# Why the equation $3\cdot0=0$ needs to be proven

In Algebra by Gelfand Page 21 ( for anyone owning the book).
He tries to prove that: $$3\cdot(-5) + 15 = 0$$.
Here's his proof: $$3\cdot(-5) + 15 = 3\cdot(-5) + 3\cdot5 = 3\cdot(-5+5) = 3\cdot0 = 0$$. After that he said:

The careful reader will asky why $$3\cdot0 = 0$$.

Why does this equation need to be proven ?
I asked somewhere and was told that $$a\cdot0=0$$ is an axiom which maybe Gelfand didn't assume was true during his proof.
But why does it need to be an axiom, it's provable:
In the second step of his proof he converted 15 to $$3\cdot5$$ so multiplication was defined so
$$a\cdot0 = (0 + 0 + \cdots)$$ x times $$= 0$$.
I'm aware multiplication is defined as repeated addition only for integers,
but 3 is an integer so this definition works in my example.

In case my question wasn't clear it can be summed up as:
Why he takes $$3\cdot5=15$$ for granted but thinks $$3\cdot0=0$$ needs an explanation?

• I think you could regard his statement as saying: If $3\times 5 = 15$ then $3 \times (-5) + 15 = 0$. Sep 26, 2018 at 20:58
• Does Gelfand really use * for multiplication like a Fortran programmer? Apart from that, your question doesn't have enough context to be answerable by anyone who doesn't have Gelfand's book to hand. To make your question answerable and useful for others, please tell us more about the context in the book: is Gelfand talking about a specific ring like the real numbers or rings in general? What axioms has he presented for the ring (or class of rings) in question? Sep 26, 2018 at 21:07
• @RobArthan No, he uses "." but I thought it would be confused with period so I changed it. He's using this question to prove that when we multiply two negative numbers we get a positive result and before this question he only explained associative, commutative and distributive laws and he didn't mention rings or axioms. Sep 26, 2018 at 21:19
• Please use MathJax to mark up your mathematics on MSE, so you can write $3 \cdot 15$. I think Gelfand's mathematical point was asking you to think about what properties a sum of no numbers must have. Sep 26, 2018 at 21:28
• When you enclose your math in dollar signs, which you're already doing, you can write \cdot instead of * to get a nice dot multiplication symbol, or \times if you prefer the x-like multiplication symbol. Sep 26, 2018 at 21:30

Gelfand doesn't really take $$3 \cdot 5 = 15$$ for granted; in the ordinary course of events, this would need just as much proof as $$3 \cdot 0$$.
But the specific value $$15$$ isn't important here; we're really trying to prove that if $$3 \cdot 5 = 15$$, then $$3 \cdot (-5) = -15$$. That is, we want to know that making one of the factors negative makes the result negative. If you think of this proof as a proof that $$3 \cdot (-5) = -(3 \cdot 5)$$, then there's no missing step.
The entire proof could be turned into a general proof that $$x \cdot (-y) = -(x\cdot y)$$ with no changes; I suspect that the authors felt that this would be more intimidating than using concrete numbers.
If we really cared about the specific value of $$3 \cdot 5$$, we would need proof of it. But to prove that $$3 \cdot 5 = 15$$, we need to ask: how are $$3$$, $$5$$, and $$15$$ defined to begin with? Probably as $$1+1+1$$, $$1+1+1+1+1$$, and $$\underbrace{1+1+\dots+1}_{\text{15 times}}$$, respectively, in which case we need the distributive law to prove that $$3 \cdot 5 = 15$$. Usually, we don't bother, because usually we don't prove every single bit of our claims directly from the axioms of arithmetic.
Finally, we don't usually make $$x \cdot 0 = 0$$ an axiom. For integers, if we define multiplication as repeated addition, we could prove it as you suggest. But more generally, we can derive it from the property that $$x + 0 = x$$ (which is usually taken as a definition of what $$0$$ is) and the other laws of multiplication and addition given in this part of the textbook.
• In the specific case of $3$ and $5$, $3 \cdot 0 = 0$ is not an exciting thing to prove. But if we're interested in the general claim $x \cdot (-y) = -(x\cdot y)$, then that statement becomes $x \cdot 0 = 0$, which requires justification. Sep 26, 2018 at 21:50