Section 1.4 Closing Ideas
We have been implicitly using some very core mathematical ideas throughout this section. Specifically, we have been working with the axioms of equality. All of the axioms are built on the idea that if you start with two quantities that are equal, and then perform the same manipulation to both of them, that the results should be equal. Here's what this formally looks like:
Definition 1.4.1. Axioms of Equality.
Let \(a\text{,}\) \(b\text{,}\) and \(c\) be real numbers. The axioms of equality state that
If \(a = b\text{,}\) then \(a + c = b + c\text{.}\)
If \(a = b\text{,}\) then \(a - c = b - c\text{.}\)
If \(a = b\text{,}\) then \(ac = bc\text{.}\)
If \(a = b\text{,}\) then \(\frac{a}{c} = \frac{b}{c}\text{,}\) provided that \(c \neq 0\text{.}\)
This may look intimidating at first, but it's much less so once you recognize that this is what you've been doing the entire section. It just says that if we add to, subtract from, multiply, or divide (as long as we don't divide by zero) both sides by the same value, then we maintain the equality. And this is what allows all of our algebraic manipulations to make sense.
It should also be fairly intuitive that if we started with an equality but treated the two sides differently, we would break the equality. For example, if we start off with \(5 = 5\text{,}\) but then add 3 to the left side and subtract 2 from the right side, the two sides will simply be different values.
The important takeaway from this is that these axioms simply make sense. It's hard to imagine a world where we each start with five apples, we both eat one of our own apples, but we somehow end up with two different numbers of apples. And that's supposed to be true about math in general. Math is supposed to make sense. In some ways, mathematics represents the epitome of strict logical reasoning.
Unfortunately, for many students, math is anything but that. There are so many symbols, so many rules, and so many manipulations that they have been forced to memorize that the whole thing is an incomprehensible mess. This book is our attempt to change that. The focus here is not going to be just on manipulating symbols, but really understanding what's going on. The emphasis will be placed on making sure you are clear in your thinking and clear in your communication. These things are far more important than just running you through the gamut of algebraic manipulations for the second, third, or fourth time in your life.
The hope is that as you start to think about math differently, you will find more and more of it simply makes sense.