Section 31.4 Closing Ideas
We have now completed a tour of all four arithmetic operations: addition, subtraction, multiplication, and division. We have seen that addition and subtraction are closely related to each other, and that multiplication and division are closely related to each other. This touches on an idea that was discussed much earlier, but is worth reviewing now that we have more knowledge and experience.
In the early parts of this book (Definition 3.4.1 and Definition 7.4.1), we introduced certain properties of addition and multiplication:
The commutative properties: \(a + b = b + a\) and \(a \cdot b = b \cdot a\)
The associative properties: \((a + b) + c = a + (b + c)\) and \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
The fact that these properties hold for addition and multiplication, but not for subtraction and division, is a signal that addition and multiplication are somehow more "basic" or more "fundamental" than the other operations. And if we think back to the work that we've done over the last several sections, we can start to see how that has played out in our analysis.
Addition is closely tied to counting. When we add numbers, it can be viewed as a process of counting up by a certain amount. We saw this when we were combining groups of objects together (whether using blocks, integer chips, and even movements on the number line). But when we subtracted, we were still required to count up to a number in some form. We had to count out the right number of negative chips, or we had to count up to the right number of objects to take away, or we had to count up to the right number of movements. And so in all situations, our core addition concept was part of the process of subtracting.
Something similar happened with multiplication and division. The process of division required us to make groups that built up to a specific value. We saw this explicitly when we were counting out multiples of numbers in the division problems. We also saw this visually when we took individual objects and turned them into groups of objects (multiplication is represented as groups of objects).
As you continue onward in your college level mathematics, you may start to see further hints of this idea. For example, in the study of logarithms, you'll see that the primary relationship is between addition and multiplication, and the relationship between subtraction and division turns out to be nothing more than a fancy way of rewriting that relationship. Those calculations are included below. It's okay if you do not really understand it right now. Just focus on the relationships between addition and multiplication and how they relate to the relationships between subtraction and division.
The sum of logarithms: \(\log(a) + \log(b) = \log(a \cdot b)\)
The difference of logarithms: \(\log(a) - \log(b) = \log( \frac{a}{b} )\)