Pressing Toward the Prize

Posts Tagged ‘property’

What are you thinking?

Posted on: October 6, 2010

How much do we do without thinking? We take for granted that the algebraic equation $x+6=15$ can be solved for $x$, but do we even consider the principles that make this possible? We know that $x$ must equal $9$ in order for this statement to be true, but how do we know this? In our Abstract Algebra class, we are looking at “abstracting” the essence of various mathematical structures in order to find commonalities among them. Part of that process has been identifying certain properties that can exist for a given structure, although they may not.

Consider the simple equation mentioned above. We can solve this because of certain properties that exist in the integers. Let’s take a look at what is actually involved in finding the solution:

\begin{aligned}x+6&=15\\ (x+6)+(-6) & = 15+(-6)\quad \textbf{(inverse)}\\ x+(6-6) &=(15-6)\quad\textbf{(associativity)}\\x+0&=9\qquad\textbf{(inverse)}\\x&=9\qquad\textbf{(identity)}\\ \end{aligned}
In reality, this simple equation can only be solved because the associative, inverse, and identity properties hold for addition on the integers. So the next time you are able to solve an algebraic equation, just say, “Thank you.”