# Numeric facts on fractions, that few are aware of.

Assume a fraction k/m, with 0≤k≤m and look at its digits written out

1. The fraction has a finite number of digits (less m) OR is a repeated sequence with at the most m-1 digits per group
2. A repeated sequence will have at the most n-1 zeroes next to each other, if the number is below 10ⁿ
3. Interesting digit-structure of the sequence times m, with many 9’s (base minus 1) .

# Diagonal to width, simple fractions. Different aspect ratios

Obviously it is easy to calculate the diagonal of e.g. a sensor from the width and height with Pythagoras.
And a bit more tedious the other way round using the aspect ratio. [*]
But for common aspect ratios it is even simpler, 1+1/n fractions within less than 0.4%

Here Is a small guide in Disqus-formatting, that I have ‘published’ several times on the old GSMArena.com blog (with tiny variation).

UPDATE: Should in 2019 finally be more or less obsolete for wide enough windows!

# Working with numbers in Q with large nominator/denominator in C#

A nerdy idea:

A C# class that quicly can supply a lot of primes (here limited to about the largest prime about 50G – as arrays with over 2G entries are not currently supported without splitting the arrays)

A struct that supports basic arithmetic for large accurately represented numbers in Q,  by factorization into primes.

A test program that tries to find solutions for a!b!=c! with 1<a<b<c

# Length of round (carpet) roll

For some reason, surprisingly few know this extremely simple formula, please share it.

I have actually seen in a carpet store an employee unrolling the carpet to measure the length!!

[The formula is simple to prove too… r=kφ, L=∫r dφ=∫kφ dφ=½k(φ2²-φ1²)=½k(φ21)×(φ21)=½(r2+r1) ×n2π=(r2+r1) × nπ , q.e.d.]