Chapter 1 — Fundamental Concepts¶
1.9 — Graphing Calculators and Computers¶
A graph proves nothing; it suggests what to prove.
This section is different from the previous ones. It introduces no new algebraic technique; instead it reflects on the place that technology occupies, and the place it does not, in the work done so far. Everything developed in this chapter, from the factoring of §1.3 to the circle of §1.8, can be visualized with a graphing calculator or with mathematical software, and that visualization is genuinely useful. But it is useful as a complement to reasoning, not as a replacement. It is worth explaining that difference now, before the book begins to rely on graphs as often as it will in the following chapters.
§1. What a graph can do for algebra¶
A graph gives two things that algebra alone does not give with the same immediacy: an overall picture of the behavior of an expression and a quick way to check a computation already done. Graphing the equation \(y = x^2 - x - 6\) means drawing the curve of §1.8 formed by the points \((x, y)\) that satisfy it. If the sign analysis of §1.7 told us that \(x^2 - x - 6 > 0\) exactly on \((-\infty, -2) \cup (3, \infty)\), the graph must lie above the \(x\)-axis precisely on those intervals. Looking to see whether this happens is an almost immediate check, and if it does not agree with what was computed, that is a clear signal that the computation must be reviewed.
The relation works in both directions. Just as algebra predicts the shape of the graph, the graph can suggest what to look for with algebra: for example, how many real solutions an equation seems to have and roughly where they are, before finding them exactly with the methods of §1.5.
§2. What it cannot do¶
A graph shows a finite window of the plane, and what falls outside the window does not appear, even though it exists. A part of the curve that heads off toward very large values, a solution located far from the origin, or two strokes that the chosen scale makes look like one are frequent errors of interpretation. The machine does not commit them; hasty use of it induces them.
More important: a graph is not a proof. Let us compare \(y = x^2 - 2x + 1\) with \(y = x^2 - 2x + 1.0001\). On almost any screen the two curves look the same, and both seem to touch the \(x\)-axis at \(x = 1\). But the discriminant of the first is \(4 - 4 = 0\) and that of the second is \(4 - 4 \cdot 1.0001 = -0.0004\). By §1.5, the first equation \(x^2 - 2x + 1 = 0\) has exactly one solution, and the second none: the second curve touches the axis at no point at all. The difference is real and algebra decides it; the screen does not have enough resolution to show it. Likewise, the fact that two curves seem not to cross does not rule out their crossing at points too close together to tell apart.
To this is added a limit of the numbers themselves. A calculator works with a finite number of digits, so an irrational number such as \(\sqrt{2}\), which by §1.1 has an infinite non-repeating decimal expansion, can only appear rounded. If a numerical computation gives \(0.0000000001\), the machine cannot decide whether the exact result is that number or is \(0\) and the difference comes from rounding. That question, again, is answered by algebra.
Reminder. A graph suggests and checks; algebra proves. No result in this book is taken as established because it "can be seen" on a graph or because a calculator returns a certain number, but because it follows from the axioms and definitions already laid down.
§3. Reasoned use of technology¶
When the book suggests visualizing an expression or an equation, that suggestion should be read as a complement to the algebraic solution. Choosing the window of the graph so that it shows what matters, such as the solutions, the highest or lowest values, or what happens as the variable grows or decreases without bound, is itself an exercise that requires having anticipated with algebra, at least roughly, what one expects to find. A calculator does not choose the right window by itself.
There is also software that manipulates expressions exactly rather than with rounded decimals: it factors, simplifies and solves equations with symbols. It is a valuable aid, but its answer must also be read with the criteria of this chapter. Such a program may simplify \(\dfrac{x^2 - 1}{x - 1}\) to \(x + 1\) without warning that the equality holds only for \(x \neq 1\), which is exactly the warning of §1.4. The machine carries out the rules; the conditions under which those rules hold remain the responsibility of whoever uses it.
Reminder. The window of a graph is chosen knowing what one is looking for, and a simplification done by software is reviewed with the domain conditions of §1.4.
With this methodological caveat made, the chapter returns to algebraic development with the simplest geometric object in the plane after the point: the line, the subject of the next section.