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.


About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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