Fruits of procrastination

I worry about “small” things.

Let us take a ring $R(+,*)$ . It is easy to prove that that $0*a=0$ . The standard proof works this way: $0*a=(0+0)*a=0*a+0*a$ . Adding $-(0*a)$ on both sides, we get $0=0*a$ .

This fact could be used to prove that $(-a)*b=-(a*b)$ . How? $(-a)*b+a*b=(-a+a)*b=0*b=0$ . Hence, as $(-a)*b$ is the additive inverse of $a*b$ , and as every element has a unique additive inverse, we have $(-a)*b=-(a*b)$ .

Although proving the above is a fairly uninvolved exercise, it was never intuitive to me, unless I started to imagine $+$ as arithmetic addition and $*$ as arithmetic multiplication. This blog post is an attempt to make the aforementioned facts more easy to visualize, and perhaps intuitive.

What is $0*a$ ? To answer this, we need to understand what $0$ is in the context of distributivity of addition over multiplication. It is $(-a+a)$ . It is $(-2a+2a)$ . It is the combination of matter and anti-matter. It takes away as much as it gives. This I feel is the easiest way to think about it in the given context. I would love to elaborate on why this is the best way some other time.

Now what is $(-a)*b$ ? It is $(a+-2a)*b$ . It is $(2a+-3a)*b$ . If $(2a+-3a)*b=(a+-2a)*b$ , then $a*b+(-a)*b=0$ . This proves that $(-a)*b=-(a*b)$ , as the additive inverse of every element is unique.