Mathe Uffzgi

Once in a while, I’m supporting Lea with her math homework (Swiss: Uffzgi). Here are some tools that help me (remember, that I prefer F# to other languages. Whenever possible, I use interactive Notebooks and make them available on github Notebooks.

Mitternachtsformel

The quadractic formula is called Mitternachtsformel in German speaking schools, since the student is supposed to know it by heart even when suddenly woken up after midnight.

It goes like this:

$$ (x_1, x_2) = \frac{ - b \pm \sqrt {b^2 - 4ac} }{2a} $$

when

$$ ax^2 + bx + c = 0 $$

It can easily be computed in F# using this piece of code:

let mitternacht (a: float) b c =
    let cof = sqrt (b * b - 4.0 * a * c) 
    let den = 2.0 * a
    ( (-1.0 * b + cof) / den, (-1.0 * b - cof) / den)

Try it:

mitternacht 1.0 -4.0 -5.0
|> printfn "%A"
// returns (5.0, -1.0)

Microsoft Math Solver

A simple way is to use the Microsoft Math Solver. You can type in the mathematical equation like

x^2 - 4x - 5 = 0

and it will return the solution and visualizes the graph.

Symbolic algebra and F#

There is an interesting library out there called Math.NET Symbolics. It can be used to do algebraic calculations.

First, the variables are declared as symbols:

open System.Numerics
open MathNet.Numerics
open MathNet.Symbolics
open Operators

let x = symbol "x"
let y = symbol "y"
let a = symbol "a"
let b = symbol "b"
let c = symbol "c"
let d = symbol "d"

let aa = a + a
let kompliziert = (a/b/(c*a))*(c*d/a)/d 

Now it can be used to simplify formulas. They can be printed using Infix.format.

a + a       // return 2*a
kompliziert |> Infix.format     // returns 1/(a*b)

Formulas can also be turned int LaTeX:

kompliziert |> LaTex.format
// returns \frac{1}{ab} 

And this LaTeX can then be used in markdown documents. How to do this with Jekyll, is shown in the post Liquid tricks.

let symbols = Map.ofList [ "a", FloatingPoint.Real 2.0; "b", FloatingPoint.Real 3.0 ]
Evaluate.evaluate symbols (1/(a*b))  // Returns Real 0.1666666667 (as float)

Unfortunately, this crashed for me. See also as notebook on github

Mathematical graphs

For visualization, I typically use Plotly.NET.

An example of plotting herbizide influence on plant growth can be found as an interactive F# notebook.