TITLE: Distributive Law and how it works
QUESTION [13 upvotes]: Quick question, which seems to irritate some people as it apparently strikes them as a nonsensical question: Why does the distributive law only work in one direction (in $(ℝ,+,*)$)? Why does it work this way:
$$ a * (b + c) = (a * b) + (a * c) $$
But not this way:
$$ a + (b * c) = (a + b) * (a + c) $$
What is it about the way addition and multiplication are constructed/composed that inhibits this form of usage? (in ℝ specifically)

REPLY [9 votes]: My way of interpreting the distributive law in an image:

The reason why people get annoyed with stuff like this is because they see the operations $+$ and $\times$ as symbols, not actual processes.
Standard index laws when using $\times$ allows us to relate multiplication with dimensions.
Standard addition laws regarding 'like terms' means we can't just simplify two entities from these 'separate' dimensions.
From image:
"The output of the * operator can be visualised by adding dimensions (as per the index laws, x^2 * x^3 = x^5"
"The output of the + operator can be visualised as lumping two of the same dimension objects together."
"If you imagine x=a=b=c and apply the distributive law, you'll end up with non-similar terms. i.e. you can't group x^2 and x terms"