A week or two ago I noticed some students in the math lounge solving these type of equations and . Abstract algebra is above my pay grade but I spent a couple of lunch hours exploring equations .
If we solve , we get so . I built this table of all possible fractions .
This is simply a rearrangement of the multiplication table . It shows that all rational numbers can be put into seven equivalence classes not just the integers. That’s pretty cool!
What about quadratic equations? Here is a table with solutions.
Some of these equations have no solution, so let’s extend the field like we do with complex numbers where we define to be a solution to . Let one solution to be . Then the other solution would be . The solutions to would be and and the solutions to would be and . Also cool.