Proof that the Product of Any Number and Zero is Zero

We continue the series of proofs of the basic theorems in Algebra.  In this post, we use the Axioms of Real Numbers to prove that the product of any number and zero is zero. You may go back the the preceding link to review the axioms stated below.

Theorem

For any number a, 0a = 0.

Proof

We know that 0 + 0 = 0. Multiplying both sides by a, we have

(0 + 0)a = 0a.

By the Distributive Property (Axiom 4), we get

0a + 0a = 0a.

Since by Axiom 5A, any number has an Additive Identity (recall c = c + 0 for any c), we can add 0, the right hand side of the equation giving us

0a + 0a = 0a + 0.

By the Cancellation Law, we eliminate one 0a on both sides of the equation resulting to

0a = 0

which is what we want to prove.