Express all possible positive integer values of m, n in terms of a third variable, k. (Use modular arithmetic)
We can solve this with "normal" algebra . . .
Express all possible positive integer values of in terms of a third variable,
Solve for .
Since is an integer, must be a multiple of 4.
. . That is: . for some integer
Substitue into : .
And we have parametric equations for all solutions:
. . . for any integer
How would you express them if you could only have them as natural numbers?
Would you have: . ? . . . . Yes!
What are parametric equations?
A parameter is an "extra variable".
Instead of having as a function of ,
. . we can have: .
This opens the door to an entire universe of fascinating functions and graphs
. . . . . curves with loops, that intersect itself, etc.