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Section 26.5 Going Deeper: Subtraction as Displacement

In Worksheet 3 of this section, we looked at the idea of using subtraction to measure distance. We're going to push that idea further and look at the signed distance between numbers, and see some applications of the idea.

Let's look at the calculation \(5 - 3 = 2\text{.}\) Using motion on the number line, we can interpret this as starting at 5 and moving 3 to the left to get to 2.

However, we can get another interpretation of this by inverting the movement and turning it into a question: If we're starting at 2 and ending up at 5, what is the movement that we must make? And by looking at the picture, we can see that we need to move 3 to the right.

Displacement is the movement from one position to another, and it can be expressed in terms of subtraction. To move from \(a\) to \(b\) the required displacement is \(b - a\text{.}\) We can check this algebraically:

\begin{equation*} \underbrace{a}_{\text{start}} + \underbrace{(b - a)}_{\text{displacement}} = \underbrace{b}_{\text{end}} \end{equation*}

This concept gets applied in a number of ways at many different levels of mathematics.

In Section 13.1, we looked at the concept of slope. The main definition we presented was purely geometric:

\begin{equation*} m = \frac{\Delta y}{\Delta x} = \frac{\text{Rise}}{\text{Run}} = \frac{\text{The change of $y$}}{\text{The change of $x$}} \end{equation*}

We're going to take a closer look at this idea through the lens of displacement to understand the algebraic formula for slope.

Consider a line that is passing through the indicated points \((x_1, y_1)\) and \((x_2, y_2)\) as shown below:

Notice that both \(\Delta x\) and \(\Delta y\) can be interpreted as displacements. They describe a motion to get from one location to another. Furthermore, these two displacements are independent of each other, meaning that as we change our \(x\)-coordinate the \(y\)-coordainte is fixed, and the same is true the other way around. In fact, we can look at both of these displacements relative to the corresponding axes:

By viewing the movements in this way, we can see that we are looking at displacement on the number line, and can use the formula for displacement for each to get the algebraic formula for slope:

\begin{equation*} m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}. \end{equation*}

This formula was given in a margin comment (The Line in the Sand), but we didn't explain it at the time because the focus was on the geometric interpretation and building basic intuition and experience. But now that we have the concept of displacement, this formula is not as mysterious.

As you continue forward in mathematics (and also science and statistics classes), you will see lots of situations where subtraction is being used to represent a quantity like acts like a displacement (though the language will vary depending on the topic). More generally, these concepts appear in vectors, which can be used to represent displacement in multiple dimensions. The following are a few examples of this. Don't worry if you don't understand these, as they're not relevant to this class. But if you've seen them before, it will help you to put this idea into context.

  • The Distance Formula: The distance formula is the Pythagorean Theorem applied to points on a plane. The lengths of the two sides of the right triangle are found using the same \(\Delta x\) and \(\Delta y\) concepts that we used with slope.

  • Torque: Torque is the result of applying a force around a pivot point. The torque is equal to the force multiplied by the displacement from the pivot. The location of the objects is often given relative to an underlying coordinate system. In fact, it is this idea that leads to the formula for the center of mass of an object.

  • Deviation from the Mean: In statistics, we often want to know where a particular data point is located relative to the mean of the sample or population. Concepts like this relate the standard deviation, and correlations.

The more familiar you are with the basic structures of mathematics, the more capacity you will have for understanding mathematical applications. Instead of seeing collections of random symbols, you will start to see concepts and ideas, which will help everything make more sense.