In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren’t. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn’t. My question: Later on, (from my M.Sc. onwards) I noticed that $\subset$ is used for general containment and $\subsetneq$ for strict. The symbol $\subseteq$ wasn’t […]

It’s straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first $n$ natural numbers count the number of ways I can choose a pair out of $n+1$ objects? What’s the intuition behind this?

There are a lot of great “approximations” that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what these statements strictly mean makes me rather uncomfortable. My question would be: What is the definition of “approximation” in terms of calculus or algebra? Are there […]

Is there any toy for learning algebraic manipulation of fractions? If you don’t know of any, how would you design one? What I’m imagining is something similar to a Rubik’s cube whose manipulation produces only true equations in some number of variables, for example: $\frac{a}{b} = \frac{c}{d}$ (turn a knob) $a = \frac{b c}{d}$ (twist […]

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection is linear. Proof Let $X,Y$ be vector spaces over a common field. Let $f : X \rightarrow Y$ be a linear bijection. We denote by $f^{-1}$ the inverse of […]

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I’ve seen some pretty complicated integrals and infinite products and infinite series and other math equations that have been reduced to simple closed form expressions using special functions […]

I am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name. So, I would like to ask, what does the operator $\propto$ mean? What is its name, and how is it […]

I am planning on taking Math 55 at Harvard this fall. I have a pretty strong background in multivariable calculus, and I would like to do some reading this summer to get prepared. I’ve heard that Analysis and Group Theory are good starting points. Does anyone know of some great books that would help me?

Without workin in a rigorous formal system, how can one intuitively establish that multiplication and addition are associative operations on the real line (including negative numbers)?

How do you define discrete math to a 5 years old kid in a nontechnical way? It seems to me even the formal definition of discrete math is vague for me.

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