My older son moved on to a new chapter in his algebra book today – complex numbers. We’ve done work with complex numbers before, and even by coincidence this weekend in our Mandelbrot set projects:

My son chose an interesting problem about simplifying complex numbers for our video today. The problem ask you to simplify the expression (5 + 3i) / (3 – 4i).

I was pleasantly surprised by his approach to the problem:

We had not discussed any of the problems or examples in the section, so it probably isn’t much of a surprise that my guess about the approach he would take to the problem was wrong. I thought it would be useful to show him what an approach similar to “rationalizing the denominator” would look like:

I love how different these two approaches look at first glance. In fact, it is pretty surprising that you arrive at the same answer. Feels like this would be a fun example to introduce matrix inversion.

My two favorite introductions to matrix inversion:

1) Draw the Hasse diagram of a finite poset P four times, in pairs. Not a tree.
Label copy 1A with some integers.
To get 1B, each number is the sum of all the numbers below (without repetition, in the case of multiple chains, which will happen because it’s not a tree).
Now label 2B with some integers, and challenge the audience to find 2A, undoing the operation above.
The matrix one must (M\”obius) invert is the incidence matrix of the poset.

My two favorite introductions to matrix inversion:

1) Draw the Hasse diagram of a finite poset P four times, in pairs. Not a tree.

Label copy 1A with some integers.

To get 1B, each number is the sum of all the numbers below (without repetition, in the case of multiple chains, which will happen because it’s not a tree).

Now label 2B with some integers, and challenge the audience to find 2A, undoing the operation above.

The matrix one must (M\”obius) invert is the incidence matrix of the poset.

2) Dual photography https://www.youtube.com/watch?v=p5_tpq5ejFQ