general topology – Giving the property of invertibility to zero using the Riemann sphere

From Tristan Needham’s Visual Complex Analysis, I learn that we can define $frac{1}{0} =infty$ by consider the rienman sphere / union-ing the $mathbb {R^2}$ with the point at infinity. But, doesn’t this lead to paradox:

$$ 0 cdot 1 = 0 cdot 2$$

Assuming invertibility of zero:
$$ 1 = 2$$

Clearly, this breaks some parts of multiplication and is clearly illogical, hence how is the newly defined inverse for zero consistent with the rienman sphere?