>$\left(5-2 \sqrt{6}\right) \left(5+2 \sqrt{6}\right)=1$</p>
<p>So $5-2 \sqrt{6}=\dfrac{1}{5+2 \sqrt{6}}$</p>
<p>substitute $\left(2 \sqrt{6}+5\right)^{x^2-3}=t$ so that $\left(5-2 \sqrt{6}\right)^{x^2-3}=\dfrac{1}{t}$</p>
<p>and solve $t+\dfrac{1}{t}=10$ which gives $t_1= 5-2 \sqrt{6},\;t_2=5+2 \sqrt{6}$</p>
<p>... |
3,163,342 | <p>Find all the ring homomorphisms <span class="math-container">$f$</span> : <span class="math-container">$\mathbb{Z}_6\to\mathbb{Z}_3$</span>.</p>
<p>definition of ring homomorphism:</p>
<p>The function f: R → S is a ring homomorphism if:</p>
<p>1) <span class="math-container">$f(1)$</span> = <span class="math-cont... | Alessio Del Vigna | 639,470 | <p>When you multiply both sides by <span class="math-container">$3$</span>, you made a mistake in the RHS.</p>
|
204,150 | <p>If I had a list of let's say 20 elements, how could I split it into two separate lists that contain every other 5 elements of the initial list?</p>
<p>For example:</p>
<pre><code>list={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
function[list]
(*
{1,2,3,4,5,11,12,13,14,15}
{6,7,8,9,10,16,17,18,19,20}
*)
</... | kglr | 125 | <pre><code>ClearAll[f1, f2]
f1[lst_, k_] := Join @@ Partition[lst[[# ;;]], k, 2 k, 1, {}] & /@ {1, k + 1}
{list1, list2} = f1[list, 5]
</code></pre>
<blockquote>
<p>{{1, 2, 3, 4, 5, 11, 12, 13, 14, 15},<br>
{6, 7, 8, 9, 10, 16, 17, 18, 19, 20}}</p>
</blockquote>
<p>An alternative way using the (still undocum... |
1,761,668 | <p>Wikipedia says about logical consequence:</p>
<blockquote>
<p>A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.</p>
</blockquote>
<p>But if φ and ψ are both true under some interpretations, then... | Olivier Oloa | 118,798 | <p>We have the following inverse image
$$
M^c=\|\cdot\|^{-1}\left([1,\infty) \right)
$$ the subset $[1,+\infty)$ is closed in $\mathbb{R}$ thus the subset $M^c$ is closed in the metric space $\mathbb{R}^{n+1}$ ($\|\cdot\|$ is a continuous function over $\mathbb{R}$), that is $M=\left(M^c \right)^c$ is an open subset i... |
1,761,668 | <p>Wikipedia says about logical consequence:</p>
<blockquote>
<p>A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.</p>
</blockquote>
<p>But if φ and ψ are both true under some interpretations, then... | Jean Marie | 305,862 | <p>I propose a proof in the spirit of what you have attempted. Let us define</p>
<p>$$g((x_1, x_2, \dots, x_{n+1})) := x_1^2 + \dots + x_{n+1}^2 -1 $$</p>
<p>then </p>
<p>$$M=g^{-1}((-1,1))$$</p>
<p>Thus $M$ is the reciprocal set of the open set $\mathbb{R}$, therefore an open set of $\mathbb{R}^n$.</p>
|
130,502 | <p>I obtained a numerical solution from the following code with <code>NDSolve</code></p>
<pre><code>L = 20;
tmax = 27;
\[Sigma] = 2;
myfun = First[h /. NDSolve[{D[h[x, y, t], t] +
Div[h[x, y, t]^3*Grad[Laplacian[h[x, y, t], {x, y}], {x, y}], {x, y}] +
Div[h[x, y, t]^3*Grad[h[x, y, t], {x, y}], {x, y}] == 0,
h[x, y, ... | ubpdqn | 1,997 | <p>I only post this as a way (without distortion or dealing with multiple scales)to illustrate the initial Gaussian (flat relative to final range) with <code>MeshFunctions</code> and using <code>ColorFunction</code>. I have voted for Nasser's answer.</p>
<pre><code>fun[t_] := Legended[Show[
Plot3D[myfun[x, y, t], {... |
2,480,528 | <blockquote>
<p>Find a formula for $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)$ then prove it. </p>
</blockquote>
<p>I assumed that $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)=\frac{2n}{2n-1}$ after doing a few cases from above then I tried to prove it with induction would this be a fair approach or ... | Community | -1 | <p>Hint: Let $z=a+ib$, then $|\frac{1+z}{1-i\overline{z}}|=\frac{|1+z|}{|1-i\overline{z}|}=\frac{|(1+a)+ib|}{|(1-b)-ia|}=\frac{\sqrt{(1+a)^2+b^2}}{\sqrt{(1-b)^2+a^2}}=\frac{\sqrt{1+a^2+2a+b^2}}{\sqrt{1+b^2-2b+a^2}}=1$ </p>
<p>Now this equality is true if and only if $a=-b$ which gives us that if $z=a+ib$ then $iz=ai-b... |
3,452,493 | <p>I remember hearing / reading about a scenario during WW2 where the US / Western Powers were thinking about attacking Japan via an overland route from India. The problem involved how to leapfrog the supplies from India over to China and there begin the fight with the Japanese. There’s a logic to it as the distance i... | Hugo | 562,826 | <p>By AM-GM with <span class="math-container">$(n-1)$</span> ones and one <span class="math-container">$a$</span>,
<span class="math-container">$$ \sqrt[n]{a} \leq \frac{1}{n} [(n-1)+a] \leq 1 + \frac{a}{n}. $$</span></p>
<p>Then, by letting <span class="math-container">$a' = 1/a$</span> you get the other inequality!<... |
3,382,241 | <p>I am trying to find the smallest <span class="math-container">$n \in \mathbb{N}\setminus \{ 0 \}$</span>, such that <span class="math-container">$n = 2 x^2 = 3y^3 = 5 z^5$</span>, for <span class="math-container">$x,y,z \in \mathbb{Z}$</span>. Is there a way to prove this by the Chinese Remainder Theorem?</p>
| donguri | 670,114 | <p>Since all <span class="math-container">$25$</span> balls are indistinguishable, we can put a ball in each box. There are <span class="math-container">$20$</span> remaining balls and <span class="math-container">$5$</span> boxes.</p>
<p>From here, you calculate using "stars and bars" <span class="math-container">$$\... |
2,322,294 | <p>I am trying to follow K.P. Hart's course <a href="http://fa.its.tudelft.nl/~hart/37/onderwijs/old-courses/settop" rel="nofollow noreferrer">Set-theoretic methods in general topology</a>. In <a href="http://fa.its.tudelft.nl/~hart/37/onderwijs/old-courses/settop/rudin.pdf" rel="nofollow noreferrer">Chapter 6</a>, Rud... | Mike V.D.C. | 114,534 | <p>I dont know how to <strong>remove</strong> answer or mark it as non-answer (without deleting it)!</p>
<p>The $\mathbb{Z}(X)$ is localization of $\mathbb{Z}[X]$, is absolutely correct. However the underlying multiplicative set is not $\{X,X^2,\ldots\}$, but $\mathbb{Z}$. </p>
<p>When you invert all non-costant poly... |
208,830 | <p>I have the plot of two surfaces given by</p>
<pre><code> pN = ParametricPlot3D[{0,u,v},{u,-0.25,0.25},{v,-0.5,0.5},\
Mesh->None,PlotStyle->Directive[Gray,Opacity[0.4]]
];
pS = ParametricPlot3D[{u,v^2,v},{u,-0.25,0.25},{v,-0.5,0.5},\
Mesh->None... | Sjoerd Smit | 43,522 | <p>The easiest way is, I think, to use the definition of a Bézier curve to define a <code>ParametricRegion</code> and then use <code>RegionDistance</code> to find out how well the curve approximates the points. Here's my suggestion. Define the points:</p>
<pre><code>n = 5;
p = Table[{Cos[\[CurlyPhi]], Sin[\[CurlyPhi]]... |
208,830 | <p>I have the plot of two surfaces given by</p>
<pre><code> pN = ParametricPlot3D[{0,u,v},{u,-0.25,0.25},{v,-0.5,0.5},\
Mesh->None,PlotStyle->Directive[Gray,Opacity[0.4]]
];
pS = ParametricPlot3D[{u,v^2,v},{u,-0.25,0.25},{v,-0.5,0.5},\
Mesh->None... | xzczd | 1,871 | <p>There're 2 issues here:</p>
<ol>
<li><p>The evaluation order should be properly controlled.</p></li>
<li><p>The argument of <code>BezierFunction</code> should be between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>, so we need to add constraints to <code>NMinimize</code>.</p><... |
2,800,015 | <p>Prove $p(x)=\frac{6}{(\pi x)^2}$ for $x=1,2,...$where $p$ is a probability function. and $E[X]$ doesn't exists.</p>
<p><b> My work </b></p>
<p>I know $\sum _{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$</p>
<p>Moreover,</p>
<p>$p(1)=\frac{6}{\pi^2}$<br>
$p(2)=\frac{6}{\pi^24}$<br>
$p(3)=\frac{6}{\pi^29}$<br>
$p(4)... | Arnaud Mortier | 480,423 | <p>To have a probability mass function of a discrete RV what you need is not $\lim_{x\to \infty}p(x)=1$ but rather $\sum_{x\in\Bbb R}p(x)=1$, where the sum over real numbers is well defined because $p(x)$ is non-zero only countably many times.</p>
<p>Then for the expectation use a comparison with the harmonic series.<... |
1,842,826 | <blockquote>
<p>Explain why the columns of a $3 \times 4$ matrix are linearly dependent</p>
</blockquote>
<p>I also am curious what people are talking about when they say "rank"? We haven't touched anything with the word rank in our linear algebra class.</p>
<p>Here is what I've came up with as a solution, will th... | Doug M | 317,162 | <p>Why are the colums of</p>
<p>$\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix}$
linearly dependent?</p>
<p>Because there exists non-zero $x$ such that </p>
<p>$\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix} x = 0$</p>
<p>i.e. </... |
306,588 | <p>I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is a request for references to where it has already been done.</p>
<p><b>Ordinary Mobius Inversion</b> Let $P$ be a fi... | David E Speyer | 297 | <p>Sami Assaf and I prove this in section 5 of our paper <a href="https://arxiv.org/abs/1809.10125" rel="nofollow noreferrer">Specht modules decompose as alternating sums of restrictions of Schur modules</a>. It is surprising that we couldn't find a reference!</p>
|
246,589 | <p>Solve the boundary value problem
$$\begin{cases} \displaystyle \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2} \\ \ \\u(0,t) = 10 \\ u(3,t) = 40 \\ u(x, 0) = 25 \end{cases}$$</p>
| doraemonpaul | 30,938 | <p>Let $u(x,t)=X(x)T(t)$ ,</p>
<p>Then $X(x)T'(t)=2X''(x)T(t)$</p>
<p>$\dfrac{T'(t)}{2T(t)}=\dfrac{X''(x)}{X(x)}=-\dfrac{n^2\pi^2}{9}$</p>
<p>$\begin{cases}\dfrac{T'(t)}{T(t)}=-\dfrac{2n^2\pi^2}{9}\\X''(x)+\dfrac{n^2\pi^2}{9}X(x)=0\end{cases}$</p>
<p>$\begin{cases}T(t)=c_3(n)e^{-\frac{2n^2\pi^2t}{9}}\\X(x)=\begin{c... |
3,542,885 | <p>Let <span class="math-container">$P(x, y) = ax^2 + bxy + cy^2 + dx + ey + h$</span> and suppose <span class="math-container">$b^2 - 4ac > 0.$</span></p>
<p>I know that we can re-write <span class="math-container">$P(x, y)$</span> as a polynomial of <span class="math-container">$x:$</span> <span class="math-conta... | Narasimham | 95,860 | <p><a href="https://i.stack.imgur.com/njdOd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/njdOd.png" alt="enter image description here"></a></p>
<p>From similar triangles <span class="math-container">$ ADE,AOB $</span></p>
<p><span class="math-container">$$ \tan A = \frac{x_1}{y_1}, \cot A =\fra... |
2,099,516 | <p>For independent Gamma random variables $G_1, G_2 \sim \Gamma(n,1)$, $\frac{G_1}{G_1+G_2}$ is independent of $G_1+G_2$. Does this imply that $G_1+G_2$ is independent of $G_1-G_2$? Thanks!</p>
| madprob | 34,305 | <p>Let $X_1=G_1+G_2$ and $X_2=G_1-G_2$.
\begin{align*}
M_{X_1,X_2}(t_1,t_2) &= E[\exp(t_1(G_1+G_2)+t_2(G_1-G_2))] \\
&= E[\exp((t_1+t_2)G_1+(t_1-t_2)G_2)] \\
&= E[\exp((t_1+t_2)G_1)]E[\exp((t_1-t_2)G_2)] \\
&= (1-(t_1+t_2))^{-n}(1-(t_1-t_2))... |
2,403,608 | <p>I was asked to solve for the <span class="math-container">$\theta$</span> shown in the figure below.</p>
<p><a href="https://i.stack.imgur.com/3Yxqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3Yxqv.png" alt="enter image description here" /></a></p>
<p>My work:</p>
<p>The <span class="math-con... | trying | 309,917 | <p>This answer makes use of analytic geometry, as an alternative to other answers.</p>
<p>Setting a cartesian coordinate system with origin in $A$ and $x$-axis parallel to $AB$ and $y$-axis parallel to $AH$ you have:</p>
<p>$y_F=\frac{\sqrt{3}}{2}X$</p>
<p>$\tan\frac{\theta}{2}=\frac{X/2}{y_H-y_F}=\frac{1}{2}\frac{1... |
1,803,589 | <p>I'm stuck at this. How is RHS rearranged? Is it a change of index?</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n}
- \sum_{n=1}^{N} \frac{1}{n}
= \sum_{n=N+1}^{2N} \frac{1}{n}
$$</p>
<p>I'm stuck here:</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{2N}
$$
$$
\sum_{n=1}^{N} \frac... | SchrodingersCat | 278,967 | <p>This is what it should be like:</p>
<p>$$\sum_{n=1}^{2N} \frac{1}{n}- \sum_{n=1}^{N} \frac{1}{n}$$
$$= \left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{2N}\right) -\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{N}\right)$$
$$=\frac{1}{N+1}+\frac{1}{N+2}+\frac{1}{N+3}+\dots+ \frac{1}{2N}$$
$$=\su... |
1,803,589 | <p>I'm stuck at this. How is RHS rearranged? Is it a change of index?</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n}
- \sum_{n=1}^{N} \frac{1}{n}
= \sum_{n=N+1}^{2N} \frac{1}{n}
$$</p>
<p>I'm stuck here:</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{2N}
$$
$$
\sum_{n=1}^{N} \frac... | Eff | 112,061 | <p>You can also simply derive it from the sigma-notation. You have that
$$\sum\limits_{n=1}^{2N}\frac1n = \sum\limits_{n=1}^N\frac1n + \sum\limits_{n=N+1}^{2N}\frac1n, $$
and hence
$$\require{cancel}\sum\limits_{n=1}^{2N}\frac1n - \sum\limits_{n=1}^{N}\frac1n = \left(\cancel{\sum\limits_{n=1}^N\frac1n} + \sum\limits_{... |
65,691 | <p>The question of generalising circle packing to three dimensions was asked in <a href="https://mathoverflow.net/questions/65677/">65677</a>. There is a clear consensus that there is no obvious three dimensional version of circle packing.</p>
<p>However I have seen a comment that circle packing on surfaces and Ricci ... | Roberto Frigerio | 6,206 | <p>The following paper </p>
<p>F. Luo, A combinatorial curvature flow for compact 3-manifolds with boundary,</p>
<p><a href="http://arxiv.org/abs/math/0405295" rel="noreferrer">http://arxiv.org/abs/math/0405295</a>
(now published in electronic research announcements, AMS, Volume 11, Pages 12--20)</p>
<p>provides a c... |
65,691 | <p>The question of generalising circle packing to three dimensions was asked in <a href="https://mathoverflow.net/questions/65677/">65677</a>. There is a clear consensus that there is no obvious three dimensional version of circle packing.</p>
<p>However I have seen a comment that circle packing on surfaces and Ricci ... | Joseph O'Rourke | 6,094 | <p>Not yet mentioned is the interesting definition of Ricci curvature by
<a href="http://www.yann-ollivier.org/" rel="noreferrer">Yann Ollivier</a>, a definition especially suited to discrete spaces, such as graphs.
His definition "can be used to define a notion of 'curvature at a given scale' for metric
spaces." For... |
78,311 | <p>Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product
$$\langle f,g \rangle_{H^1(\mu)} := \int f g\, d\mu + \int \nabla f \cdot \nabla g\, d\mu.... | Community | -1 | <p>Nate, I once needed this result, so I proved it in <a href="http://www.stat.ualberta.ca/people/schmu/preprints/japonica.pdf"><em>Dirichlet forms with polynomial domain</em></a> (Math. Japonica <strong>37</strong> (1992) 1015-1024). There may be better proofs out there, but you could start with this paper. </p>
|
1,292,759 | <blockquote>
<p>Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that
$$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$</p>
</blockquote>
<p>This isn't hard problem. I have already solved it in following way:<br/>
Let $x=\frac1a,y=\frac1b,z=\frac1c$, then $xyz=1$. Now, it is enought to prove that... | xpaul | 66,420 | <p>You can use this way to do. Your inequality $L\ge \frac32$ is equivalent to
$$ 2[x^2(x+y)(x+z)+y^2(x+y)(y+z)+z^2(x+z)(y+z)]\ge 3(x+y)(x+z)(y+z). $$
Let
$$ F(x,y,z)=2[x^2(x+y)(x+z)+y^2(x+y)(y+z)+z^2(x+z)(y+z)]-3(x+y)(x+z)(y+z)-\lambda(xyz-1). $$
Then set
$$ \frac{\partial F}{\partial x}=0, \frac{\partial F}{\partial... |
1,600,054 | <p>The graph of $y=x^x$ looks like this:</p>
<p><a href="https://i.stack.imgur.com/JdbSv.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/JdbSv.gif" alt="Graph of y=x^x."></a></p>
<p>As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$.</p>
<p>... | Brian Fitzpatrick | 56,960 | <p>Let $f(x)=x^x$. Note that $f(x)$ is only defined for $x>0$.</p>
<p>Then
$$
\ln f(x)=x\cdot\ln x\tag{1}
$$
Differentiating (1) gives
$$
\frac{1}{f(x)}f^\prime(x)=x\frac{1}{x}+\ln x=1+\ln x
$$
Note that we have used the chain rule and the product rule.</p>
<p>Solving for $f^\prime(x)$ gives
$$
f^\prime(x)=f(x)(1+... |
365,128 | <p>How does one find the Inverse Laplace transform of $$\frac{6s^2 + 4s + 9}{(s^2 - 12s + 52)(s^2 + 36)}$$ where $s > 6$?</p>
| Amzoti | 38,839 | <p>Hints:</p>
<ol>
<li><p>Write out the partial fraction expansion.</p></li>
<li><p>Put the partial fraction into the forms that let you use the inverse table.</p></li>
</ol>
<p>Why do they put the restriction on s (you'll see it if you do 1 and 2)?</p>
<p>Clear?</p>
<p><strong>Update</strong></p>
<p>We are given ... |
2,013,115 | <p>I tried to do this problem in the following way:</p>
<p>As, $x^2+1 + \langle 3 , x^2+1 \rangle= 0 + \langle 3 , x^2+1 \rangle \implies x^2+1 \equiv 0 \implies x^2 \equiv -1.$</p>
<p>Also, $3+ \langle 3 , x^2+1 \rangle=0 +\langle 3 , x^2+1 \rangle \implies 3 \equiv 0$.</p>
<p>Now, any element of $\mathbb{Z}[x]/\l... | iam_agf | 196,583 | <p>Well, note that all your polynomials have the condition that their coefficients are $0,1,2$ and are of degree $0,1$, because $x^2+1=0$. So you have that the elements without $x$ are $0,1,2$. Then you consider the monic elements with $x$, that are $x,2x$. Finally you sum all the possible combinations of the firsts wi... |
1,840,159 | <blockquote>
<p>Question: Prove that a group of order 12 must have an element of order 2.</p>
</blockquote>
<p>I believe I've made great stride in my attempt.</p>
<p>By corollary to Lagrange's theorem, the order of any element $g$ in a group $G$ divides the order of a group $G$.</p>
<p>So, $ \left | g \right | \mi... | awllower | 6,792 | <p><strong>Hint:</strong><br>
Consider the Sylow $2$-subgroups of $G,$ which have order $4.$ </p>
<p>Hope this helps.</p>
|
3,380,998 | <p>Is it possible to express the cube root of "i" without using "i" itself?</p>
<p>If this is possible can you show me how to arrive at it?</p>
<p>thanks</p>
| Mohammad Riazi-Kermani | 514,496 | <p>On the unit circle mark the <span class="math-container">$30$</span> degree, <span class="math-container">$150 $</span> degree and <span class="math-container">$270$</span> degree points.
These are the cube roots of <span class="math-container">$i$</span> </p>
|
705,829 | <p>I'm trying to solve a problem here.</p>
<p>It says: "Prove that a triangle is isoceles if $\large b=2a\sin\left(\frac{\beta}{2}\right)$."
$B-\beta$
I've tried to prove it but I can't</p>
<p>Can anyone help me?</p>
| DeepSea | 101,504 | <p>Square both sides and use the law of cosine: $$b^2 = 4a^2\cdot (sin(B/2))^2 = 2a^2(1 - cosB) = 2a^2(1 - (a^2 + c^2 - b^2)/2ac)$$ . Simplify this equation we get:</p>
<p>$$(a - c)(a^2 - ac - b^2) = 0$$. </p>
<p>Case 1: If $a = c$, we're done.</p>
<p>Case 2: If $a \lt c \Rightarrow a - c \lt 0$, and $a^2 - ac - b^... |
4,181,524 | <p><span class="math-container">$$\text{I need to prove the following lemma : }\frac{\zeta'(s)}{\zeta(s)} = - \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$$</span></p>
<p><strong>My attempt:</strong></p>
<p><span class="math-container">$$\text{We know that }\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s} $$</span><span clas... | Infinity_hunter | 826,797 | <p>It is easy to show that <span class="math-container">$\log n = \sum_{d | n } \Lambda(d)$</span>. So by <a href="http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula" rel="nofollow noreferrer">Mobius inversion</a> it follows that <span class="math-container">$\Lambda(n) = \sum_{d|n} \mu(d) \log(n/d)$</span>. We... |
1,238,481 | <p>Let $\vec p, \vec q$ and $\vec r$ are three mutually perpendicular vectors of the same magnitude. If a vector $\vec x$ satisfies the equation $\begin{aligned} \vec p \times ((\vec x - \vec q) \times \vec p)+\vec q \times ((\vec x - \vec r) \times \vec q)+\vec r \times ((\vec x - \vec p) \times \vec r)=0\end{aligned}... | Sam | 221,113 | <p>To summarize, and to answer the question, it seems that we have an unknown problem. </p>
<p>The reason being: the OP has established "If $ZFC\nvdash$ $\neg$Con$(ZFC)$, then $ZFC\nvdash$ Con$(ZFC)$ $\implies SM$." in the statement of his question. Also, Asaf has established (the almost trivial) "If $ZFC\vdash$ $\ne... |
2,703,639 | <p>a) $f: L^1(0,3) \rightarrow \mathbb{R}$</p>
<p>b) $f: C[0,3] \rightarrow \mathbb{R}$</p>
<p>for part a I got $\|f\| = 1$ because $\|f(x)\|=|\int_0^2x(t)dt| \leq \int_0^2|x(t)|dt \leq\int_0^3|x(t)|dt = \|x(t)\|_1$ so $\|f\|=1$</p>
<p>for b, I think its similar: $\|f(x)\|=|\int_0^2x(t)dt| \leq \int_0^2|x(t)|dt \le... | user284331 | 284,331 | <p>b) is not correct. There is some $c>0$ such that $\|x\|_{1}\leq c\|x\|_{\infty}$, then $\|f\|\leq c$ for this $c$, it does not mean that for every $c>0$, $\|f\|\leq c$.</p>
<p>Rather, the question needs the exact norm of the operator. Actually one has $|f(x)|=\left|\displaystyle\int_{0}^{2}x(t)dt\right|\leq\d... |
3,136,568 | <p><a href="https://i.stack.imgur.com/STONY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/STONY.jpg" alt="enter image description here"></a></p>
<p>I seem to be struggling with this particular question. It is my understanding that in this situation, where du does not equal dx, that you must manipu... | Paras Khosla | 478,779 | <p><strong>Hint</strong>:</p>
<p><span class="math-container">$$\text{Let }\begin{bmatrix}u \\ \mathrm du \\ a\end{bmatrix}=\begin{bmatrix}5t \\ \mathrm dt \\ 2\end{bmatrix}$$</span></p>
|
604,070 | <p>While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not understand from the proof where we are using this idea as we just make equivalence classes of asymptotic Cauchy sequence... | rewritten | 43,219 | <p>For a metric space $\langle T, d\rangle$ to be complete, all Cauchy sequences must have a limit. So we add that limit by defining it to be an "abstract" object, which is defined by "any Cauchy sequence converging to it".</p>
<p>We have two cases:</p>
<ol>
<li><p>The Cauchy sequence already had a limit in $T$. In t... |
1,364,430 | <p><strong>Problem</strong></p>
<p>How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers?</p>
<p><strong>My thoughts</strong></p>
<p>I have no idea where to begin. I see no immediate connection between a factorial and a possible square. Much less for such ridiculously high numbers as $2015!$.</p>
<p... | Machining Machine | 90,278 | <p>Of all $n!$ only $0!$ and $1!$ are perfect squares.</p>
<p><a href="http://mathforum.org/library/drmath/view/55881.html" rel="nofollow">http://mathforum.org/library/drmath/view/55881.html</a></p>
<blockquote>
<p>To prove that a factorial bigger than 1 can't be a perfect square,
first think about breaking down... |
2,873,520 | <p>I want to find out how interference of two sine waves can affect the output-phase of the interfered wave. </p>
<p>Consider two waves,</p>
<p>$$ E_1 = \sin(x) \\
E_2 = 2 \sin{(x + \delta)}
$$</p>
<p>First off, I don't know how to prove it, but I can see visually (plotting numerically) that the sum of these waves... | Mohammad Riazi-Kermani | 514,496 | <p>Yes the polynomial associated to $$ ay'' + by' +cy =0$$ is $$P(\lambda )= a \lambda ^2 + b \lambda +c$$ which is called the charateristic polynomial. </p>
<p>This polynomial plays a very important role in finding the solutions to your differential equation. </p>
<p>The genera solution to the differential equation... |
65,480 | <p>The example question is </p>
<blockquote>
<p>Find the remainder when $8x^4+3x-1$ is divided by $2x^2+1$</p>
</blockquote>
<p>The answer did something like</p>
<p>$$8x^4+3x-1=(2x^2+1)(Ax^2+Bx+C)+(Dx+E)$$</p>
<p>Where $(Ax^2+Bx+C)$ is the Quotient and $(Dx+E)$ the remainder. I believe the degree of Quotient is d... | Américo Tavares | 752 | <p>EDIT to add these short answers.</p>
<blockquote>
<p>I believe the degree of Quotient is derived from degree of $8x^4+3x-1$
- degree of divisor.</p>
</blockquote>
<p>That's right.</p>
<blockquote>
<p>But for remainder?</p>
</blockquote>
<p>The degree of the remainder is less than the degree of the divisor,... |
743,988 | <p>If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get</p>
<p>$$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$</p>
<p>Applying this operator to a real analytic function, we have</p>
<p>$$\begin{align*}e^\frac{d}{dx} f(x) &... | Community | -1 | <p>You can just see it as an identity: the shift operator can be expressed in terms of a Taylor series, and then we just compute its closed form.</p>
<p>There are other visualizations for this, though. You can think of $1 + \frac{d}{dx}$ as an <em>infinitesimal</em> shift operator, and exponentiation accumulates all o... |
3,460,426 | <p>I tried to take the <span class="math-container">$Log$</span> of <span class="math-container">$\prod _{m\ge 1} \frac{1+\exp(i2\pi \cdot3^{-m})}{2} = \prod _{m\ge 1} Z_m$</span>, which gives </p>
<p><span class="math-container">$$Log \prod_{m\ge 1} Z_m = \sum_{m \ge 1} Log (Z_m) = \sum_{m \ge 1} \ln |Z_m| + i \sum_{... | Henry | 6,460 | <ul>
<li>If there are <span class="math-container">$n$</span> families then there are <span class="math-container">$1.8 n$</span> children</li>
<li>If there are <span class="math-container">$n_k$</span> families with <span class="math-container">$k$</span> children then <span class="math-container">$\sum_k n_k =n$</spa... |
1,456,444 | <p>How can I go about solving this Pigeonhole Principle problem? </p>
<p>So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$</p>
<p>I am trying to put this in words...</p>
| Jacob Westman | 273,234 | <p>If you pick five numbers from the set, you could pick exactly one from every pair you listed (and all numbers in the set are listed in exactly one pair). If you then pick another number then that must be from one of the five pairs you already have a number from. So you are guaranteed to end up with a pair that sums ... |
2,855,339 | <p>What would be the complement of...</p>
<p>$\{$x:x is a natural number divisible by 3 and 5$\}$</p>
<p>I checked it's solution and it kind of stumped me...</p>
<p>$\{$x:x is a positive integer which is not divisible by 3 <em>or</em> not divisible by 5$\}$</p>
<p>Why the word <em>or</em> has been used in the solut... | Oldboy | 401,277 | <p>(Many thanks to <strong>Empy2</strong> and <strong>gandalf61</strong> for spotting a mistake in the first version! Hopefully they won't find another one :)</p>
<p><strong>Not an answer</strong>, just a ballpark estimate:</p>
<pre><code>In[141]:= hourAngle[h_, m_, s_] := 2*Pi*(h*3600 + m*60 + s)/43200;
minuteAngle[... |
1,059,427 | <p>What is a good method to number of ways to distribute $n=30$ distinct books to $m=6$ students so that each student receives at most $r=7$ books?</p>
<p>My observation is: If student $S_i$ receives $n_i$ books, the number of ways
is: $\binom{n}{n_1,n_2,\cdots,n_m}$.</p>
<p>So answer is coefficient of $x^n$ in $n!(... | hardmath | 3,111 | <p>Since the Question does not state that each student must receive at least one book, I have included below in the 46 ways (that six nonnegative integers sum to 30) those with parts equal zero, limiting however summands not to exceed 7:</p>
<p>$$ 30 = s_1 + s_2 + s_3 + s_4 + s_5 + s_6 $$</p>
<p>such that $ 7 \ge s_1... |
849,093 | <p>After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me:
Can the non-elementary functions be decomposed to elementary ones? For instance, the logarithm, an elementary, can be decomposed into multiplication (e.g. $\ln x=y$ is the ... | Gina | 102,040 | <p>No. Doesn't matter which logical language you try to use and the interpretation, number of function that can be defined using that language is at most the number of finite string that can be formed using the symbols of that language. Since symbols set is finite, number of possible string is countable. Number of func... |
819,830 | <p>Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the conclusion)? Or can it simply be an absurdity that you know is false but through your derivation comes out true?</p>
| Vladhagen | 79,934 | <p>Evaluating your integral:</p>
<p>$$2 \pi \int_0^4 x\sqrt{x} dx = 2 \pi \int_0^4 x^{3/2} dx = 2\pi \frac{2}{5}x^{5/2}\bigg|_0^4 = \frac{128\pi}{5}$$</p>
<p>Hopefully this is what you were getting.</p>
<p>BUT. If we rotate around the $x$-axis (not the $y$-axis) then we will get $8\pi$ as follows:</p>
<p>$$2\pi\int... |
1,303,772 | <blockquote>
<p>Show that $$-2 \le \cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$$ for all value of $\theta$.</p>
</blockquote>
<p>Trial: I know that $0\le \sin^2 \theta \le1 $. So, I have $\sqrt3 \le \sqrt{\sin ^2 \theta +3} \le 2 $. After that I am unable to solve the problem. </p>
| math110 | 58,742 | <p>Use this well known inequality
$$-\dfrac{a^2+b^2}{2}\le ab\le\dfrac{a^2+b^2}{2},a,b\in R$$
so
$$-\dfrac{\cos^2{\theta}+4\sin^2{\theta}}{2}\le\cos{\theta}\cdot 2\sin{\theta}\le\dfrac{\cos^2{\theta}+4\sin^2{\theta}}{2}\tag{1}$$</p>
<p>$$-\dfrac{4\cos^2{\theta}+\sin^2{\theta}+3}{2}\le2\cos{\theta}\cdot\sqrt{\sin^2{\t... |
1,303,772 | <blockquote>
<p>Show that $$-2 \le \cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$$ for all value of $\theta$.</p>
</blockquote>
<p>Trial: I know that $0\le \sin^2 \theta \le1 $. So, I have $\sqrt3 \le \sqrt{\sin ^2 \theta +3} \le 2 $. After that I am unable to solve the problem. </p>
| juantheron | 14,311 | <p><strong>My Solution::</strong> Given $$f(\theta) = \cos \theta \left(\sin \theta + \sqrt{\sin^2 \theta + 3}\right)$$</p>
<p>Now let $$y=\sin \theta \cdot \cos \theta +\cos \theta \cdot \sqrt{\sin^2 \theta + 3}$$</p>
<p>Now using the <strong>Cauchy-Schwarz</strong> inequality, we get</p>
<p>$$\left(\sin^2 \theta... |
4,248,766 | <p>How many functions <span class="math-container">$f: \{1,...,n_1\} \to \{1,...,n_2\}$</span> are there such that if <span class="math-container">$f(k)=f(l)$</span> for some <span class="math-container">$k,l \in \{1,...,n_1\}$</span>, then <span class="math-container">$k=l$</span>?</p>
| Youem | 468,504 | <p>You are looking for the number of injective functions from <span class="math-container">$\{1,\ldots,n_1\}$</span> to <span class="math-container">$\{1,\ldots,n_2\}$</span>. This is <span class="math-container">$$n_1!\binom{n_2}{n_1}$$</span> if <span class="math-container">$n_1\le n_2$</span> and <span class="math-c... |
2,500,961 | <p>I've been able to find formulas all over the place for the sum and product of roots, but I haven't found anything that explains the significance of what they mean or how to interpret them to further gain understanding of the polynomial under evaluation. Is there any physical meaning? Do the values have any significa... | Ken Wei | 243,183 | <p>The roots of a polynomial in $x$ are the values you can plug in for $x$ such that the polynomial takes the value $0$. So the first part of your last paragraph doesn't make much sense, since roots are special values of $x$.</p>
<p>The sum and product of roots of a quadratic polynomial $ax^2 + bx + c$ are $-b/a$ and ... |
4,548,329 | <p>Find the first derivative of <span class="math-container">$$y=\sqrt[3]{\dfrac{1-x^3}{1+x^3}}$$</span></p>
<p>The given answer is <span class="math-container">$$\dfrac{2x^2}{x^6-1}\sqrt[3]{\dfrac{1-x^3}{1+x^3}}$$</span> It is nice and neat, but I am really struggling to write the result exactly in this form. We have ... | aarbee | 87,430 | <p>You got <span class="math-container">$$y'=-\dfrac{2x^2}{(1+x^3)^2}\left(\dfrac{1+x^3}{1-x^3}\right)^\frac23$$</span></p>
<p>You can leave till here if you want. If you want to match the given answer, multiply and divide by <span class="math-container">$\left(\dfrac{1+x^3}{1-x^3}\right)^\frac13$</span>, thus,</p>
<p>... |
253,966 | <p>Just took my final exam and I wanted to see if I answered this correctly:</p>
<p>If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$
have the following relations:</p>
<p>$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$</p>
<p>does it follow that $A \cong Z_{3} \tim... | Amr | 29,267 | <p>Since $7x+5y+2z=0=3x+3y$, therefore $7x+5y+2z+2(3x+3y)=13x+11y+2z=0$, hence the last relation is not important. We also note that $7x+5y+2z-2(3x+3y)=x-y+2z$</p>
<p>Consider the group $G$={$ix+jy+kz|i,j,k\in Z$} (with addition defined as: ($i_1x+j_1y+k_1z)+(i_2x+j_2y+k_2z)=(i_1+i_2)x+(j_1+j_2)y+(k_1+k_2)z$).</p>
<p... |
253,966 | <p>Just took my final exam and I wanted to see if I answered this correctly:</p>
<p>If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$
have the following relations:</p>
<p>$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$</p>
<p>does it follow that $A \cong Z_{3} \tim... | Gerry Myerson | 8,269 | <p>A colleague of mine has written some <a href="http://web.science.mq.edu.au/~chris/groups/CHAP10%20Finitely-Generated%20Abelian%20Groups.pdf" rel="nofollow">notes</a> that we use in a course here. They should help you understand how to do this kind of question. </p>
|
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| David Holden | 79,543 | <p>square it and simplify - gives period</p>
|
397,347 | <p>I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies J=-I'(1)$, but I couldn't figure out what $I(s)$ was. My other idea was contour integration, but I'm not sure how to deal... | Ron Gordon | 53,268 | <p>How about pulling factors of $e^{-x}$ from both the denominator and log terms? Then you end up with two separate integrals:</p>
<p>$$\int_0^{\infty}dx \frac{x^4 \, e^{-x}}{1-e^{-x}} + \int_0^{\infty}dx \frac{x^3 \, e^{-x}}{1-e^{-x}} \log{(1-e^{-x})}$$</p>
<p>In both cases, you Taylor expand the denominator in $e^... |
2,971,143 | <p>Let me choose <span class="math-container">$n=1$</span> for my induction basis: <span class="math-container">$2 > 1$</span>, true.</p>
<p>Induction Step : <span class="math-container">$2^n > n^2 \rightarrow 2^{n+1} > (n+1)^2 $</span></p>
<p><span class="math-container">$2^{n+1} > (n+1)^2 \iff$</span></... | mathematics2x2life | 79,043 | <p>You have an error. You had <span class="math-container">$n^2-2^n+1+2n-2^n<0$</span>, by in the induction hypothesis, you do know that <span class="math-container">$n^2-2^n<0$</span>. But that does not mean that <span class="math-container">$1+2n-2^n<0$</span>. It could be that <span class="math-container">$... |
4,487,489 | <p>I'm currently working on completing their first unit on calculus ab and I've encountered this roadblock. That's probably an exaggeration but I honestly can't figure out what they mean by "for negative numbers". I did the math and got the right number (at least the right absolute value) but the missing nega... | emacs drives me nuts | 746,312 | <blockquote>
<p>I honestly can't figure out what they mean by "for negative numbers".</p>
</blockquote>
<p>It means that the equation holds for negative <span class="math-container">$x$</span>, that is if <span class="math-container">$x<0$</span>. Reason is that the real square-root is non-negative by d... |
1,908,844 | <p>The following example is taken from the book "Introduction to Probability Models" of Sheldon M. Ross (Chapter 5, example 5.4).</p>
<blockquote>
<p>The dollar amount of damage involved in an automobile accident is an
exponential random variable with mean 1000. Of this, the insurance
company only pays that amou... | syusim | 138,951 | <p>The answer is ${n+m \choose k}$ and the best possible intuition is that both this and the big summation are counting all possible ways of choosing $k$ things from two sets of things, one of size $m$ and one of size $n$.</p>
<p>I find it easier to do combinatorial identities by thinking about what the formulas might... |
1,466,198 | <p>I was solving some mathematical questions and have come across the situation, where I need to divide 3900/139. Here is my question, </p>
<p>a. Can I assume 139 to 140 for the ease of division?</p>
<p>If so, how will I know what percentage of error I am introducing? How can I ensure that I am adding very less value... | PTDS | 277,299 | <p>In general, if y = c/x (c is a constant) and you make a small change in x, say by h (> 0), then the following happens:</p>
<ol>
<li>The exact relative error is (-h/(c+h))</li>
<li>This is approximately equal to (-h/c) [If you take the log and then the differential, that will be evident].</li>
</ol>
|
4,220,972 | <p>I'm studying a for the GRE and a practice test problem is, "For all real numbers x and y, if x#y=x(x-y), then x#(x#y) =?</p>
<p>I do not know what the # sign means. This is apparently an algebra function but I cannot find any such in several searches. I'm an older student and haven't had basic algebra in over 4... | RobertTheTutor | 883,326 | <p>P(any random variable being within k standard deviations of its mean) <span class="math-container">$\geq 1-1/k^2$</span>.</p>
<p>The variance is <span class="math-container">$1/4$</span>, the standard deviation is <span class="math-container">$1/2$</span> and the error is <span class="math-container">$1$</span>, whi... |
3,637,283 | <p>How would I find the fourth roots of <span class="math-container">$-81i$</span> in the complex numbers? </p>
<p>Here is what I currently have: </p>
<p><span class="math-container">$w = -81i$</span> </p>
<p><span class="math-container">$r = 9$</span> </p>
<p><span class="math-container">$\theta = \arctan (-81)$</... | Physical Mathematics | 592,278 | <p><span class="math-container">$$w:= -81i = 81 e^{-i\pi/2},81 e^{3i\pi/2}, 81e^{7i\pi/2},81 e^{11i\pi/2},$$</span>
So the 4th roots of <span class="math-container">$w$</span> are:
<span class="math-container">$$\sqrt[4]{81} e^{-i\pi/8}, \sqrt[4]{81}e^{3i\pi/8}, \sqrt[4]{81} e^{7i\pi/8}, \sqrt[4]{81} e^{11i\pi/8},$$</... |
3,637,283 | <p>How would I find the fourth roots of <span class="math-container">$-81i$</span> in the complex numbers? </p>
<p>Here is what I currently have: </p>
<p><span class="math-container">$w = -81i$</span> </p>
<p><span class="math-container">$r = 9$</span> </p>
<p><span class="math-container">$\theta = \arctan (-81)$</... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>It is much simpler: the (real) fourth root of <span class="math-container">$81$</span> is <span class="math-container">$3$</span>. So you simply have to determine the fourth roots of <span class="math-container">$-i$</span>. For that, use the complex exponential notation:
<span class=... |
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