https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2022-09-26T03:23:52Zhttps://math.codidact.com/posts/287063Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$.Carefree Explorerhttps://math.codidact.com/users/568512022-09-24T14:21:59Z2022-09-26T03:23:52Z<blockquote>
<p>Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$.</p>
</blockquote>
<p>This was a simple observation I made when playing around and I came up with the follow...https://math.codidact.com/posts/287002Proving that $p\mid (p+1776)$ if $p$ is a prime and $p(p+1776)$ is a perfect squareSnoopyhttps://math.codidact.com/users/568172022-09-14T18:23:50Z2022-09-15T19:21:22Z<p><strong>Problem</strong>: Suppose $p$ is a prime number and $p(p+1776)$ is a perfect square. Prove that $p\mid (p+1776)$.</p>
<p>From the assumption of the problem, $p(p+1776)=k^2$ for some pos...https://math.codidact.com/posts/286985Given two angles of a triangle, finding an angle formed by a medianSnoopyhttps://math.codidact.com/users/568172022-09-11T21:44:35Z2022-09-13T11:38:33Z<blockquote>
<p><strong>Problem</strong>: Suppose in $\triangle ABC$, $\angle BAC = 30^\circ$ and $\angle BCA = 15^\circ$. Suppose $BM$ is a <a href="https://en.wikipedia.org/wiki/Median_(geometry...https://math.codidact.com/posts/286991organizing a librarymsh210https://math.codidact.com/users/81062022-09-12T15:36:11Z2022-09-13T05:58:16Z<p>Suppose you have $n>1$ books lined up on a shelf, numbered $1$ to $n$, not in the correct order, and you wish to put them in order. Here's your method: Choose a misplaced book<sup>[1]</sup> a...https://math.codidact.com/posts/286961Is the really only one solution?Grovehttps://math.codidact.com/users/534522022-09-06T15:22:23Z2022-09-10T21:57:17Z<p>Going through some old notes, I stumbled upon this question, I don't remember where it take from/what inspired me to look at it. If this is an obvious copy of something posted/covered elsewhere,...https://math.codidact.com/posts/286956If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension?Snoopyhttps://math.codidact.com/users/568172022-09-04T01:50:34Z2022-09-05T05:34:32Z<p>It is known that $\mathbf{R}$, as a vector space over the field of <em>real</em> numbers, has the dimension $1$. I know that $\mathbf{Q}$ is also a field.</p>
<p><strong>Question</strong>: If $...https://math.codidact.com/posts/286926Inhomogenous differential equations in sciencesMissMulanhttps://math.codidact.com/users/541072022-08-24T17:14:46Z2022-08-24T17:15:04Z<p>A first order inhomogenous differential equation is :</p>
<p>$$ \frac{dy}{dx}+p(x)y(x) = q(x)$$</p>
<p>with $$ q(x)\neq0$$</p>
<p>I like studying applied math but I dont like pure math so im ...https://math.codidact.com/posts/286907Prove $e^x \ge x+1 \\\; \forall x \in \mathbb{R}$ using inductionCarefree Explorerhttps://math.codidact.com/users/568512022-08-20T20:14:35Z2022-08-21T06:56:53Z<blockquote>
<p>(How) can we prove $e^x \ge x+1 \; \forall x \in \mathbb{R}$ using induction (without using the derivative of $e^x$ at any stage)? Comments on my attempt are appreciated.</p>
</bl...https://math.codidact.com/posts/286848Is there a way to encode a unique arrangement of vertices of a graph with a unique short word?Carefree Explorerhttps://math.codidact.com/users/568512022-08-11T12:52:39Z2022-08-13T09:12:57Z<p>I call graphs $G_1$ and $G_2$ <em>distinct</em> iff (i) $G_1$ has a different arrangement<sup>1</sup> of vertices than $G_2$ <strong>and</strong> (ii) $G_1$ and $G_2$ have the same number of ver...https://math.codidact.com/posts/286829Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$Carefree Explorerhttps://math.codidact.com/users/568512022-08-02T20:32:05Z2022-08-03T05:05:08Z<blockquote>
<p>Show that $$f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2} \quad \forall x \in (-\infty, -1)\large\cup (0, \infty)$$</p>
</blockquote...https://math.codidact.com/posts/286808What's the common ratio for this geometric sequence?General Sebast1anhttps://math.codidact.com/users/541142022-07-28T07:58:04Z2022-07-28T12:49:03Z<p>Delayed learning some math, now I'm back at it.</p>
<p>Geometric sequences. Basically a sequence where it has a common ratio.</p>
<p>An example:</p>
<pre><code>Sequence = {14, 28, 56, 112}
R...https://math.codidact.com/posts/286788The gcd of powers in a gcd domainSnoopyhttps://math.codidact.com/users/568172022-07-22T18:12:11Z2022-07-25T14:30:34Z<p><strong>Question:</strong></p>
<p>$\def\gcd{\operatorname{gcd}}$Let $R$ be a <a href="https://en.wikipedia.org/wiki/GCD_domain">gcd domain</a>. Does it always hold that $\gcd(x^m,y^m)=\gcd(x,y)...https://math.codidact.com/posts/286728Isn't "any, some, or all" redundant? Why not write just "any"?Chgg Clouhttps://math.codidact.com/users/535642022-07-16T01:02:32Z2022-07-16T08:18:24Z<p>Please see the title of this post. In the following quotations, what changes — if anything — if you replace "<strong>any, some[,] or all</strong>" with just "<code>any</code>"? Don't these autho...https://math.codidact.com/posts/286709What are the 2 arithmetic means of $x + y$ and $4x - 2y$?General Sebast1anhttps://math.codidact.com/users/541142022-07-12T07:21:05Z2022-07-12T11:44:58Z<p>I'm currently learning arithmetic sequences, and I've gotten to the means. I'm answering an activity as a test to see if what I'm doing is right.</p>
<p>Here's an example through format:</p>
<...https://math.codidact.com/posts/286612Notation for nested exponentsJRNhttps://math.codidact.com/users/529962022-06-20T01:18:16Z2022-07-07T07:41:14Z<p>An expression such as $a^{b^c}$ is usually interpreted as $a^{(b^c)}$ and not as ${(a^b)}^c$. (See, for example, the Wikipedia entry for <a href="https://en.wikipedia.org/wiki/Double_exponentia...https://math.codidact.com/posts/286150equilateral triangle inscribed in an ellipsemsh210https://math.codidact.com/users/81062022-03-24T19:58:14Z2022-07-05T16:36:36Z<p>A high-schooler I know was given the following problem:</p>
<blockquote>
<p>In the ellipse $x^2+3y^2=12$ is inscribed an equilateral triangle. One of the triangle's vertices is at the point $(...https://math.codidact.com/posts/286655Do the Faber partition polynomials have integer coefficients?Peter Taylorhttps://math.codidact.com/users/363562022-06-30T17:23:47Z2022-06-30T17:58:06Z<p>The Online Encyclopedia of Integer Sequences includes <a href="https://oeis.org/A263916">A263916: Coefficients of the Faber partition polynomials</a>. Perhaps the clearest definition given is</p...https://math.codidact.com/posts/286572Endomorphisms on an entropic structure whose pointwise product is the identity automorphism - entropic idempotent structure?Prime Moverhttps://math.codidact.com/users/566792022-06-10T10:07:24Z2022-06-10T11:20:28Z<p>Context: self-study from Warner's "Modern Algebra (1965): Exercise 16.27.</p>
<blockquote>
<p>Let $\alpha$ and $\beta$ be endomorphisms of an entropic structure $(S, \odot)$ such that $\alpha ...https://math.codidact.com/posts/281006In "if and only if" proofs, why's 1 direction easier to prove than the other? Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:27:22Z2022-06-10T09:57:37Z<p><a href="https://math.stackexchange.com/q/3069488">This list</a> on Math StackExchange instantiates (biconditional) logical equivalences where one direction can be proved swimmingly, but the oth...https://math.codidact.com/posts/286561Optimising a 3 value problem (well 2 really)KalleMPhttps://math.codidact.com/users/81142022-06-06T15:01:52Z2022-06-09T14:07:49Z<p>I have tried to find a way to optimise value selections so a range of results based on 3 values will be as evenly spaced as possible as close as possible to a geometric series.</p>
<p>I have a ...https://math.codidact.com/posts/286527Notation for one-sided hypothesis testingtommihttps://math.codidact.com/users/534072022-05-30T11:36:06Z2022-05-30T11:36:06Z<p>I see the following notation for one-sided hypothesis testing:</p>
<ul>
<li>$H_0$: $K = 2$</li>
<li>$H_1$: $K > 2$</li>
</ul>
<p>I would find it more natural to write:</p>
<ul>
<li>$H_...https://math.codidact.com/posts/286166Complex numbers in 2D, quaternions in 4D, why nothing in 3D? Fred Wamsleyhttps://math.codidact.com/users/550432022-03-27T21:22:28Z2022-05-28T06:36:52Z<p>I'm just trying to understand why quaternions are necessary. If I understand right, first off please check me on this, Hamilton kept trying to find ways to multiply triplets and found something ...https://math.codidact.com/posts/285984How $ijk=\sqrt{1}$?deleted user#2022-02-18T15:34:12Z2022-05-27T19:32:10Z<p>In the <a href="https://en.wikipedia.org/wiki/Quaternion#History">Wikipedia page</a>, I can clearly see that
$$i^2=j^2=k^2=ijk=-1$$</p>
<p>But if we consider them separately</p>
<p>$$i=\sqrt{...https://math.codidact.com/posts/286428$g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$Udi Fogielhttps://math.codidact.com/users/564582022-05-10T21:34:01Z2022-05-19T15:41:49Z<p>Recently in my ordinary differential equations class we were given the following problem:</p>
<blockquote>
<p>Suppose $g:(0,\infty)\to\mathbb{R}$ is an increasing function of class $C^{1}$ suc...https://math.codidact.com/posts/286472Are vectrices useful for calculations as opposed to formalism?Fred Wamsleyhttps://math.codidact.com/users/550432022-05-17T01:22:34Z2022-05-17T23:22:17Z<p>The context here is calculations about satellite attitude control and motion. There is a frequent need to convert between reference frames, and a vectrix is a clever notation to assist.</p>
<p>...https://math.codidact.com/posts/285432How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2?DNBhttps://math.codidact.com/users/536282021-12-31T08:31:07Z2022-05-09T13:01:11Z<p>Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and ...https://math.codidact.com/posts/286398Matrices with rotational symmetryPeter Taylorhttps://math.codidact.com/users/363562022-05-05T22:24:34Z2022-05-05T22:46:21Z<p>I've seen a <a href="https://mathoverflow.net/a/418547">claim without proof</a> that the characteristic polynomials of matrices with rotational symmetry (i.e. $n \times n$ matrices $A$ with $A_{...https://math.codidact.com/posts/286292Why does the Dottie number $=\sqrt{1-\left(2\text I^{-1}_\frac12\left(\frac12,\frac32\right)-1\right)^2}$?Tyma Gaidashhttps://math.codidact.com/users/565582022-04-22T04:26:17Z2022-04-22T04:37:45Z<p><strong>Introduction:</strong></p>
<p>For some background information on the <a href="https://en.m.wikipedia.org/wiki/Dottie_number"><strong>Dottie Number D</strong></a>,</p>
<p>Cross posted f...https://math.codidact.com/posts/286188A discrete-time queue Markov Chain problemJoehttps://math.codidact.com/users/565152022-04-05T08:20:31Z2022-04-05T08:20:31Z<p>Hello everyone, I hope somebody can help me with this question, only for parts c and d, because I have already figured out parts a and b. Here is the question:</p>
<p>Items arrive in a queue du...https://math.codidact.com/posts/286140Finding distance to parabola's focus, given some pointsmsh210https://math.codidact.com/users/81062022-03-21T20:44:16Z2022-03-21T21:21:24Z<p>A high school student I know has the following problem:</p>
<blockquote>
<p>A parabola is given by $y^2=2px$ with $p>0$. The point $D$ is on the parabola in the first quadrant at a distance...https://math.codidact.com/posts/286130How can I justify whether I'm able to apply the Gaussian elimination method to this system?dumplingshttps://math.codidact.com/users/547102022-03-20T11:01:16Z2022-03-21T12:15:58Z<p>I'm currently practicing some Gaussian elimination questions for my discrete math topic, I came across this question and I need help. How can I justify whether I'm able to apply the Gaussian eli...https://math.codidact.com/posts/286004Sample skewness consistencyGuilherme Costahttps://math.codidact.com/users/534532022-02-22T10:55:44Z2022-03-04T08:54:25Z<p>Is sample skewness a consistent estimator of skewness?</p>
<!-- g: js, mdit -->https://math.codidact.com/posts/285670Why 1. multiply the number of independent options? 2. add the number of exclusive options?DNBhttps://math.codidact.com/users/536282022-01-24T03:31:04Z2022-02-27T05:05:31Z<p>Ironically, this textbook highlights understanding over memorization, but it doesn't expatiate the two WHY's in the question title!</p>
<blockquote>
<p>When faced with a series of independent ...https://math.codidact.com/posts/285977Maximize Independent Variable of Matrix MultiplicationJosh Hyatthttps://math.codidact.com/users/536292022-02-17T20:47:41Z2022-02-18T20:23:40Z<p>Let $T$ be an $m\times n$ matrix with column vectors $\vec{T_i}$:</p>
<p>$$
\vec{T_i}=\begin{bmatrix}
\vec{T_1} & ... & \vec{T_n}
\end{bmatrix}
$$</p>
<p>Let $\vec{x}$ be an unknow...https://math.codidact.com/posts/285819Is it impossible to prove Jensen's inequality by way of step functions?whybecausehttps://math.codidact.com/users/550222022-02-04T01:03:31Z2022-02-11T21:32:00Z<p>Jensen's Inequality: Let $\varphi:\Bbb R\to \Bbb R$ be convex, and $f:[0,1]\to\Bbb R$ be integrable, and suppose $\varphi\circ f$ is integrable over [0,1]. Then
$$ \varphi\left(\int_{[0,1]} f\...https://math.codidact.com/posts/285925"Pointwise equicontinuity implies uniform" implies compactwhybecausehttps://math.codidact.com/users/550222022-02-11T05:02:37Z2022-02-11T20:05:08Z<p>Suppose $(M,d)$ is a metric space such that every sequence $f_n:M\to \Bbb R$ which is pointwise equicontinuous is also uniformly equicontinuous. Does this imply that $M$ is a compact metric spa...https://math.codidact.com/posts/285679How to intuit, construe multiplicands and multiplicators $\le 10$ resulting from $\dbinom pc$, WITHOUT division or factorials?DNBhttps://math.codidact.com/users/536282022-01-25T05:46:05Z2022-02-01T10:24:37Z<p>I grok, am NOT asking about, the answers below. Rather — <strong>how can I deduce and intuit the multiplicands and multiplicators $\le 10$, resulting from simplifying $\dbinom {p \text{ people...https://math.codidact.com/posts/285756$\{f_n\}\to f$ in $X$ implies $||f||\le\liminf||f_n||$ whybecausehttps://math.codidact.com/users/550222022-01-31T02:27:52Z2022-01-31T16:28:15Z<p>Let $X$ be a normed linear space and suppose that, for each $f\in X$ there exists a bounded linear functional $ T\in X^* $ such that $T(f)=||f||$ and $||T||_*=1$. Prove that if $\{f_n\}\to f$ in...https://math.codidact.com/posts/285753Missing a solution: Are A and B always equal if A-B=0re89jhttps://math.codidact.com/users/553482022-01-30T23:10:46Z2022-01-31T09:01:14Z<p>I just came across this problem on an online learning app:</p>
<p>4x(5−x)−12(5−x)+100=100</p>
<p>I tried to solve it by subtracting 100 from each side then inferring if something minus somethi...https://math.codidact.com/posts/282608Out of 4 people, why does ways to choose a 2-person committee overcount by 2 the ways to divide the 4 into 2 teams of 2?DNBhttps://math.codidact.com/users/536282021-07-09T05:04:41Z2022-01-28T15:52:59Z<ol>
<li>
<p>Please see the sentence alongside my red line below. Why does part (a) overcount part (b) by a factor of c?</p>
</li>
<li>
<p>Scilicet, why aren't the answers to parts (a) and (b...https://math.codidact.com/posts/285671How can you "easily see that such squares [of side length $\sqrt{13}$ and $\sqrt{18}$] will not fit into the [4 × 4] grid"?DNBhttps://math.codidact.com/users/536282022-01-24T03:45:23Z2022-01-28T12:29:42Z<blockquote>
<p>Problem 2.4: How many squares of any size can be formed by connecting dots in the grid shown in Figure 2.2.</p>
</blockquote>
<p>I skip p 31, but apprise me if you want me to inc...https://math.codidact.com/posts/285680How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?DNBhttps://math.codidact.com/users/536282022-01-25T06:07:44Z2022-01-25T19:24:52Z<p>Paradoxically, though <a href="https://math.stackexchange.com/q/3572304">Rothe-Hagen Identity</a> (henceforth RHI)</p>
<p>$\sum\limits_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+...https://math.codidact.com/posts/285672To correct for overcounting, why can't we divide by the sum of the over-counts?DNBhttps://math.codidact.com/users/536282022-01-24T08:02:35Z2022-01-24T08:52:47Z<p>The book doesn't expatiate the sentence underlined in red below. 1. Why "it isn't 2 + 2"?</p>
<ol start="2">
<li>Note that $2 \times 2 = 2 + 2 = 4$! Is this a coincidence?</li>
</ol>
<blockq...https://math.codidact.com/posts/285673How can you prognosticate to rewrite the same sum backwards, then add the same sum (twice)? DNBhttps://math.codidact.com/users/536282022-01-24T08:22:11Z2022-01-24T08:23:40Z<p><a href="https://math.stackexchange.com/questions/924203/reversing-the-order-of-summation-in-sum-i-1n-i#comment1907997_924214">This comment doesn't fulfill me</a>, because it doesn't demystify t...https://math.codidact.com/posts/285443Why's Pr(the running total of a fair dice rolled repeatedly = n) = $\frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})$?DNBhttps://math.codidact.com/users/536282022-01-01T05:01:33Z2022-01-17T09:18:39Z<p>Where did $\color{red}{p_n = \frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})}$ spring from? The solution doesn't expatiate, and makes it appear out of the blue? Can you pleas...https://math.codidact.com/posts/285433Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4?DNBhttps://math.codidact.com/users/536282021-12-31T08:41:57Z2022-01-17T08:54:19Z<ol>
<li>
<p>Why can this problem "be thought of as a gambler's ruin where each player starts out
with $2"?</p>
</li>
<li>
<p>Please see my red arrow. Why is this exponent 4? I quote op. cit....https://math.codidact.com/posts/285524How's it possible to arrange 0 objects? How can 0! = 1?DNBhttps://math.codidact.com/users/536282022-01-10T00:06:59Z2022-01-12T16:28:38Z<ol>
<li>
<p>I don't understand the following explanation. Isn't it physically impossible to "arrange 0 objects", or nothing? It's senseless to raise the concept of arrangement when you have not...https://math.codidact.com/posts/285431If C = Calvin wins the match, and $X \thicksim Bin(2, p) =$ how many of the first 2 games he wins — then why P(C|X = 1) = P(C)?DNBhttps://math.codidact.com/users/536282021-12-31T08:24:45Z2022-01-05T11:14:44Z<p>The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$? <a href="https://math.stackexchange.com/q/4288685">This similar question on Math Stack Exchange</a> has 0 answers, a...https://math.codidact.com/posts/285344How can you analogize mathematical induction to dominoes falling, if some domino can fail to topple?Chgg Clouhttps://math.codidact.com/users/535642021-12-21T00:33:33Z2022-01-05T10:00:41Z<p>This analogy doesn't convince me, because what if some domino (after b, the base case) fails to topple? In real life, a domino can remain standing upright if it got placed too far apart from the...https://math.codidact.com/posts/280847How can I re-write or supplant “basis point” UNambiguously?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:29:34Z2022-01-05T07:29:51Z<p>For this question, please DON'T use or even allude to "basis point" in your rewording, because I'm trying to replace "basis point"! "basis point" befuddles me! <a href="https://www.investopedia....