In some situations, using geometry makes math easier to understand. I don’t know if this is the case in the field that Stephen works in though. But here is an example.

Famous example is that of imaginary numbers or the square root of -1.

It is difficult to imagine a number that, when squared, results in -1.

I can see the number ‘3’ as a process that takes a line segment from 0 to 1 and makes it 3 times longer.

Then 3^2 is just this process twice.

Then 1/3 is a process that takes a line segment and makes it smaller.

-1 is a process that takes a line segment from the right side and moves it to the left side.

Or takes a negative number on the left and moves it to the right.

This way I can think of all numbers as processes that do something on this line.

If I imagine all line segments from 0 to infinity I am working on the right side of zero.

Next to the right side of zero I can imagine the left side of zero.

All numbers on this line (left and right of zero) when squared result in a number on the right side.

So therefore it is difficult to imagine a process, when done 2 times results in numbers on the left side.

A line has only 1 dimension.

However; if I extend the line from 1 dimension to 2 dimensions I am working in a plane.

Now it is easy to imagine a process that when done 2 times results in numbers on the left.

It is a rotation by 90 degrees!

Think about it. Take a line segment, let’s say from zero to 3 and rotate it in the plane.

This is a process, when done 2 times, that takes any number on the right side of zero to the left side.

How easy is that.

What we have done is some simple geometry to understand imaginary numbers.

Sometimes things are easier using geometry.