Here’s how dividing by zero leads to results like 1=2:
You may have heard that functions must be well-defined, which means x=y ⇒ f(x)=f(y). This property of functions is what allows you to apply any function to both sides of an equation and preserve truth doing it. If the function is one-to-one (ie x=y ⇔ f(x)=f(y)), truth is preserved both ways and you can un-apply a function from both sides of an equation as well. Multiplication by a factor c is one-to-one iff c isn’t 0. Therefore, un-applying multiplication by 0 is not in general truth-preserving.
Here’s how dividing by zero leads to results like 1=2:
You may have heard that functions must be well-defined, which means x=y ⇒ f(x)=f(y). This property of functions is what allows you to apply any function to both sides of an equation and preserve truth doing it. If the function is one-to-one (ie x=y ⇔ f(x)=f(y)), truth is preserved both ways and you can un-apply a function from both sides of an equation as well. Multiplication by a factor c is one-to-one iff c isn’t 0. Therefore, un-applying multiplication by 0 is not in general truth-preserving.