Intereting Posts

Proof of injective and continuous
Distribution of the digits of Pi
L1 regularized unconstrained optimization problem
If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 – 4\mathbf{A} – 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible
Derivative of matrix with respected to vector (matrix in se(3))
Preimage orientation.
Unit sphere in $\mathbb{R}^\infty$ is contractible?
Definition of $K$-conjugacy classes
Is there a Stokes theorem for covariant derivatives?
Proof of the five lemma
Roadmap to study Atiyah–Singer index theorem
Calculating the norm of an element in a field extension.
Proof of the single factor theorem over an arbitrary commutative ring
proving identity for statistical distance
Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers?

- If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
- complex and decimal tetration
- If $\sum\limits_{j=1}^n | w_j|^2 \leq 1$ implies $ \left| \sum\limits_{j=1}^n z_j w_j \right| \leq 1$, then $\sum\limits_{j=1}^n | z_j|^2 \leq 1$
- Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$
- Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?
- Write an expression for $(\cos θ + i\sin θ)^4$ using De Moivre’s Theorem.
- Understanding imaginary exponents
- Find solution of equation $(z+1)^5=z^5$
- Do odd imaginary numbers exist?
- Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

The most obvious property that we lose is the linear (or total) ordering of the real line.

We lose equality of the complex conjugate and total order.

So:

$x+i y \ne x-i y$ for complex numbers which are not also reals.

And you can’t say wether $ i > -i $ or $i < -i$, etc.

All you have is the magnitude which, in the given example, is equal.

- If every $0$ digit in the expansion of $\sqrt{2}$ in base $10$ is replaced with $1$, is the resulting sequence eventually periodic?
- Linear independency before and after Linear Transformation
- Smallest number of points on plane that guarantees existence of a small angle
- In algebraic geometry, why do we use $\mathbb C$ instead of the algebraic closure of $\mathbb Q$?
- why $\mathbb Z \ncong \mathbb Z$?
- Showing $\pi\int_{0}^{\infty}^{-n}dx={(2n-2)!!\over (2n-1)!!}\cdot{2\over 2^n}$
- Probability distribution and their related distributions
- Kind of basic combinatorical problems and (exponential) generating functions
- Stuck on proving uniform convergence
- Intuitive explanation of variance and moment in Probability
- how to show that a subset of a domain is not in the range
- Proof of Gelfand formula for spectral radius
- Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.
- Expectation conditioned on an event and a sigma algebra
- Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent