help! i suck at math!
Out of context: Reply #11
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- ukit0
Now a more interesting, but still very basic, example:
Suppose we have a set {0, 1, 2, 3, 4}, with operators ┼, ●. The ┼ is addition, but the result always brought back to {0, 1, 2, 3, 4} by going modulo 5. E.g. 1 ┼ 4 = 0, 4 ┼ 3 = 2. The ● is multiplication, but also modulo 5, e.g. 3●4 = 2, 2●2=4, 2●3=1.
Now suppose we want to solve the equation x●x = 4. This has two solutions, being 2 and 3 (3●3 is 9 modulo 5, which is 4). Now suppose we have the equation x●x = 2 and x must be in the set {0, 1, 2, 3, 4}. This equation has no solution. Again, we can introduce a new symbol , with the abstract property Λ●Λ = 2. Now we extend the solution space to the set {0,1,2,3,4,Λ}. Now we created a new number system and with the operators ┼, ● we can have expressions like 2┼Λ, Λ●4, etc. By introducing this new symbol and keeping it as a purely abstract entity, with the property Λ●Λ=2, we add a lot of new arithmetic to our simple system and we allow the solution of all equations of the form x●x=b, with b any number from {0, 1, 2, 3, 4}. E.g. 2●Λ is a solution for x●x= 3. (2●Λ●2●Λ = 4●Λ●Λ = 4●2 = 3), the other solution for this is 3●Λ. The original solution space has 5 elements, the extended solution space has 25 elements.