Alternative Proof to show any integer multiplied with 0 is 0

[Seydlitz]

In his book, Spivak did the proof by using the distributive property of integer. I am wondering if this, I think, simpler proof will also work. I want to show that $a \cdot 0 = 0$ for all $a$ using only the very basic property (no negative multiplication yet).

For all $a \in \mathbb{Z}$, $a+0=a$.

We just multiply $a$ again to get $a^2+(a \cdot 0) = a^2$. Then it follows $a \cdot 0 = 0$. (I remove $a^2$ by adding the additive inverse of it on both side)

 

[jgens]

That is essentially the same proof as the one given in Spivak. I have no idea what simplification you think it affords.

 

[Seydlitz]

That is essentially the same proof as the one given in Spivak. I have no idea what simplification you think it affords.

I'm glad then that it's the same. Because I thought it's fallacious because I haven't showed if the integers are closed with multiplication, and Spivak's proof is the more appropriate one.

 

[jgens]

Because I thought it's fallacious because I haven't showed if the integers are closed with multiplication, and Spivak's proof is the more appropriate one.

Closure is not necessary in this argument.

 

[CellCoree]

Your proof is indeed correct and valid. It is a simple and elegant way to prove that any integer multiplied with 0 is 0. However, it is worth noting that Spivak's proof using the distributive property is also valid and perhaps more widely used because it can be extended to more complicated mathematical concepts. Both proofs demonstrate the fundamental property of multiplication with 0, which is that any number multiplied by 0 results in 0. This is an important concept in mathematics and is used in many different areas, such as algebra, calculus, and number theory. Your proof is a great example of how using basic properties can lead to a simple and effective proof.