Can anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any atten...

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there anot...

We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root...

Write the following in polar form:$$\frac{1}{z}$$ $$\frac{1}{z}=\frac{1}{x+yi}$$ $$\frac{1}{x+yi}\cdot \frac{x-yi}{x-yi}=\frac{x-yi}{x^2+y^2}=\frac{x-yi}{...

Since $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sq...

I have been reading a research paper. The author has used a variable '$x$' with an underline beneath '$x$'. I know that a variable wit...

People often play around with the word "perfect circle" but I have always been curious as to what exactly is a "perfect circle". It is...

I have an imaginary bank account with an initial balance $x$. I can only withdraw $0\% < w\% < 100\%$ (of the current balance) or deposit $0\% < ...

Show that for the de Rham cohomology, $H^{1}_{dR}({T^{2}})$ is isomorphic to $\mathbb{R}^{2}$ by showing that the following map: $[\alpha] \to (\int_{S^{1...

When we say $1,2,3...$ are natural numbers, why don't we include rational and irrational numbers? Isn't $\pi$ something natural? Shouldn't ...