## Modulo 7

A week or two ago I noticed some students in the math lounge solving these type of equations $2x=3\mod{7}$ and $x^2=-3\mod{7}$. Abstract algebra is above my pay grade but I spent a couple of lunch hours exploring equations $\mod{7}$.

If we solve $2x=3\mod{7}$, we get $5$ so $5 = \frac{3}{2}\mod{7}$. I built this table of all possible fractions $\mod{7}$.

Fractions Modulo 7

This is simply a rearrangement of the multiplication table $\mod{7}$.  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.

Quadratic Equations Modulo 7

Some of these equations have no solution, so let’s extend the field like we do with complex numbers where we define $i$ to be a solution to $x^2+1=0$.  Let one solution to $x^2-3=0$ be $\sqrt{3}$.  Then the other solution would be $-\sqrt{3} = -1\sqrt{3}=6\sqrt{3}\mod{7}$.  The solutions to $x^2-5=0$ would be $2\sqrt{3}$ and $5\sqrt{3}$ and the solutions to $x^2-6=0$ would be $3\sqrt{3}$ and $4\sqrt{3}$.  Also cool.