Answer to Question #145705 in Algebra for Ruchika Sharma

Question

How would you convince a child that any number multiplied by 0 is 0, using a
teaching aid.

Answer 1

The multiplication property of zero: Regardless of what the other number is, multiplying by zero always results in an answer of zero. That zero manages to be both a non-negative and non-positive integer yet is neither negative nor positive is just one of the unique properties of the number.

This is my proof:

We have to show that 0⋅0=0

0⋅0=0

Since we know that

a−a=0

a−a=0

By substitution,

We have (a−a)(a−a)=0

(a−a)(a−a)=0

Then by simplifying, 

a²−a²+a²−a²=0

and the we have 0−0=0.

Therefore, 0=0

But we can solve it elementarily.

Observe this tautology:

∀a a−a=0

Thus, multiplying any number x with 0 means :

0.x=(a−a).x=a.x−a.x

Now, it is obvious that a.x=a.x

a.x=a.x no matter what source of mathematical axioms you invoke.

Therefore,

a.x−a.x=0

and 0.x=0. Thus, the problem is solved.

But again, I played tricks because real numbers are a field.