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How to answer mathematical questions in the proper way
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44 votes

You mention mathematical writing and symbols/notation. Though I’ve never taken a linear algebra course, here are some relevant tips: Make sure every line of work has a meaningful relation (symbol) in ...

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How to pronounce $\mathcal{E}$?
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13 votes

It depends on your audience and the context. In physics, $\mathcal E$ usually denotes emf, so you would say that. In set theory, $\mathcal P$ usually denotes a power set, so you would say that. ...

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What is the significance of deciding the convention of $1 \text{ radian} = 180 \text{ degrees}$ over $\pi$?
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11 votes

Historically speaking, it is not likely that beautifying the Taylor expansion of sine was “the” reason for defining the radian. After all, the radian was defined by the relationship $r\theta=s$, which ...

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Trouble understanding only one way implication truth
7 votes

$$x=-1 \implies x^2 = 1$$ but $$x=-1 \not\Longleftarrow x^2=1$$ because it could be that $x=1$.

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Why is there no product/quotient rule for integration?
7 votes

Edit: In light of Carmeister’s comment, I would like to clarify that (in my opinion) there does exist an integral analogue to the product rule, and that analogue is integration by parts. ...

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Six dice blank on five sides. How to roll as one?
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6 votes

Roll the dice and eliminate any that show up blank. Continue rolling until one die remains. If all of your remaining dice get eliminated in one throw, then reroll. Edit (thanks Rahul): Eliminate ...

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What is the number of segments in the picture
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6 votes

Look at the middle line of points formed by the intersections of the line segments. Count how many segments run through each point, counting the left- and rightmost points as having one segment ...

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How to show that the infinite sum of $(-1)^x$ diverges
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5 votes

It’s really quite simple: $\lim_{n\to\infty}\left[ (-1)^n\right]$ does not approach $0$; hence $\sum_{n=1}^\infty\left[ (-1)^n\right]$ diverges.

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Best practice for the notation of conditions in cases: 'if' vs 'for' vs ','
5 votes

I agree that there is no formal difference and that either is acceptable provided you are consistent. I also think that punctuation creates unnecessary clutter. However, it is important to keep your ...

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What are Hyperbolic Trig Functions Functions of?
4 votes

The connecting factor amongst the hyperbolic and circular functions and the unit circle and hyperbola is not the angle subtended by the curve but rather the area bounded by it. Here are two images I ...

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Mathematical Entity That Takes a Function and Returns a Function
4 votes

These are often called operators. Some operators take two arguments, called binary, like $()\times()$ or $()-()$; others take only one, called unary, such as $()^2$. Some examples of operators or “...

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How to find the quadratic approximation of a quotient?
4 votes

I’m not entirely sure if this is what you’re asking, but I suppose you could use the first few terms of the Taylor series $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} +\cdots $$ for $|x|<\pi/2$....

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How can points that have length zero result in a line segment with finite length?
4 votes

Perhaps you create a mapping from the length of each segment to the number of segments necessary, namely $n= (1 \ \mathrm{cm})/{\ell}$. Graph this function with $n$ on the vertical axis and $\ell / \...

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Legit Math Software for Mac Users that is Free for H.S. Students
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3 votes

Desmos.com all the way. It does everything you want and more. They also have a mobile app that will sync with your account on desktop. Go check it out.

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e^-it as t approaches infinity
3 votes

Hint: $$e^{iat}=\cos at + i \sin at$$

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Solve $2\log_bx + 2\log_b(1-x) = 4$
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3 votes

You are correct to transform $2\log_b x + 2\log_b (1-x) = 4$ into $x^2-x+b^2=0$. These are the restrictions implicated by the original equation: $b$ is a positive number not equal to $0$ nor $1$ $x&...

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Logical fallacy that suggests $3/9=3/10$
3 votes

The issue here is the difference between equality $=$ and approximation $\approx$. Here is what’s actually happening: $$\frac39 = \frac13 = 0.33333\cdots \approx 0.3$$ The approximation comes when ...

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How to take the derivative of a power.
3 votes

Formulaic approach via the substitution $u=t^3$: $$\begin{align} {d \over dt}\left(2^{t^3}\right) &= {d \over dt}\left( 2^u \right) \\ &= 2^u\ln(2){du \over dt} \\ &= \ln(2)2^{\left(t^3\...

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Find the measure of ∠PRQ, with points $P(0, 3, 0) \;\;\; Q(-3, 4, 2) \;\;\; R(-2, 9, 1) \;$
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3 votes

We can calculate the angle by using the properties of dot products. Since $R$ is the vertex of the angle, $\angle PRQ = \Theta$ is between $\overrightarrow{RP}$ and $\overrightarrow{RQ}$. $$\...

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Product rule, help me understand this proof
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3 votes

When we divide by $\Delta x$, why do we put the $\Delta x$ under the other $\Delta$? Why does it not go under the constant v or u? If you’re asking why we write $u\frac{\Delta v}{\Delta x}$ ...

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Complex Numbers conceptual question
3 votes

Say $Z_1 = a+bi$ and $Z_2 = x+yi$ where $\{a,b,x,y\}\subset\Bbb{R}$. Say $Z_1\overline{Z_2} = R+Ji$ where $\{R,J\}\subset\Bbb R$. The notation $\overline Z$ means the complex conjugate of $Z$. The ...

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Two vectors are in the same direction if?
3 votes

Two vectors $\mathbf v$ and $\mathbf w$ are in the same direction if and only if $$\frac{\mathbf{v}}{v}\cdot\frac{\mathbf{w}}{w}=1$$ One of the many ways your can rephrase this is $\mathbf{\hat v}=\...

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Visible Portion of the Earth's Surface
3 votes

Adding up the sections of $A_h$ is a lot like stacking up incrementally smaller onion rings and adding up the surface areas of their exteriors. a very dextrous sketch of a stack of onion rings in the ...

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How to prove a polynomial can be written as Taylor-style?
3 votes

Polynomials are already in this Taylor form. If you want to start with a polynomial, then convert it into this Taylor form, you can do that simply by differentiating the polynomial up to $n$ times ...

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How to find the quadratic approximation of a quotient?
3 votes

$$\begin{align} f(x) &= \tan x \\ f’(x) &= \sec^2x \\ f’’(x) &= 2\sec(x)\tan(x) \\ \end{align}$$ You should just memorize $f’(x)$, and $f’’(x)$ comes from chain rule. Assuming we expand ...

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Can we integrate without integration rules?
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3 votes

I believe you are asking if integration can be defined in a way other than setting an identity (e.g., with a limit like differentiation). This is how the Riemann integral is defined. The integral is ...

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Trigonometry and vectors
3 votes

If I understand the problem correctly, these two vectors will outline two of the four sides of something that looks like a slanted rectangle. Therefore, the length of the diagonal will be the segment ...

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How to denote that an integer doesn't have zero digit?
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2 votes

Call the unique base-ten expansion of a finite number $X$ that doesn’t involve insignificant zeroes the series $$X = \sum_{k=a}^{b} 10^k x_k$$ Therefore, in base ten, the $k$th digit of $X$ is $x_k$ (...

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How to graph $x^2+y^2=4$ on a TI-$84$?
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2 votes

You can parametrise: get the equation in terms of $t$. Or you could convert to polar coordinates: use the form $r$ as a function of $\theta$. However, in general, this can be complicated. Instead you ...

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What's the modulus of $(\sqrt{11}-i)^{1000}$?
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2 votes

Take $\newcommand{\e}{\mathrm e}\newcommand{\i}{\mathrm i} z= \sqrt{11}-\i = r\e^{\theta\i}$, where $r=\lvert z\rvert$. Then $$\begin{align} z^{1000} &= (r\e^{\theta\i})^{1000} \\ &= r^{1000}\...

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