Intereting Posts

How many expected people needed until 3 share a birthday?
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
Set of all injective functions $A\to A$
Find a surjective function $f:\mathbb{N}\to \mathbb{Q}$
Is every finite group of isometries a subgroup of a finite reflection group?
How to prove that a set R\Z is open
Continuity of a monotonically increasing function
Exercise concerning the Lefschetz fixed point number
Is the image of a null set under a differentiable map always null?
Why is $\gcd(a,b)=\gcd(b,r)$ when $a = qb + r$?
Matrix non-identity
A is recursive iff A is the range of an increasing function which is recursive
Is Kaplansky's theorem for hereditary rings a characterization?
How can you derive $\sin(x) = \sin(x+2\pi)$ from the Taylor series for $\sin(x)$?
Does the Rational Root Theorem ever guarantee that a polynomial is irreducible?

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a – bi$ does the image end up meaningfully different from the field I started with? Or when we write out complex numbers are we arbitrarily choosing which of the non-real solutions to $z^4 = 1$ to call $i$ and which to call $-i$?

- Set of prime numbers and subrings of the rationals
- The free abelian group monad
- A ring problem in Bhattacharya's book “Basic Abstract Algebra”
- Can this quick way of showing that $K/(Y-X^2)\cong K$ be turned into a valid argument?
- Torsion Subgroup (Just a set) for an abelian (non abelian) group.
- Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
- Irreducible Polynomials: How many elements are in E?
- How to learn commutative algebra?
- On the Factor group $\Bbb Q/\Bbb Z$
- When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Yes, one of the roots of the polynomial $z^2+1$ is called $i$, the other $-i$; since one is the negative of the other. You can switch the names if you really want to and say that “one root is called $-i$ and the other is called $i$”, but this would not make much difference, right? Traditionally, $i$ is drawn in the upper half-plane and $-i$ in the lower, but this is only a tradition. I am not sure what else would you want to know.

If you have some previous notion of orientation for the plane — some notion of “clockwise” and “counter-clockwise” — then you can specify which solution of $z^2+1=0$ is which. And vice-versa: given a choice of $i$ for $\mathbb C$, you get a corresponding orientation for the plane $\mathbb R^2$ using the usual bijection.

- Sniper probability question
- Given that $xyz=1$, prove that $\frac{x}{1+x^4}+\frac{y}{1+y^4}+\frac{z}{1+z^4}\le \frac{3}{2}$
- Why is this limit $\lim_{n \to \infty}(1 + \frac{1}{n})^n$ equal to $e$?
- What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?
- Find the point on the curve farthest from the line $x-y=0$.
- Intuitive approach to topology
- Multiplying Binomial Terms
- What are a few examples of noncyclic finite groups?
- Prove these two segments are equal
- How to show that $H \cap Z(G) \neq \{e\}$ when $H$ is a normal subgroup of $G$ with $\lvert H\rvert>1$
- Column Vectors orthogonal implies Row Vectors also orthogonal?
- Evaluation of $\int_{0}^{1} \frac{dx}{1+\sqrt{x}}$ for $n\in\mathbb{N}$
- Special case of the Hodge decomposition theorem
- How does one create a partition of unity for a complex manifold?
- Computing the product of p/(p – 2) over the odd primes