# How to prove formally that division is not commutative?

by Alvise Sembenico   Last Updated March 14, 2019 20:20 PM

I am studying quantitative reasoning and I am wondering how to prove formally that division is not commutative. Everyone knows that 7/3 is not the same as 3/7, but is there a formal way to conceptualize it? Thank you

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Asserting that $$\frac73\neq\frac37$$ is a full proof of the fact that division is not commutative. A simpler proof would be the fact that $$\frac21\neq\frac12$$.
Let us define $$\frac{a}{b}:=a b^{-1}$$ for $$b\neq0$$ and $$a\neq bk$$ for any $$k\in\Bbb Z$$. Here, $$b^{-1}$$ just denotes the element that produces unity upon multiplication by $$b$$ (stated succinctly, $$b\cdot b^{-1}=1$$). Since $$ab^{-1}\neq a^{-1}b$$, we have that division is not commutative. Indeed, $$ab^{-1}=a^{-1}b\Longrightarrow (ab^{-1})^2=1\Longrightarrow a=\pm b,$$which is a contradiction to our assumption that $$a\neq bk$$. We have used the commutative property of multiplication here.
An alternative, but essentially equivalent method: if $$ab^{-1}=a^{-1}b$$, then we have $$a^2=b^2$$. Now all we need to do is to demonstrate a pair of elements $$a,b$$ whose squares are not equal; in particular, $$3^2=9\neq 49=7^2$$, so $$\frac 37\neq\frac73$$.