TITLE: Is there math with non-commutative multiplication of real numbers?
QUESTION [7 upvotes]: I'm wondering is there a math with non-commutative multiplication of real numbers. For example, we could define operator ⊗ for $ n, m ≥ 0$:
$$ n⊗ m = n\times m  $$
$$ n⊗ (-m) = n\times m  $$
$$ -n⊗ m = -(n\times m) $$
$$ -n⊗ (-m) = -(n\times m) $$
Or we can choose some other rules of multiplication. 
How could this math be applied?

REPLY [2 votes]: Did someone ask for Grassman variables? Why have boring commutative multiplication where $xy = yx$ when you can have exciting anticommutative multiplication where $xy = -yx$? Exponentiation becomes super easy since $x^n = 0$ for $n\ge 2$. Differentiation and integration are now the same thing. Quantum field theory makes marginally more sense. You'll wonder why you even bothered with $\mathbb R$ in the first place.