Is there a sort of "axiomatic high school maths" course, that builds up from the basics without handwaving? i.e. that covers the actual reasons and proofs for things, not just the mechanics/tricks/techniques. The way geometry is taught.

Some might be beyond high school level, like fundamental theorems of arithmetic (unique prime factorization) and algebra (polynomial factors/zeros). There's also non-integer exponents, defining reals etc.

All this stuff is used in High School but must be taken on faith. Good for believers, bad for skeptics.

Or is this just Mathematics 101? [my engineering maths didn't cover it]