Bill Dubuque raised an excellent point here: Coping with abstract duplicate questions.
I suggest we use this question as a list of the generalized questions we create.
I suggest we categorize these abstract duplicates based on topic (please edit the question). Also please feel free to suggest a better way to list these.
Also, as per Jeff's recommendation, please tag these questions as faq.
Arithmetic arithmetic
Laws of signs (minus times minus is plus): Why is negative times negative = positive?
Order of operations in arithmetic: What is the standard interpretation of order of operations for the basic arithmetic operations?
Algebra/Precalculus algebra-precalculus
Solving equations with multiple absolute values: What is the best way to solve an equation involving multiple absolute values?
Extraneous solutions to equations with a square root: Is there a name for this strange solution to a quadratic equation involving a square root?
Principal $n$-th roots:
$0! = 1$: Prove $0! = 1$ from first principles
Partial fraction decomposition of rational functions: Converting multiplying fractions to sum of fractions
Highest power of a prime $p$ dividing $N!$, number of zeros at the end of $N!$ and related questions: Highest power of a prime $p$ dividing $N!$
Exponentiation exponentiation
Solving $x^x=y$ for $x$: Is $x^x=y$ solvable for $x$?
What is the value of $0^0$? Zero to the zero power – is $0^0=1$?
Calculus calculus
Indefinite integrals obtained using different techniques appear different (but are actually the same up to a constant): Getting different answers when integrating using different techniques
Integrating polynomial and rational expressions of $\sin x$ and $\cos x$: Evaluating $\int P(\sin x, \cos x) \text{d}x$
Integration using partial fractions: Integration by partial fractions; how and why does it work?
Intuitive meaning of Euler's constant $e$: Intuitive Understanding of the constant "$e$"
Evaluating limits of the form $\lim_{x\to \infty} P(x)^{1/n}-x$ where $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ is a monic polynomial: Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
Finding the limit of rational functions at infinity: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials
$\zeta(2)$: Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
Divergence of the harmonic series: Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Universal Chord Theorem: Universal Chord Theorem
Nested radical series: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
Derivative of a function expressed as $f(x)^{g(x)}$: Differentiation of $x^{\sqrt{x}}$, how?
Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic? ; What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?
Removable discontinuity: How can a function with a hole (removable discontinuity) equal a function with no hole?
Calculus Meets Geometry
- Volume of intersection between cylinders
- Two cylinders, same radius, orthogonal. This post is not particularly good but there are many existing duplicate-links. Note that this can be done without calculus.
- Two cylinders variation: different radii (orthogonal), non-orthogonal (same radius), and elliptic cylinders (essentially unsolved).
- Three cylinders: same radius and orthogonal.
Combinatorics combinatorics
Number of permutations of $n$ elements where no number $i$ is in position $i$
How many ways can N elements be partitioned into subsets of size K?
Seating arrangements of four men and three women around a circular table
How to use stars and bars for solving number of integral solutions to an equality?
How many different spanning trees of $K_n \setminus e$ are there? (or Spanning Trees of the Complete Graph minus an edge)
extended stars-and-bars problem(where the upper limit of the variable is bounded)
Functional equations functional-equations
Graph theory graph-theory
How to tell whether two graphs are isomorphic?
Group theory group-theory
For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
An element of a group has the same order as its inverse
Linear algebra linear-algebra
- Definition of Matrix Multiplication: (Maybe there should just be one canonical one?)
- On the determinant:
- Determinants of special matrices:
- Eigenvectors and Eigenvalues
- Gram-Schmidt Orthogonalization
- Prove that A + I is invertible if A is nilpotent
- A generalization for non-commutative rings
Logic logic
Number Theory elementary-number-theory
- Modular exponentiation: Modular exponentiation by hand ($a^b\bmod c$)
- Solving the congruence $x^2\equiv1\pmod n$: Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
- Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
- Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
- What is the period of the decimal expansion of $\frac mn$?
- When does $p^2$ divide $an^k + bp$?
- Posts related to proving that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$
Sequences/Series sequences-and-series (also summation summation)
Geometric Series: Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Summing series of the form $\sum_n (n+1) x^n$: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
Finding the limit of rational functions at infinity: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials
$\zeta(2)$: Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
Divergence of the harmonic series: Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Nested radical series: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
Limit of exponential sequence and $n$ factorial: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Elementary set theory elementary-set-theory
The set of all finite subsets / $n$-element subsets / 2-element subsets of $\Bbb N$ is countable
There are different sizes of infinity: What Does it Really Mean to Have Different Kinds of Infinities?
Trigonometry trigonometry
Solving triangles: Solving Triangles (finding missing sides/angles given 3 sides/angles)
(Confusing) notation for inverse functions ($\sin^{-1}$ vs. $\arcsin$): $\arcsin$ written as $\sin^{-1}(x)$