Intereting Posts

Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$
Inner Product Spaces over Finite Fields
Integral inequality (Cauchy-Schwarz)
Convolution with Gaussian, without dstributioni theory, part 3
Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann
finding $\int {\tan^{3/2} 3x\sec 3x\,dx}$
Equivalence of the definition of the Subbasis of a Topology
Tables and histories of methods of finding $\int\sec x\,dx$?
Limit problem: $\sqrt{x+1} – \sqrt{x}$ as $x$ approaches infinity
Show $m(A \cap I) \leq (1-\epsilon)m(I)$ for every $I$ implies $m(A) = 0$
Number to the exponent divided by exponent value
Can we turn $\mathbb{R}^n$ into a field by changing the multiplication?
Integration with respect to Dirac measure
$M$ is a flat $R$-module if and only if its character module, i.e. $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$, is injective.
Beautiful cyclic inequality

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it’s just 0). For example, 1/3 in base 10 is 0.33333…, in base 5 it’s 0.131313…, and in base 3 it’s just 0.1.

A less common number system uses the Fibonacci sequence as its base, so the first few digit places represent 1, 2, 3, 5, 8, 13, 21, and so on (instead of the decimal 1, 10, 100, 1000, etc.). In this system:

$$

\begin{aligned}

17_{10} &= 100101_F \\

40_{10} &= 10001001_F

\end{aligned}

$$

- Show that $f(2n)= f(n+1)^2 - f(n-1)^2$
- Fibonorial of a fractional or complex argument
- Prove Divisibility In Fibonacci Sequence Over A Prime Number
- Conjecture: Only one Fibonacci number is the sum of two cubes
- Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$
- Applications of the Fibonacci sequence

This can easily be extended to digits after the radix point, so $0.1_F = \frac{1}{2}_{10}$, $0.01_F = \frac{1}{3}_{10}$, $0.001_F = \frac{1}{5}_{10}$, and so on.

$$

\begin{aligned}

\frac{5}{6}_{10} &= 0.11_F \\

\frac{7}{10}_{10} &= 0.101_F

\end{aligned}

$$

Some numbers aren’t as easy to write. Greedily adding up the first places that are smaller than the remainder of the number we’re trying to write:

$$

\frac{1}{4} = \frac{1}{5}+\frac{1}{21}+\frac{1}{610}+\frac{1}{1597}… = 0.001001000000101…_F \\

\frac{2}{3} = \frac{1}{2}+\frac{1}{8}+\frac{1}{34}+\frac{1}{89}… = 0.10010010100001…_F

$$

Do all rational numbers end in a repeating sequence in Fibonacci coding?

(For the specific case of $\frac{1}{4}$, see here.)

- How to find all rational points on the elliptic curves like $y^2=x^3-2$
- Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?
- Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
- H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$
- Tautological line bundle over rational projective space
- Are there any natural proofs of irrationality using the decimal characterization?
- Prove or disprove that ${F_{n}}^2 + 41$ is always a composite
- Proving the rationals are dense in R
- Applications of the Fibonacci sequence
- Why is $\frac{987654321}{123456789} = 8.0000000729?!$

- Find the spectrum of the linear operator $T: \ell^2 \to \ell^2$ defined by $Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$
- Pacman on a Mobius Strip
- rational functions on projective n space
- Every ideal of an algebraic number field can be principal in a suitable finite extension field
- Why Composition and Dihedral Group have reverse order of operation?
- What is the formal definition of a variable?
- Computing the variance of hypergeometric distribution using indicator functions
- Why are properties lost in the Cayley–Dickson construction?
- How to do a change of variable in an ODE
- Does a finite ring's additive structure and the structure of its group of units determine its ring structure?
- Bounded operator that does not attain its norm
- Morphism from a line bundle to a vector bundle
- Why is Skolem normal form equisatisfiable while the second order form equivalent?
- How to construct a line with only a short ruler
- incidence matrix of a digraph with a self loop