KAlgebra/Homework

This page shows some uses of KAlgebra in real world problems.

Combinatorial example

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

We know 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 item rotates its position by 1, the fifth rotates position by 2, the fourth rotates position by 3, the third rotates position by 4, the second rotates position by 5 and first rotates 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

and the answer returned is

=720

This kind of arrangement of things around some position, where the position number is equal to the number of things, is called "permutation".

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

factorial(6)

and we get

=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 appearing.

We can define positive probability, the result of the event being favourable to us, and negative probability, the result of the event being unfavourable 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 chose will 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 instance 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 previous value of x
4. - Take third value of x and add the previous value of x

....

N - Take the last value of x and add the last value of x

Electronic

Example 1:

Let's take a simple circuit a and port with two inputs and one output. To resolve it in KAlgebra we will write

and(variable1, variable2)

from which we will get the and value of the input as output.

Example2:

We have a simple circuit: a battery of 3V and two eletrical resistances (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 electric resistance expressed according to the law:

TotalResistance = (1/R1 + 1/R2)^-1

Current = Voltage/TotalResistance

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

totalresistance:=(R1,R2)->(1/R1+1/R2)^-1

current:=(voltage,totalresistance)->voltage/totalresistance

Let's see what we get:

current(3, totalresistance(3, 3))

=2