https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 25 Aug 2026 10:02:53 -0400https://pop.com.sg/what-is-sqrt-36.htmlhttps://pop.com.sg/what-is-sqrt-36.htmlI am working on a math problem $(x^{2} - 6x + 45)$ where I must utilize the "completing the square" method, and was able to break down the problem to $(x-3)^{2} = -36.$ I know I have to ......Tue, 25 Aug 2026 13:55:05 -0400https://pop.com.sg/definition-of-exp-x.htmlhttps://pop.com.sg/definition-of-exp-x.htmlI've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ......Tue, 25 Aug 2026 13:54:25 -0400https://pop.com.sg/expressing-logarithms-as-ratios-of-natural-logarithms.htmlhttps://pop.com.sg/expressing-logarithms-as-ratios-of-natural-logarithms.html$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an abstract ...Tue, 25 Aug 2026 13:42:14 -0400https://pop.com.sg/explanation-of-the-rational-root-theorem.htmlhttps://pop.com.sg/explanation-of-the-rational-root-theorem.htmlI just discovered the rational root theorem and I feel like I can understand it if I can get past the notational jargon presented in Wikipedia. Here's what I think I understood. If $r$ is a rational ...Tue, 25 Aug 2026 13:39:12 -0400https://pop.com.sg/limited-function.htmlhttps://pop.com.sg/limited-function.htmlLet $f:[0,+\infty) \rightarrow \mathbb{R}$ be a function bounded on each bounded interval. Prove that if $\lim_{x \rightarrow +\infty } [f(x+1)-f(x)] = L$, then $\lim_{x \rightarrow +\infty} \frac......Tue, 25 Aug 2026 13:34:51 -0400https://pop.com.sg/formula-for-probability-of-specific-die-outcomes.htmlhttps://pop.com.sg/formula-for-probability-of-specific-die-outcomes.htmlRoll some number, N, of the same kind of dice, say d6. Picking a target outcome, say 1, what is the FORMULA to determine the chance of getting 1s as a result, which is to say, how many chances of ......Tue, 25 Aug 2026 13:32:39 -0400https://pop.com.sg/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlhttps://pop.com.sg/why-balanced-ternary-exists-what-is-the-problem-with-ternary.htmlRecently, I countered a problem of representing data in ternary. I came to know there exists ternary and balanced ternary representation. It is my best understanding that balanced ternary helps in ...Tue, 25 Aug 2026 13:30:54 -0400https://pop.com.sg/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlhttps://pop.com.sg/prove-that-a-subseteq-b-and-b-is-an-ideal-of-r.htmlLet $R$ be a commutative ring with unity, let $A$ be an ideal of $R$ and let $a \in R$. Define $B = \{b \in R ~|~ ab \in A \}$. Prove that $A \subseteq B$ (where $A$ does not have to be the proper ......Tue, 25 Aug 2026 13:03:25 -0400https://pop.com.sg/probability-of-duplicate-guid.htmlhttps://pop.com.sg/probability-of-duplicate-guid.htmlA GUID (globally unique identifier) is a 32 character hexadecimal string: http://en.wikipedia.org/wiki/Globally_Unique_Identifier If you randomly generate 2, the chance of them being the same is ...Tue, 25 Aug 2026 13:02:18 -0400https://pop.com.sg/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlhttps://pop.com.sg/how-do-you-solve-an-equation-involving-the-weierstrass-p-function.htmlWhat $z\in \mathbb{C}$ satisfy \begin{eqnarray*} \wp(z) = \wp(nz) \end{eqnarray*} if $n\in \mathbb{N}$? The Weierstrass p-function is defined as \begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{\omega......Tue, 25 Aug 2026 12:56:30 -0400https://pop.com.sg/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlhttps://pop.com.sg/sufficient-conditions-on-the-isomorphism-of-two-groups.htmlLet $G_1, G_2$ be two groups with at least one nontrivial proper subgroup each. Let $S_1, S_2$ be the sets of proper subgroups of, respectively $G_1, G_2$. Suppose there exists a bijective functi......Tue, 25 Aug 2026 12:43:18 -0400https://pop.com.sg/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlhttps://pop.com.sg/can-the-entropy-of-a-random-variable-with-countably-many-outcomes-be-i.htmlConsider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possibl......Tue, 25 Aug 2026 12:15:31 -0400https://pop.com.sg/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlhttps://pop.com.sg/where-can-i-find-a-list-of-large-prime-numbers-closed.htmlA repository of say 13 digit prime, 15 digit primes etc....Tue, 25 Aug 2026 11:58:54 -0400https://pop.com.sg/is-basis-change-ever-useful-in-practical-linear-algebra.htmlhttps://pop.com.sg/is-basis-change-ever-useful-in-practical-linear-algebra.htmlIn layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis?...Tue, 25 Aug 2026 11:55:21 -0400https://pop.com.sg/how-to-factor-x-2-2xy-y-2.htmlhttps://pop.com.sg/how-to-factor-x-2-2xy-y-2.htmlI know the formula $x^2 + 2xy + y^2 = (x + y)^2$ by heart and in order to understand it better I am trying to solve the formula myself; however, I don't understand how step 4 is derived from step 3......Tue, 25 Aug 2026 11:47:22 -0400https://pop.com.sg/integral-of-1-x-2-without-power-rule.htmlhttps://pop.com.sg/integral-of-1-x-2-without-power-rule.htmlI was wondering if it was possible to evaluate the following integral without using the power rule for negative exponents \begin{equation*} \int \frac{1}{x^2} \; dx \end{equation*} When using ...Tue, 25 Aug 2026 11:39:09 -0400https://pop.com.sg/math-logic-symbol-for-instance-of.htmlhttps://pop.com.sg/math-logic-symbol-for-instance-of.htmlI saw a discussion of a possible symbol for "example," but I need "example/instance of." There is of course $a \in S$ which is a is a member of/in S, but is there a specific way of stating $a "...Tue, 25 Aug 2026 11:32:29 -0400https://pop.com.sg/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlhttps://pop.com.sg/i-want-an-intutive-theoretical-understanding-of-the-sin-omega-t-f-clos.htmlWhat is the meaning of $x(t)=x_0\cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ phase constant? I know how to solve the mathematical problems whic......Tue, 25 Aug 2026 11:22:58 -0400https://pop.com.sg/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlhttps://pop.com.sg/translation-of-weber-039-s-lehrbuch-der-algebra-vol-1-2-3.htmlI have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would ......Tue, 25 Aug 2026 11:17:26 -0400https://pop.com.sg/what-039-s-the-difference-between-a-b-and-a-b.htmlhttps://pop.com.sg/what-039-s-the-difference-between-a-b-and-a-b.htmlIs there any difference between $]a,b[$ and $(a,b)$? If there is no difference, what would be the motivation of using $]a,b[$ over $(a,b)$?...Tue, 25 Aug 2026 11:16:57 -0400https://pop.com.sg/axiomatic-definition-of-groups.htmlhttps://pop.com.sg/axiomatic-definition-of-groups.htmlTo define natural numbers one can either: use the Peano axioms in second-order logic; encode them in set theory as von Neumann ordinals. The relation between those two definitions of natural num......Tue, 25 Aug 2026 11:15:22 -0400https://pop.com.sg/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlhttps://pop.com.sg/use-laplace-transform-to-solve-the-following-initial-value-problems.htmlUse Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(......Tue, 25 Aug 2026 11:06:40 -0400https://pop.com.sg/how-much-does-the-owner-lose.htmlhttps://pop.com.sg/how-much-does-the-owner-lose.htmlA thief stole a $\$100$ from the owner without him knowing. The same thief comes back inside the store $5$ minutes later and spends $\$70$ of the money. The owner then gives him back $\$30$ in chan......Tue, 25 Aug 2026 11:04:44 -0400https://pop.com.sg/simple-double-summation.htmlhttps://pop.com.sg/simple-double-summation.htmlI've seen how nesting works with a simple $(i+j)$ but this problem below is tripping me up. It's either because of the multipliers or because they each start at zero but I get 60, and the answer I ...Tue, 25 Aug 2026 10:59:18 -0400https://pop.com.sg/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlhttps://pop.com.sg/probability-density-of-the-maximum-of-samples-from-a-uniform-distribut.htmlSuppose $$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$ and suppose $$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$ How would I find the probability density of $\hat\theta$? Thank you!...Tue, 25 Aug 2026 10:55:53 -0400https://pop.com.sg/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlhttps://pop.com.sg/how-to-quickly-tell-if-a-discontinuity-is-removable-or-non-removable.htmlGiven any function where you're asked to find all points of discontinuity and to classify those points, is there a quick way to look at it and tell if that discontinuity is removable or not?...Tue, 25 Aug 2026 10:52:43 -0400https://pop.com.sg/reverse-of-modulo-operator.htmlhttps://pop.com.sg/reverse-of-modulo-operator.htmlI have posted this question originally in Stack Overflow. The question is, "Is there a mathematical approach in getting the reverse of the Modulo Operator with given result $r$ and divisor $d$?" ......Tue, 25 Aug 2026 10:45:02 -0400https://pop.com.sg/tan-2x-sec-2x-express-in-terms-of-sin-cos.htmlhttps://pop.com.sg/tan-2x-sec-2x-express-in-terms-of-sin-cos.htmlI am trying to express this problem in terms of sin/cos and simplify. I couldn't figure out where to go, I tried as best I could. I know the answer is -1 but I am more interested to know how to do ......Tue, 25 Aug 2026 10:39:40 -0400https://pop.com.sg/meaning-of-occurs-for-infinitely-many-and-all-but-finitely-many.htmlhttps://pop.com.sg/meaning-of-occurs-for-infinitely-many-and-all-but-finitely-many.htmlSuppose $P(n)$ is certain statement. I hear all the time that my teachers say $$ P(n) \; \; \text{occurs for infinitely many} \; \; \;n $$ $$ P(n) \; \; \text{for all but finitely many} \; \; n ......Tue, 25 Aug 2026 10:32:08 -0400https://pop.com.sg/rewrite-double-integral-in-polar-coordinates-int-0-3-int-x-3-1-fdydx.htmlhttps://pop.com.sg/rewrite-double-integral-in-polar-coordinates-int-0-3-int-x-3-1-fdydx.htmlI have the following... $$\int_{0}^{3}\int_{x/3}^{1}fdydx$$ I need to rewrite this in polar coordinates. I graphed the triangle and can see that... $$\tan(\theta) = \frac{1}{3}$$ But I do not kn......Tue, 25 Aug 2026 10:26:33 -0400https://pop.com.sg/is-a-decimal-with-a-predictable-pattern-a-rational-number.htmlhttps://pop.com.sg/is-a-decimal-with-a-predictable-pattern-a-rational-number.htmlI'm starting as a private Math tutor for a high school kid; in one of his Math Laboratories (that came with an answer sheet) I was stumped by an answer I encountered in the True or False section (I'm ...Tue, 25 Aug 2026 10:24:46 -0400https://pop.com.sg/non-brute-force-proof-of-multiplicative-property-of-determinants.htmlhttps://pop.com.sg/non-brute-force-proof-of-multiplicative-property-of-determinants.htmlDoes anybody have a "non brute" force way to prove the following for non-singular matrices $A, B$: $$\det(AB) = \det(A) \det(B)$$...Tue, 25 Aug 2026 10:18:44 -0400https://pop.com.sg/what-does-it-mean-for-a-function-to-be-riemann-integrable.htmlhttps://pop.com.sg/what-does-it-mean-for-a-function-to-be-riemann-integrable.htmlI need to come up with a precise mathematical definition of what a Riemann integrable function is. I know what the Riemann integral is but when I look for definitions all I find are proofs of how to ...Tue, 25 Aug 2026 10:04:43 -0400https://pop.com.sg/step-2-in-sard-039-s-theorem-proof-countably-many-neighborhoods.htmlhttps://pop.com.sg/step-2-in-sard-039-s-theorem-proof-countably-many-neighborhoods.htmlI'm studying Sard's Theorem in prof john lee's 'smooth manifolds' The proof contains three step and there is a second step: F($C_k$ \ $C_{k+1}$) is measure zero for smooth map F and $C_k$ = {$x \i......Tue, 25 Aug 2026 10:01:45 -0400https://pop.com.sg/expected-value-for-blackjack.htmlhttps://pop.com.sg/expected-value-for-blackjack.htmlIn the game of blackjack, the odds of winning each hand are slightly less than 50 percent. As you play an infinite amount of hands, you would always lose money because you would win less than 50 pe......Tue, 25 Aug 2026 09:46:33 -0400https://pop.com.sg/continuous-function-on-0-1-f-0-f-1.htmlhttps://pop.com.sg/continuous-function-on-0-1-f-0-f-1.htmlI came across this very interesting question, which seems to be partially answered in a couple posts around here: Let $f:[0,1]\rightarrow\mathbb{R}$ continuous such that $f(0)=f(1)$. Then for all $......Tue, 25 Aug 2026 09:38:36 -0400https://pop.com.sg/integrating-cubic-spline.htmlhttps://pop.com.sg/integrating-cubic-spline.htmlSuppose that I have a finite number of samples of the function $f(x)$ for the range $a < x < b$. I don't have the expression for $f(x)$ but I can do a cubic spline interpolation. If I want to ...Tue, 25 Aug 2026 09:36:05 -0400https://pop.com.sg/math-symbol-for-where.htmlhttps://pop.com.sg/math-symbol-for-where.htmlPretty much any math text I've read introduces notation through a similar format of equation-notation or notation-equation form e.g ax+b+c where a = ..., b = ..., etc. or Let a = ..., b =......Tue, 25 Aug 2026 09:35:37 -0400https://pop.com.sg/finding-real-and-imaginary-part-of-exponential-function.htmlhttps://pop.com.sg/finding-real-and-imaginary-part-of-exponential-function.htmlCan someone explain to me how I find the real and the imaginary part of $e^{\theta i}$? I'm learning complex numbers but I don't quite understand how $e$ is intertwined in all this....Tue, 25 Aug 2026 09:34:47 -0400https://pop.com.sg/if-there-are-7-runners-in-a-race-how-many-different-orderings-can-ther.htmlhttps://pop.com.sg/if-there-are-7-runners-in-a-race-how-many-different-orderings-can-ther.htmlIf there are 7 runners in a race, how many different orderings of 1st, 2nd, and 3rd place can there be? I did 7*6*5 to get 210. I did this because there are 7 runners that can get first, after one ...Tue, 25 Aug 2026 09:18:13 -0400https://pop.com.sg/proving-standard-properties-of-sine-and-cosine-defined-by-their-power-.htmlhttps://pop.com.sg/proving-standard-properties-of-sine-and-cosine-defined-by-their-power-.htmlDefinition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x......Tue, 25 Aug 2026 09:08:10 -0400https://pop.com.sg/what-allows-us-to-use-imaginary-numbers.htmlhttps://pop.com.sg/what-allows-us-to-use-imaginary-numbers.htmlWhat axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers? In the beginning, when there were just reals, these operations were defined for them. T......Tue, 25 Aug 2026 09:02:41 -0400https://pop.com.sg/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-d.htmlhttps://pop.com.sg/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-d.htmlWe add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers). Why this trick i......Tue, 25 Aug 2026 09:02:33 -0400https://pop.com.sg/why-is-the-period-of-sin-2x-is-pi.htmlhttps://pop.com.sg/why-is-the-period-of-sin-2x-is-pi.htmlIn my textbook, it is stated that the period of $\sin(2x)$ is $\pi$. The author justifies this using a mathematical statement which I cannot understand. He writes that, since $\sin(2x) = \sin(2x+2\......Tue, 25 Aug 2026 08:56:27 -0400https://pop.com.sg/concrete-category-and-abstract-category.htmlhttps://pop.com.sg/concrete-category-and-abstract-category.htmlFrom Analysis and Its Foundations By Eric Schechter: A concrete category consists of a collection of objects and a collection of morphisms. I am curious what an "abstract category" is? I foun......Tue, 25 Aug 2026 08:56:08 -0400https://pop.com.sg/given-the-exponential-equation-4-x-64-what-is-the-logarithmic-form-of-.htmlhttps://pop.com.sg/given-the-exponential-equation-4-x-64-what-is-the-logarithmic-form-of-.htmlGiven the exponential equation $4^x = 64$, what is the logarithmic form of the equation in base $10$? Would it be $\frac{\log_{10}64}{\log_{10}4}$?...Tue, 25 Aug 2026 08:50:58 -0400https://pop.com.sg/why-is-a-constant-sequence-a-cauchy-sequence.htmlhttps://pop.com.sg/why-is-a-constant-sequence-a-cauchy-sequence.htmlI think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence: The sequence $\{q_n\}$ of rationals is a Cauchy sequence if, for every $\epsilon>0$, where $\e......Tue, 25 Aug 2026 08:40:16 -0400https://pop.com.sg/proving-two-lines-don-039-t-intersect.htmlhttps://pop.com.sg/proving-two-lines-don-039-t-intersect.htmlThe line $l_1$ passes through the position vector $i-j-2k$ and is parallel to the vector $3i-4j-2k$. The line $l_2$ passes through the position vector $(1+5\cos t)i-(1+5\sin t )j-14k$ where $0≤ t≤......Tue, 25 Aug 2026 08:36:30 -0400https://pop.com.sg/all-girls-are-liked-by-some-boys-no-boy-likes-a-sulk-no-girl-is-a-sulk.htmlhttps://pop.com.sg/all-girls-are-liked-by-some-boys-no-boy-likes-a-sulk-no-girl-is-a-sulk.htmlAll girls are liked by some boys. No boy likes a sulk. No girl is a sulk. How do you write the above in predicate logic statements without starting with a negation? My guess is below. Let me kno......Tue, 25 Aug 2026 08:36:14 -0400https://pop.com.sg/maclaurin-series-of-ln-1-z.htmlhttps://pop.com.sg/maclaurin-series-of-ln-1-z.htmlI'm not sure how to start on this problem other than the definition of $\ln(z)$, which is: $$\ln(1-z) = \ln|1-z| + i(\arg z)$$ or I don't know what to make of $\ln(1-z)$....Tue, 25 Aug 2026 08:29:09 -0400https://pop.com.sg/exact-values-of-hyperbolic-trig-functions.htmlhttps://pop.com.sg/exact-values-of-hyperbolic-trig-functions.htmlThere are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the ...Tue, 25 Aug 2026 08:29:01 -0400https://pop.com.sg/what-is-the-sum-of-all-real-numbers-from-0-to-1-closed.htmlhttps://pop.com.sg/what-is-the-sum-of-all-real-numbers-from-0-to-1-closed.htmlI wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it....Tue, 25 Aug 2026 08:26:44 -0400https://pop.com.sg/consistent-matlab-plot-size-w-different-axis-labels.htmlhttps://pop.com.sg/consistent-matlab-plot-size-w-different-axis-labels.htmlI'm trying to specify the sizes of two MATLAB figures in such a way that the graphs appear to be the same size, regardless of extra space used outside the axes. Specifically, I'm want the white re......Tue, 25 Aug 2026 08:17:01 -0400https://pop.com.sg/unary-predicates-theory-with-no-principal-types.htmlhttps://pop.com.sg/unary-predicates-theory-with-no-principal-types.htmlDefinition Take a satisfiable set of $L$-sentences $\Sigma$ and variables $x = x_1, \ldots, x_n$. Denote by $S_x(\Sigma)$ the set of all $\Sigma$-realizable $x$-types in $L$. A type $p(x) \in S_x(......Tue, 25 Aug 2026 08:14:18 -0400https://pop.com.sg/compound-interest-formula-with-regular-deposits-solve-for-time.htmlhttps://pop.com.sg/compound-interest-formula-with-regular-deposits-solve-for-time.htmlI cant seem solve the following problem due to my bad memory of logarithms if someone opens an account with initial balance of $B$ interest rate of $R$ and does a constant monthly deposit of $D$ ......Tue, 25 Aug 2026 08:09:07 -0400https://pop.com.sg/how-to-calculate-the-perceived-frequency-of-two-sinusoidal-waves-added.htmlhttps://pop.com.sg/how-to-calculate-the-perceived-frequency-of-two-sinusoidal-waves-added.htmlIf you add together two sinusoidal waves of different frequencies, how do you calculate the frequency of the resulting function as perceived by a human?...Tue, 25 Aug 2026 08:08:22 -0400https://pop.com.sg/differences-between-real-and-complex-analysis.htmlhttps://pop.com.sg/differences-between-real-and-complex-analysis.htmlTo start with, real analysis deals with numbers along the (one dimensional) number line, while complex analysis deals with numbers along two dimensions, real and imaginary, Cartesian style. Could t......Tue, 25 Aug 2026 08:03:16 -0400https://pop.com.sg/l-039-hospital-039-s-rule-doesn-039-t-converge-for-a-function-with-squ.htmlhttps://pop.com.sg/l-039-hospital-039-s-rule-doesn-039-t-converge-for-a-function-with-squ.htmlI was trying to find the limit of a function of the form $\frac{x}{\sqrt{(x+a)(x+b)}}$. When I apply L'Hospital's rule and differentiate the numerator and denominator, after simplification I end up......Tue, 25 Aug 2026 07:56:44 -0400https://pop.com.sg/what-is-the-name-of-the-answer-to-exponentiation.htmlhttps://pop.com.sg/what-is-the-name-of-the-answer-to-exponentiation.htmlWhat is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to exponenti......Tue, 25 Aug 2026 07:55:56 -0400https://pop.com.sg/ferris-wheel-problem.htmlhttps://pop.com.sg/ferris-wheel-problem.htmlOne of the largest ferris wheel ever built is in the british airways london eye which was completed in 2000. The diameter is 135 m and passengers get on at the bottom 4 m above the ground. The wheel ...Tue, 25 Aug 2026 07:46:55 -0400https://pop.com.sg/quadratic-lyapunov-function.htmlhttps://pop.com.sg/quadratic-lyapunov-function.htmlConsider the nonlinear model of a chemical process: $$ \begin{align} \dot{x}_1 &= x_2\\ \dot{x}_2 &=-5x_1 -3x_2\left(1 - \frac{0.6x_2}{1 + x_2^2}\right). \end{align} $$ Writing this system in ...Tue, 25 Aug 2026 07:45:23 -0400https://pop.com.sg/which-of-the-following-statements-are-false.htmlhttps://pop.com.sg/which-of-the-following-statements-are-false.htmlAll of the following sets are subsets of positive integers. $A = \{x\mid x\ \text{is divisible by 2}\} \\ B = \{x\mid x\ \text{is divisible by 4}\} \\ C = \{x\mid x\ \text{is divisible by 6}\}$ B......Tue, 25 Aug 2026 07:37:22 -0400https://pop.com.sg/annualized-probability-of-loan-default.htmlhttps://pop.com.sg/annualized-probability-of-loan-default.htmlCan someone help with how to calculate the annualized probability of a loan default given: 70% probability of survival (30% default) over the next 20 months? Edit: I should have been more specif......Tue, 25 Aug 2026 07:16:14 -0400https://pop.com.sg/prove-disprove-that-if-two-sets-have-the-same-power-set-then-they-are-.htmlhttps://pop.com.sg/prove-disprove-that-if-two-sets-have-the-same-power-set-then-they-are-.htmlI am really sure that if two sets have the same power set, then they are the same set. I just am wondering how does one exactly go about proving/showing this? I'm usually wrong, so if anyone can s......Tue, 25 Aug 2026 07:06:37 -0400https://pop.com.sg/continuity-essential-supremum-and-supremum.htmlhttps://pop.com.sg/continuity-essential-supremum-and-supremum.htmlTake $f: \mathbb{R}\to \mathbb{R}$ be continuous, then w.r.t. Lebesgue measure $L$, we have $$\operatorname{ess sup} |f|=\sup |f|$$ w.r.t. $L$. I have been provided a proof in a Measure theory co......Tue, 25 Aug 2026 06:54:15 -0400https://pop.com.sg/runge-kutta-and-butcher-table.htmlhttps://pop.com.sg/runge-kutta-and-butcher-table.htmlIn the Wikipedia article on Runge-Kutta methods, there is a notation explained using a Butcher table with a $c_{i}$ vector (nodes), a $b_{i}$ vector (weights) and a runge-kutta matrix $a_{ij}$. My ...Tue, 25 Aug 2026 06:52:22 -0400https://pop.com.sg/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlhttps://pop.com.sg/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlWhen rolling two dice, which of the following events are independent of the event that the first die is 4: (a) The second is 2 (b) The sum is 6 (c) The sum is 7 (d) The sum is even I thought only o......Tue, 25 Aug 2026 06:49:59 -0400https://pop.com.sg/proof-of-polya-039-s-theorem-probability-theory.htmlhttps://pop.com.sg/proof-of-polya-039-s-theorem-probability-theory.htmlI know a theorem called Polya's theorem: $X_n \rightarrow X$ in distribution as $n\rightarrow \infty$ is equivalent to $\sup_{x} | F_n(x) -F(x)| \rightarrow 0$ as $n \rightarrow \infty$, where $F_n......Tue, 25 Aug 2026 06:47:51 -0400https://pop.com.sg/two-system-of-linear-equation-have-same-solution-if.htmlhttps://pop.com.sg/two-system-of-linear-equation-have-same-solution-if.htmlTwo system of linear equation have same solution if and only if each equation in each system is a linear combination of the equations in other system? I didn't get it what are they trying to say?...Tue, 25 Aug 2026 06:43:40 -0400https://pop.com.sg/find-a-formula-for-cos-5x-in-terms-of-sin-x-and-cos-x.htmlhttps://pop.com.sg/find-a-formula-for-cos-5x-in-terms-of-sin-x-and-cos-x.htmlI was asked to find a formula for $\cos(5x)$ in terms of $\sin(x)$ and $\cos(x)$. I tried to use the formula $\cos(5x) + i\sin(5x) = (\cos(x)+i\sin(x))^5 $ and what I get is $16i\sin^5(x) - 2......Tue, 25 Aug 2026 06:42:13 -0400https://pop.com.sg/how-do-i-find-a-closed-form-of-the-characteristic-function-of-gamma-di.htmlhttps://pop.com.sg/how-do-i-find-a-closed-form-of-the-characteristic-function-of-gamma-di.htmlIt’s written in wikipedia that $(1-i \beta t)^{-\alpha}$ is the characteristic function of the Gamma distribution $Gamma(\alpha,\beta)$. I tried to prove this but I failed. So I googled it for a pr......Tue, 25 Aug 2026 06:36:52 -0400https://pop.com.sg/understanding-the-definition-of-a-local-minimizer.htmlhttps://pop.com.sg/understanding-the-definition-of-a-local-minimizer.htmlWhat does the following notations mean? As far as I understood, Suppose that $f$ is a function which takes $n$ real values and spits out one real value where those input real values are ...Tue, 25 Aug 2026 06:31:05 -0400https://pop.com.sg/what-039-s-the-different-among-the-concepts-probability-possibility-an.htmlhttps://pop.com.sg/what-039-s-the-different-among-the-concepts-probability-possibility-an.htmlCan you please explain the difference among the concepts Probability, Possibility and Belief through some simple examples?...Tue, 25 Aug 2026 06:29:25 -0400https://pop.com.sg/how-to-find-the-radius-of-convergence-and-the-interval-of-convergence-.htmlhttps://pop.com.sg/how-to-find-the-radius-of-convergence-and-the-interval-of-convergence-.htmlFor this question, I'm stuck on finding the radius of convergence and interval of convergence for the power series. Here is what I have so far. Can anyone please help me out? Find the radius of ...Tue, 25 Aug 2026 06:22:11 -0400https://pop.com.sg/what-is-the-period-and-phase-shift-of-f-x-3-sin-4-left-x-frac-pi-4-rig.htmlhttps://pop.com.sg/what-is-the-period-and-phase-shift-of-f-x-3-sin-4-left-x-frac-pi-4-rig.htmlFor the function what is the period and the phase shift respectively are:? $$f(x) = 3\sin 4\left(x +\frac{\pi}{4}\right) - 1$$ My attempt: It is wrong though. $y = a\sin b(x\pm h) \pm k$ $a$ : ...Tue, 25 Aug 2026 06:14:58 -0400https://pop.com.sg/dividing-each-term-by-the-highest-degree-in-the-denominator-of-a-limit.htmlhttps://pop.com.sg/dividing-each-term-by-the-highest-degree-in-the-denominator-of-a-limit.htmlLimits that involve infinity are not a problem for me, but there is an idea that is used in rational limits that I never really understood why. The so called "rule" says that given a rational ...Tue, 25 Aug 2026 06:07:27 -0400https://pop.com.sg/what-is-wrong-with-my-approach-to-solving-x-log25-25-log-x-10.htmlhttps://pop.com.sg/what-is-wrong-with-my-approach-to-solving-x-log25-25-log-x-10.htmlFound this equation on the web: $$x^{\log25} + 25^{\log x} = 10$$ The person solved by substitution and got $x = \sqrt{10}$ which satisfies the equation. I tried different ways after following the ......Tue, 25 Aug 2026 06:04:23 -0400https://pop.com.sg/how-do-you-calculate-the-modulo-of-a-high-raised-number.htmlhttps://pop.com.sg/how-do-you-calculate-the-modulo-of-a-high-raised-number.htmlI need some help with this problem: $$439^{233} \mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks....Tue, 25 Aug 2026 06:01:44 -0400https://pop.com.sg/what-is-the-difference-between-finite-difference-methods-finite-elemen.htmlhttps://pop.com.sg/what-is-the-difference-between-finite-difference-methods-finite-elemen.htmlCan you help me explain the basic difference between FDM, FEM and FVM? What is the best method and why? Advantage and disadvantage of them?...Tue, 25 Aug 2026 06:01:39 -0400https://pop.com.sg/getting-the-inverse-of-a-lower-upper-triangular-matrix.htmlhttps://pop.com.sg/getting-the-inverse-of-a-lower-upper-triangular-matrix.htmlFor a lower triangular matrix, the inverse of itself should be easy to find because that's the idea of the LU decomposition, am I right? For many of the lower or upper triangular matrices, often I ......Tue, 25 Aug 2026 05:51:51 -0400https://pop.com.sg/fat-cantor-set.htmlhttps://pop.com.sg/fat-cantor-set.htmlSo the standard Cantor set has an outer measure equal to $0$, but how can you construct a "fat" Cantor set with a positive outer measure? I was told that it is even possible to produce one with an ......Tue, 25 Aug 2026 05:50:55 -0400https://pop.com.sg/difference-between-nonpositive-and-negative-numbers.htmlhttps://pop.com.sg/difference-between-nonpositive-and-negative-numbers.htmlI am wondering if there is any difference between non-positive and negative numbers? I think that negative numbers mean "negative real numbers" and "Non-positive numbers" are negative real numbe......Tue, 25 Aug 2026 05:48:26 -0400https://pop.com.sg/continuous-rotation-formula-of-turning-particle.htmlhttps://pop.com.sg/continuous-rotation-formula-of-turning-particle.htmlContext I am writing a "flight" trajectory program in Mathematica that ignores all physics. I model a turning aircraft (pitching at full deflection) as a parametrised circle. I have been able to ...Tue, 25 Aug 2026 05:40:50 -0400https://pop.com.sg/how-to-use-lemniscate-sine-and-lemniscate-cosine-elliptic-integrals.htmlhttps://pop.com.sg/how-to-use-lemniscate-sine-and-lemniscate-cosine-elliptic-integrals.htmlI’ve been reading up on lemniscate sine and cosine functions and found that they are essentially the lemniscate analogues of circular sine and cosine functions. The following wiki page elaborates a......Tue, 25 Aug 2026 05:30:25 -0400https://pop.com.sg/what-does-multiple-edges-mean-in-simple-graph-definition.htmlhttps://pop.com.sg/what-does-multiple-edges-mean-in-simple-graph-definition.htmlFrom Wolfram Alpha : A simple graph, also called a strict graph, is an unweighted, undirected graph containing no graph loops or multiple edges. Unless stated otherwise, the unqualified term "...Tue, 25 Aug 2026 05:28:03 -0400https://pop.com.sg/the-shortest-distance-between-two-parallel-lines.htmlhttps://pop.com.sg/the-shortest-distance-between-two-parallel-lines.htmlI was working on a set of problems involving finding the shortest distance between two skew lines, which was fine, but then parallel lines showed up. In essence this should be much easier to solve ......Tue, 25 Aug 2026 05:05:26 -0400https://pop.com.sg/how-to-find-the-analytical-expression-for-the-supporting-hyperplane.htmlhttps://pop.com.sg/how-to-find-the-analytical-expression-for-the-supporting-hyperplane.htmlIn Stephen Boyd & Lieven Vandenberghe's Convex Optimization book we have following set $$\{x\in \mathbb R^2_+ \mid x_1x_2\geq1\}$$ Now the analytical expression for the supporting hyperplane......Tue, 25 Aug 2026 05:03:36 -0400https://pop.com.sg/when-to-use-washers-method-and-when-to-use-shells-method.htmlhttps://pop.com.sg/when-to-use-washers-method-and-when-to-use-shells-method.htmlI know that I can use either washers method or shells method. I have a few questions which I have googled but I did not receive much luck. How do I decide in which case which method is easier and......Tue, 25 Aug 2026 05:01:43 -0400https://pop.com.sg/what-is-foundations-of-mathematics-and-why-isn-039-t-it-taught-more-of.htmlhttps://pop.com.sg/what-is-foundations-of-mathematics-and-why-isn-039-t-it-taught-more-of.htmlI have just come across "Foundations of Mathematics". Wikipedia describes it as Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical ...Tue, 25 Aug 2026 04:58:25 -0400https://pop.com.sg/what-exactly-is-a-tensor.htmlhttps://pop.com.sg/what-exactly-is-a-tensor.htmlI've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...Tue, 25 Aug 2026 04:55:16 -0400https://pop.com.sg/how-to-calculate-the-norm-of-an-ideal.htmlhttps://pop.com.sg/how-to-calculate-the-norm-of-an-ideal.htmlWould someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly. For example, I know that the norm $N_\mathbb{Q(\sqrt10)}(\langle2,\sqrt10\......Tue, 25 Aug 2026 04:37:30 -0400https://pop.com.sg/help-with-using-euler-s-formula-to-prove-that-cos-2-theta-frac-cos-2-t.htmlhttps://pop.com.sg/help-with-using-euler-s-formula-to-prove-that-cos-2-theta-frac-cos-2-t.htmlI have to use Euler's Formula to prove that: $$\cos^2(\theta) = \frac{\cos(2\theta)+1}{2}.$$ I have managed to prove this using trigonometric identities but I'm not sure how to use Euler's Formul......Tue, 25 Aug 2026 04:33:12 -0400https://pop.com.sg/question-about-an-isometric-immersion.htmlhttps://pop.com.sg/question-about-an-isometric-immersion.htmlThis is the question: Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur. I thought ab......Tue, 25 Aug 2026 04:30:57 -0400https://pop.com.sg/complex-number-solutions-for-a-circle.htmlhttps://pop.com.sg/complex-number-solutions-for-a-circle.htmlI have a circle of radius 5 with its center at the origin represented as $X^2+Y^2=25$. I get that it has a solution for all values ranging from $-5$ to $+5$. My question is what does it mean when ......Tue, 25 Aug 2026 04:20:28 -0400https://pop.com.sg/is-there-a-pythagorean-triple-such-that-all-angles-of-the-correspondin.htmlhttps://pop.com.sg/is-there-a-pythagorean-triple-such-that-all-angles-of-the-correspondin.htmlObviously, the most interesting pythagorean triple $(a, b, c)$ would be one for which the corresponding triangle (with integer side lengths $a, b, c$) has angles 90°, 60° and 30° ($\frac{\pi}{2}, \......Tue, 25 Aug 2026 04:11:44 -0400https://pop.com.sg/what-is-the-output-of-a-function-which-is-not-defined-for-input-0-for-.htmlhttps://pop.com.sg/what-is-the-output-of-a-function-which-is-not-defined-for-input-0-for-.htmlMy teacher told me that every function has to be able to assign an output to every input we give to this function. What is than an output for a function which is not defined in zero? Nothing? Or do ...Tue, 25 Aug 2026 04:02:37 -0400https://pop.com.sg/hatcher-and-house-with-two-rooms.htmlhttps://pop.com.sg/hatcher-and-house-with-two-rooms.htmlOn page 4 of Hatcher's "Algebraic Topology" he constructs the "house with two rooms" space. He claims that there is some neighborhood containing this space that is homeomorphic to the unit ball (in $\...Tue, 25 Aug 2026 03:55:17 -0400https://pop.com.sg/proof-verification-for-proof-by-contraposition.htmlhttps://pop.com.sg/proof-verification-for-proof-by-contraposition.htmlprove using contraposition that if a and b are integers: $a^2-4b-2\neq0$ So the contrapositive: $a^2-4b-2=0$ implies that at least one of either a or b are not integers Working from the above, I go......Tue, 25 Aug 2026 03:50:14 -0400https://pop.com.sg/how-to-calculate-15-without-using-calculator.htmlhttps://pop.com.sg/how-to-calculate-15-without-using-calculator.htmlI am joining a maths competition and recently I am preparing for it. I came across a question that asks me to fill the blank of a number: 1_0767436_000 And this number is the product of $15!= 15\...Tue, 25 Aug 2026 03:44:10 -0400https://pop.com.sg/why-is-y-sqrt-x-4-a-function-and-y-sqrt-4-x-2-should-be-a-circle.htmlhttps://pop.com.sg/why-is-y-sqrt-x-4-a-function-and-y-sqrt-4-x-2-should-be-a-circle.htmlSo why is it a function, even though for example $x = 8$; you'll have $y = +2$ and $y = -2$. It'll fail the vertical line test. But every textbook considers it as a function. Did I misunderstand ...Tue, 25 Aug 2026 03:28:43 -0400