# Tag Info

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Clearly $|z|^5$ is non-negative and real so any solution to the equation $|z|^5 = z^5$ has that $z^5$ is non-negative and real. Furthermore, any non-negative and real number clearly satisfies the equation. So it is sufficient and necessary that $z^5$ is non-negative and real. Zero is one solution. As for the remaining solutions, recall that in the complex ...

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Here are some hints. To answer this question, you can go a couple of ways. You can solve the equation for $x$ by taking the square root of both sides. (Remember that the square root of $a^2$ is both $a$ and $-a$.) This gives you a direct test of the inequality (in terms of $x$ rather than $x^2$). Or, you can substitute in the values and test the ...

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Hint: Left side is real non-negative, so the same is true for $z^5$. Use the $r,\phi$ notation of complex number and geometric interpretation of a product.

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Given: $x^2\ge25$ By brute force, let's square each possible answer: A. 25 B. 490,000 C. 25 D. 390,625 E. 390,625 F. 15,625 Thus, all will work. Alternatively, $|x|\ge5$ which could be interpreted as $x\ge5$ for $x\ge0$ and $x\le-5$ for $x<0$ would also be another approach to solve the problem.

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HINT: $$x^2 \geq 25$$ $$\iff |x| \geq \sqrt{25}$$ $$\iff x \geq +5\ \lor\ x \leq -5$$

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$88\%$ of $x$ is $10000$, then $x=\dfrac{10000}{88\%}=10000\cdot\dfrac{100}{88}=11363.63$. Check: $88\%$ of $11363.63$ is $9999.99$.

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see its quite simple sorry for misinterpreting first time. so here we assume total amount to be $x$ thus you want $12\%$ commision of $x$ and extra $10K$. so our equation becomes $\frac{12x}{100}+10000=x$ thus on solving you get $x=11363.63..$ now i hope i have interpreted your question correctly.

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One may observe that, in general, $$x - a\%\:x=b$$ is just $$\left(1-\frac{a}{100} \right)\times x=b$$ giving, if $a \neq 100$, $$x=\frac{b}{\left(1-a/100\right)}$$

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Personally, I was STUNNED by $$1+2+3+4\cdots=-1/12$$ This undoubtedly sparked my interest in mathematics. (Although I didn't know it then, this is a zeta-regularized sum)

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I can say that it is fairly true that when people (like Ramanujan himself), does not have a proper scholar background, might have a tendency to obtain a result, let us say "empirically". Even people like Carl Frederick Gauss, with proper scholar formation, got some of his most famous results that way (I am thinking on the Prime Number Theorem, as mentioned, ...

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$\begin{bmatrix} a_{1}\\ b_{1} \\ \end{bmatrix} =\begin{bmatrix} 5 & 1\\ -1 & 3 \\ \end{bmatrix} \begin{bmatrix} a_0\\ b_0 \\ \end{bmatrix} + \begin{bmatrix} 1 \\ -1 \\ \end{bmatrix}= \begin{bmatrix} 5a_0 + b_0+1\\ -a_0+3b_0 -1 \end{bmatrix}$ $\begin{bmatrix} a_{2}\\ b_{2} \\ \end{bmatrix} =\begin{bmatrix} 5 & 1\\ -1 & 3 \\ \end{bmatrix} ... 3 I would argue, as you say, that knowledge is worth something even in absence of practical applications. However, that is not to say that theoretical mathematics has no practical applications. In my opinion the following is the main difference between applications of "theoretical" and "applied" mathematics: Pure mathematics is an investment for the far ... 2 TLDR : if the result is more important to you, go applied. If the road-to-result is more important, go pure. I'm sure you are aware that behind many world-changing inventions lie an accumulation of "pure math" results, that seemed useless at their time. Traditional examples include number theory results that are used in cryptography, or Turing's idea of a ... 5 Being successful at Olympiad mathematics is certainly correlated with being successful in later studies and research, but there is no implication in either direction. This is what you would expect a priori: coming up with creative ideas is a part of the work of a research mathematician, but by no means the only (and arguably not the most important) part. ... 8 I am not one thousandth the mathematician that Terry Tao is, but my own feeling is rather different. I had a college classmate who was far better at competition mathematics than I was, and when we went to grad school (together), he seemed good at following a prescribed path, but not so good at striking out on his own. In later professional life, he made no ... 9 Find Terry Tao's blog. He talks about his experience of learning at different levels of mathematics education. Among other insights, he writes how patterns from competition problems he later discovered to be examples of more general, deep and beautiful results. What I took away from all that was that while solving competition problems isn't directly ... 2 John Nash struggled with significant mental health issues when he should have been in the twilight of his mathematical journey. He and his wife spent many difficult years battling with this illness. Slowly, Nash started to get back in touch with the mathematical community in Princeton; engaging with the students, his passion for mathematics never died. It ... -2 Grigori Perelman who possibly proved a generalization of the Poincare Conjecture - the Thurston Geometrization Conjecture ( has anyone claimed to show any flaw in the proof yet? ). That being said, it seems that it's not very uncommon with odd behaviour amongst great mathematicians. They have a huge uphill struggle. Anything they discover could overshine ... 4 Andrew Wiles with his Fermat proof is an example of massive struggle. He worked on the topic for 9 years and got demolished when presenting an erroneous proof after 7 years. There were several additional problems, but I forgot the details. You can read about it on Wiki and in much detail in the very accessible "Fermat's Last Theorem" by Simon Singh. 8 The analytic number theorist Hua Loo-Keng overcame abject poverty, handicappedness, political persecution; for more information refer to one of his faithful biography. Charles Hermite overcame much too, but in different aspects; he failed nearly every math exam that he was to take. To supplement, the analytic number theorist Chen Jing-Run, the man closest ... 8 Srinivasa Ramanujan was such a mathematician. He failed to got admitted to college but he became one of the best mathematician of$20$th century. Evariste Galois failed to enter to Ecole Polytechnique twice. 0 Quote from On Riemann-Liouville and Caputo Derivatives: In the realm of the fractional differential equations, Caputo derivative and Riemann-Liouville ones are mostly used. It seems that the former is more welcome since the initial value of fractional differential equation with Caputo derivative is the same as that of integer differential equation; ... 3 Here are some examples: Ian Roulstone, John Norbury: Invisible in the Storm: The Role of Mathematics in Understanding Weather Vladimir Arnold: Catastrophe Theory Julian Havil: GAMMA David Harel: Computers Ltd George Szpiro: Kepler's Conjecture Malba Tahan: The Man Who Counted: A Collection of Mathematical Adventures. 2 I've only read the first couple chapters (so far), but I really like the Springer Undergrad Texts in Mathematics book Mathematics and its History by John Stillwell Also, I don't think you can do wrong with Newton's Philosophiae Naturalis Principia Mathematica - 'twas the book that first roused my interest in matters physick and mathematick. As well, I ... 0 Can't believe no one has (yet) mentioned George Polya's incomparable How to Solve It. It's very dissimilar to Feynman in that it covers very few (if any) specific subfields of mathematics - but it's very similar in that it attempts to give the student an understanding of how to approach the discipline, and to build their intuition so they can grapple with ... 2 My recommendations Taming the infinite by Ian Stewart. The great mathematical problems by Ian Stewart. Does God play dice by Mario Livio Golden Ratio by Mario Livio 1 Mathematics by David Bergamini is good. Some of it (especially the parts about computers) is dated, but much of it is just as valid today as it ever was. 4 I would recommend these books: Journey through Genius Dr. Euler's Fabulous Formula Prime Obsession The Music of the Primes Gödel's Proof (by Ernest Nagel) The Code Book 0 I think the way you read mathematical books is an interesting way to grasp all the aspects of the theory you are studying. There are often different kinds of exercises: Firstly, there are exercises made to understand the subtlety of a particular definition and these exercises don't require the knowledge of results inherent to the theory you are studying. ... 0 Just some pointers to the development of the field - there is formalized mathematics branch with the de Gryuter journal "Formalized Mathematics". There is univalent foundation and homotopical type theory as new foundation and formalization of mathematics. Coq and Agda are practical proof assistants in this field. There is also Mizra system. The question ... 0 The purpose of exercises is to solidify your understanding of the definitions and help you in thinking about the big picture. Your third plan is what we do in advanced math when we want to self study. Of course we would also spend a decent amount of time getting through a few basic problems, a mid-level problem and trying a hard problem. Your plan 2 is ... 0 An interesting paper dealing with a random walk is M. Orkin and R. Kakigi, "What is the worth of free casino credit," The American Mathematical Monthly Vol. 102 (1995). This application is also discussed in the book Understanding Probability by Henk Tijms. Chapter 5 of this book gives several other applications of random walks that are of interest in ... 2 There already is at least one such list, compiled by Stephen Smale. See here 0 Some obfuscation using linear algebra: Write$x + \frac{1}{x} = a$and let $$p(\lambda) = \left( \lambda - x \right) \left( \lambda - \frac{1}{x} \right) = \lambda^2 - a\lambda + 1$$ be a polynomial whose roots are$x$and$\frac{1}{x}$and consider the companion matrix $$A = \left( \begin{matrix} 0 & -1 \\ 1 & a \end{matrix} \right).$$ The ... 0 Not sure how to show what the continued fraction equals but here's my attempt. Again similar in spirit to the algebraic method. Solving for$x$in the first equation we have:$x = 4 - \frac{1}{x}$. Substituting this into the second equation:$ (4 - \frac{1}{x})^{2} + \frac{1}{(4 - \frac{1}{x})^{2}} = 16 - \frac{8}{4-\frac{1}{x}} + \frac{1}{ 16 - ...

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Here is a mildly disguised version of the same idea. Let $x=e^u$ Then we have $$\cosh u=2$$ And we want $$x^2+\frac {1}{x^2}=2\cosh 2u=2(2\cosh^2 u-1)=2(8-1)=14$$

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When you come across a concept in a math book/ lecture that you don't think is being defined rigorously you have two options (that I can think of): Write it down and move on. Sometimes professors/ authors are more concerned with making sure you can compute something than whether you see all of the intricacies involved. Sometimes you just don't have all ...

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Student Exercise Tasks (for Mathematics, Language Arts, etc.) - autocorrected raw url: http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html (hot link) In fact, you are free to copy the entire website and alter it to fit your individual needs, if you wish.

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Hint.-I would do it this way: You can start with a lesson about countable and uncountable sets. Rinse thoroughly that $\mathbb Q$ is countable and $\mathbb R$ is uncountable. Then tell them the numbers with finite decimal expansion form a "very small" subset of $\mathbb Q$. Play with that "very small" (it is infinite and equals $\mathbb Z[\frac ... 0 Well, depends what the meaning of "disaster" is. In the rationals,$ \mathbb Q$, every non-zero quantity is invertible. Any fraction can be expressed as a finite or infinite repeating decimal. Thus$1.5$has as inverse$0.(6)$. In a more theoretical POV, any linear transformation with coefficients in$ \mathbb Q$acting on elements of$ \mathbb Q$will ... 0 When the system is in motion the forces acting towards$O$are $$xm_2\omega^2 = 10m_1\omega^2$$ As$\omega$is equal for both masses, we can cancel$\omega$and multiply both sides by$g$, which shows, in general, that the moments about$O$, of each weight, are equal when the system is at rest. 2 One of my teachers always told me "don't know definitions, don't know math." At the time I was pretty annoyed, but he was completely right. The only way to learn math is to have the fundamentals down cold. This involves both a rigorous side, (memorizing them is a good start) and an intuitive side. So at an entry level, I strongly recommend spending a long ... 1 Intuition and logic are not the same thing. Take, for example, the idea that $$\lim_{x\to\infty} \frac{1}{x}=0$$ What does this mean? Intuitively, you can imagine a graph of the function and see that it gets closer and closer to$0$, but who's to say that the limit isn't actually$0.0001\$? To show that this isn't the case, you need a formal definition of ...

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Why not look at a multiplication table? Let's make a little one, including some negative numbers. You could of course make it bigger to make the patterns clearer. Let's start with what we already know: $$\begin{array}{|c|c|c|c|c|c|} \hline &\textbf{-2}& \textbf{-1} & \textbf{0} & \textbf{1} & \textbf{2} \\ \hline \textbf{-2} & & ... 1 Well, the way I think about it is this. We have the non-negative integers (0,1,2,3,4, etc.). We introduce the negative numbers and need to define multiplication with negative numbers so that we have internal consistency. We wish to keep the property that 0*anything = 0, negative or positive. We also want to keep the distributive property. In order to ... 0 "Do the same thing to both sides" also works for other operators - multiply, divide, powers, roots ... you could say it's the master/meta rule of all the others. 3 If your son is clear on the concept of money and knows what a credit card is, this might be a good explanation: Imagine you tell your son that you will buy him \color{green}{\mathrm{seven}} gift vouchers worth £\color{green}{5} each and pay for them using your credit card. Explain to your son that you now owe money, and say that it is 7 \times ... 4 I would go for the flipping explanation of the negative numbers: multiplying with a negative number flips from positive to negative and from negative to positive. Imagine he understands that multiplying with 1 makes no difference, then it's very simple: -1 * 1 = -1 can mean two things (for children, the fact that multiplication is commutative is obvious): ... 4 Your child may have been introduced to the minus sign by means of the word opposite. This is a great term to use in your conversations. Indeed, a and -a are opposites in that they are additive inverses. On the number line, opposite numbers are mirrored in their distances from zero, which provides a nice visual aid as well. We can use the term to describe ... 36 I think a lot of answers are either too simple or stray away from mathematics too much. Just remember that multiplication is repeated addition. When dealing with negative numbers, it becomes repeated subtraction. I'd simply put it in this context: The equation:$$\begin{equation*}\begin{array}{c} \phantom{\times9}2\\ \underline{\times\phantom{9}2}\\ ...

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