KAlgebra/Homework

This page show some use of kalgebra in real word

Combinatorial example

  We have 6 people who want to know how to get around a table with 6 chairs.

  We now that 6 people can get around the table with this configuration

  p1 p2 p3 p4 p5 p6
  p1 p2 p3 p4 p6 p5
  p1 p2 p3 p5 p4 p6
  p1 p2 p3 p5 p6 p4

  And so on

  We notice that the last  rotate position by 1, the fifth rotate position by
  2, the fourth rotate position by 3, the third rotate position by 4, the
  second rotate position by 5 and first rotate position by 6.

  So we can write down a simple formula:

  6*5*4*3*2*1

  Let's write this into kalgebra console:

  ((((1*2)*3)*4)*5)*6

=720

  This kind of arragenment of things around some position, where position
  number is equal of number of things is called "permutation"

  Let's try to call in kalgebra the permutation function:

  factorial(6)

=720

  It's the same result as you can see.

Probability example

  Let's roll a dice, we want to know the probability of one face
     
  We can define positive probability the favourble result of the event to us
  and negative probability the unfavorable result of the event to us

  So you have to pick only one face:

  probability = 1(face picked)/6(total face)

  So now we know that when a dice is rolled there is a 1/6 of probability that
  a face we choice come up

  We can set a simple function in kalgebra to take this formula in a simple 
  way:

  probability:=(favorable,total)->favorable/total

Numerical Theory

   Let's say that we want to know the sum of all numbers between a bounded
   interval for istance 1 - 100

   we have to do the sum of all numbers from 0 to 100 if we don't know the
   rule to get them
   
   kalgebra offers a great facility to this task. Let's write in console:

   sum(x: x=1.100)
   
   and we get the result

   The syntax indicate this:

   1- Bound x as variable
   2- Take first value of x
   3- Take second value of x and add the previus value of x
   4- Take third value of x and add the previus value of x
   ....
   N- Take the last value of x and add the last value of x
   

Eletronic

   Example1:

   Let's take a simple circuit a and port with two input and one output

   To resolve it on kalgebra we will write

   and(variable1, variable2)

   we will get the and value of the input as output

   Example2:
  
   We have a simple circuit: a battery of 3V and two eletrical resistence
   (R1 and R2) put on parallel of 3kohm. We want to get the current
   circulating in the circuit.

   We have first to calculate the value of the eletric resistence expressed
   as the law:

   TotalResistence = (1/R1 + 1/R2)^-1
   Current = Voltage/TotalResistence

   Let's write a simple function in kalgebra to do this:

   totalresistence:=(R1,R2)->(1/R1+1/R2)^-1
   current:=(voltage,totalresistence)->voltage/totalresistence

   let's see what we get:

   current(3, totalresistence(3, 3))
  	   =2