Articles of intuition

Is it possible to gain intuition into these trig compound angle formulas – and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don’t really like the proofs that have been given, even the geometric proofs. I feel that none of them address the following points: cos(A+B) = cosAcosB – sinAsinB If you keep one angle constant […]

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) \cong HK/H \le G/H$. The third isomorphism theorem [wiki] is as follows: Let $G$ be a group. $K \triangleleft G$. […]

Geometrical interpretation of simplices

I’ve just started doing simplices, where the $n$-simplex has been defined to be $$\Delta^n = \{x \in \mathbb{R}^{n+1}\mid x_i \geq 0, \sum x_i=1\}.$$ It’s easy to see that the $0$-simplex is the point $1$ in $\mathbb{R}^1$, the $1$-simplex is the line from $(1,0)$ to $(0,1)$ in $\mathbb{R}^2$, and the $2$-simplex is the triangle, including the […]

How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following’s source for concision. $1.$ Now look at $\Pr(A \cap B)$. We know that if $A$ has happened, then $A \cap B$ happens with probability $\Pr(B\mid A)$. $2.$ If we do NOT […]

Action of a group on itself by conjugation is faithful $\iff$ trivial center

Definition 2.15. A group action of $G$ on $X$ is called faithful (or effective) if different elements of $G$ act on $X$ in different ways: when $g_1\neq g_2$ in G, there is an $x\in X$ such that $g_1\cdot x \neq g_2\cdot x$. Example 2.17. The action of $G$ on itself by conjugation is faithful if […]

reinterpreting bolzano-weierstrass with equicontinuity

Is there anything in the Bolzano-Weierstrass proof (for points in $\mathbb{R}^k$) that is analogous to using equicontinuity for functions in Arzela-Ascoli? Does the statement: “points are always equicontinuous” make sense? I feel that I may be forcing this reinterpretation, but I also feel that there might be some connection between the topology of points and […]

Intuition – $fr = r^{-1}f$ for Dihedral Groups – Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square’s vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles’s $f$ is different. Carter fleshes out why $frf = r^{-1} $ intuitively: (1.) Can someone please unfold, like Carter, why $fr = r^{-1}f $? I see why for this […]

Trouble understanding equivalence relations and equivalence classes…anyone care to explain?

What exactly are equivalence relations and equivalence classes? The latter is giving me the most trouble; I’ve tried to read multiple sources online but it just keeps going over my head. Example question: What are the equivalence classes of 0 and 1 for congruence modulo 4?

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is topologically equivalent to the circle. It is equivalent to the upper half of the circle where the two end points are glued together – i.e. another […]

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. Where $\alpha$ is a constant. Any help will be most appreciated. Thank you.