Understanding Algebra for Calculus

Section 1.1 Addition, Subtraction, and Multiplication

Algebra is ultimately about applying some procedure, or sequence of procedures, to numbers, and asking which numbers yield some desired result. To this end, we need to conceptualize numbers more dynamically than just as a way of counting objects. In the following table we present a way to conceptualize numbers as positions on a line, and the basic operations as movements.

In order to understand what's going on here, it will be useful to draw pictures. This is a picture of the real number line that you've probably seen before:

This picture does not show us much, so let’s show one with some generic numbers \(a\) and \(b\text{,}\) where both \(a\) and \(b\) are positive with \(b>a\text{.}\)

To represent \(a+b\) and \(a-b\) geometrically, we first measure out the distance \(b\) is to the right of zero, then move that far to the right of \(a\) for \(a+b\) and move that distance to the left of \(a\) for \(a-b\text{.}\)

In Figure 1.1.4, since \(b>a\) we end up to the left of zero when we subtract \(b\) from \(a\text{.}\) This should make sense; if you take away more than you started out with, you end up with a shortage.

One of the important ideas in algebra is that of inverse operations. That is, two operations/actions that, when performed in succession on a number or variable, result in the original number or variable being unchanged. We can see this idea by looking at subtraction. Observe that if \(a\) and \(b\) are numbers, then \(a+b-b = a\text{.}\) This may seem obvious, especially if you draw it on the number line. However, the concept is very important; the inverse of adding \(b\) to \(a\) is subtracting \(b\text{.}\) We may say that subtracting “undoes” addition or that when you add and subtract the same quantity they “cancel each other”.

Multiplication of real numbers is more difficult to visualize geometrically than addition or subtraction. The fundamental idea is that of stretching, with \(1\) being a basic unit of measurement. Let's start with \(b\) being any positive number and \(a=1.5\text{.}\) We see that \(1\) fits into \(a\) exactly \(1.5\) times. Since \(ab\) must have the same size relative to \(b\) as \(a\) does to \(1\text{,}\) \(b\) must fit into \(ab\) exactly \(1.5\) times. Think of this as stretching \(b\) by a factor of \(a\) when \(a>1\text{.}\)

If \(a\) is smaller than \(1\text{,}\) for example \(a = 0.75\text{,}\) then you don't get a whole copy of \(1\) in \(a\text{,}\) you only have \(75\%\) of one. This means you will only get \(75\%\) of a copy of \(b\) in \(ab\text{.}\) In the case when \(a<1\text{,}\) we think of \(ab\) as a squeeze of \(b\) by a factor of \(a\text{.}\)

An important case happens when one of the two numbers is a whole number (an integer). If \(n\) is a positive integer, then \(1\) fits into \(n\) exactly \(n\) times. Hence, if \(b\) is positive, \(b\) will fit into \(nb\) exactly \(n\) times. Let's view this on the number line with \(n=5\text{:}\)

The key observation here is that when you add \(b\) to itself \(n\) times you get the same result as when you multiply \(b\) by \(n\text{.}\) In other words, multiplication by a positive integer represents repeated addition. This is where the terminology “\(n\) times \(b\)” comes from; we’re taking \(b\) and adding it to itself \(n\) times.

Note that this gives us two different ways of thinking of multiplication; it represents repeated addition when the multiplier is positive and whole, but represents stretching or compression in other cases. We use the mental representation that makes the most sense for a given problem. This may seem confusing at first, but being aware of having several representations of mathematical objects, and knowing when you use them, is a key skill in truly understanding mathematics.

Lastly, let's think about multiplying with negative numbers. Recall from Table 1.1.1 that if \(a\) is a number, then \(-a\) is the same distance from \(0\text{,}\) but on the other side of \(0\) from \(a\text{.}\) Multiplying negative numbers extends this idea. When \(b\) is any number and \(a\) is a negative number, then \(ab\) is the number with the same squeeze or stretch factor applied to \(b\) as if \(a\) were positive, but flipped to the other side of zero. For instance if \(b\) is positive, then \(-0.75b\) is the number \(75\%\) as far from \(0\) as \(b\text{,}\) flipped to the other side of \(0\) from \(b\text{.}\)