It's actually super easy to form a "brand new branch of mathematics". Just start with some definitions and run with them. Although you'll almost certainly end up with something inconsistent. And if you don't, it'll almost certainly be not useful. And if it is useful, it'll almost certainly turn out to be the exact same math just wearing a costume.

There are no problems with the square root of two.

> show me any physical system where an action times an action does not equal a reaction.

Show me any gazzbok where a thrushbloom minus a grimblegork does not equal a fistelblush. Haha, you can't do it, can you!? I WIN!

That is to say: you're using silly made up definitions of "action" and "times" here.

> show me any physical system where an action times an action does not equal a reaction

Not quite sure what an action times an action is, but how about rotating a 2d shape 180 degrees? Do that twice and it's the same as not rotating it at all.

You mean two reactions. Otherwise 1x1 would be 1

Are you saying you actually buy into the Terrence Howard school of mathematics? For serious?