1

I need to prove that $0*a = a*0 = 0$ using only axiomatic properties for real numbers(It's a exercise from Apostol's book.

My try: We can write $0$ as $(a - a)$. Hence we can rewrite the equation as follows: $0*a = a*0 = a*(a-a) = 0$ By distributive law we have: $aa -aa = 0$, where $aa - aa$ evaluates to $0$ because $(-aa)$ is the negative of $aa$

Note: It's given that $a(b - c) = ab - ac.$ (Theorem 1.4)

RayDansh
  • 1,073
Victor Feitosa
  • 335
  • 2
    If you have already proven that the additive inverse of $aa$ is $-aa$ then your proof works great. If you haven't proven that yet, then you probably shouldn't assume it is true. – SlipEternal Jul 18 '18 at 00:57
  • 1
    An alternative is to use $0=0+0$, multiply by $a$, use the distributive law, and inverses. – Michael Burr Jul 18 '18 at 01:09
  • I forgot to mention but it's also given that the additive inverse of aa is -aa. Thanks! I'll try to proof using it @MichaelBurr – Victor Feitosa Jul 18 '18 at 01:21

1 Answers1

2

Given the Theorem 1.4, this proof seems fine. It uses only the results available to you and the axioms.

Unfortunately, there is something bugging me in general about the proof. One has to be careful with establishing an arithmetic with minus signs. Little things can trip you up. For example, it's natural to look at $(-a)b$ and think it just means $-(ab)$, but conceptually, they're different.

Yes, the theorem does seem to cover this, but it still bothers me, because establishing Theorem 1.4 often relies on establishing $0a = 0$ first. Here's the way I'm used to the proof going: $$0a = (0 + 0)a = 0a + 0a,$$ which implies $0a = 0$ by cancellation (add the inverse of $0a$ to both sides, and simplify). Then, to establish that $(-a)b = -(ab) = a(-b)$, we observe that $$(-a)b + ab = ((-a) + a)b = 0b = 0,$$ and similarly $ab + (-a)b = 0$. Therefore $(-a)b$ fits the definition of the inverse of $ab$. Similarly, $a(-b) = -(ab)$ as well. Finally, this yields Theorem 1.4: $$ab - ac = ab + (-(ac)) = ab + a(-c) = a(b + (-c)) = a(b - c).$$ I don't really know how you'd prove Theorem 1.4 without relying on $0a = 0$.

Theo Bendit
  • 50,900