Chapter 7: Rational and Irrational Numbers

In this chapter, we begin by reviewing the "real line" model for numbers—a topic first covered in Chapter 2 of seventh grade. We'll recall how integers and rational numbers (numbers that can be written as fractions) are matched up with points on a number line. Once every rational number has a matching point on the line, a natural question arises: does every point on the line correspond to a rational number?

Remember that a point on the line represents the length of the segment from the origin (the starting point, 0) to that point. This length is negative if the point sits to the left of the origin. Using a construction involving "tilted" squares, we'll rediscover something the Pythagoreans noticed 2,500 years ago: some lengths—like the diagonal of a square with side length 1—do not correspond to any rational number. This construction produces numbers whose squares equal whole numbers, which leads us to the symbol √A, meaning "a number whose square is A." We'll also introduce the cube root, written ³√V, meaning "the side length of a cube whose volume is V." Along the way, tilted squares give us a preview of the Pythagorean theorem: a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse (the longest side, opposite the right angle).

Next, we return to a square with area 2 and prove that its side length, √2, cannot be written as a fraction. This means √2 is not a rational number—we call such numbers irrational. The same reasoning applies to √5 and other lengths built from tilted squares. In fact, there's a general rule: for any whole number N, either N is a perfect square (the square of an integer), or √N is irrational—meaning it cannot be written as a quotient of two integers.

This raises another question: can we represent lengths that aren't ratios of integers using numbers at all? The ancient Greeks couldn't, mainly because they lacked a good system for writing lengths as numerical values. Today, we have that system: decimal representation (reviewed in Chapter 1 of seventh grade).

Recall from seventh grade that a rational number has a terminating decimal (one that ends) only if its denominator, when fully reduced, is made up of only 2s and 5s multiplied together. This means many rational numbers—like 1/3, 1/7, and 1/12—don't terminate, but instead produce repeating decimals. The reverse is true too: every repeating decimal represents a rational number. We can think of a number's decimal expansion as a step-by-step process for getting as close as we want to its point on the line, by repeatedly dividing intervals into tenths. In fact, every decimal expansion—terminating, repeating, or neither—represents some point on the line, and therefore some number. If the decimal never terminates or repeats, that number is irrational.

So can decimal expansions represent all lengths? We start by exploring square roots, using Newton's method, a technique for approximating square roots. You begin with a reasonable guess and improve it using this formula:

$$a_{new} = \frac{1}{2}\left(a_{old} + \frac{N}{a_{old}}\right)$$

Through examples, we'll see that repeating this process generates the decimal expansion of √N to whatever level of accuracy we need. Finally, we'll note an important caution: when doing arithmetic with irrational numbers (as well as rational ones), achieving a certain level of accuracy in your final answer may require even greater accuracy in the numbers you start with.

Section 7.1: Representing Numbers Geometrically

Let's begin by recalling how the rational number system is represented as points on a line. Using a straight edge, draw a horizontal line. For any two points a and b on this line, we say a < b if a lies to the left of b. The portion of the line between a and b is called the interval between them. Note that for any two different points, exactly one of these must be true: either a < b or b < a. (If a < b, we can also write this as b > a.)

Choose a point on the line and label it the origin, or 0. Place a ruler with its left end at 0, then pick a point to the right—maybe the 1 cm or 1 in mark—and label it 1. The distance between 0 and 1 is called one unit. Mark off the same distance to the right of 1 and call that point 2. Continuing this way, every positive integer gets matched to a point on the line. Now mark off equally spaced points to the left of 0, labeling them −1, −2, −3, and so on. This way, every integer has a place on the line.

We can also locate fractions. The midpoint of an interval corresponds to a half-integer—so the midpoint between 3 and 4 is 3.5, and the midpoint between −7 and −6 is −6.5. If we divide the unit interval into three equal parts, the first part represents 1/3, and the first two parts together represent 2/3. In general, for any integer p, laying p copies of one-third of a unit end to end (to the right of the origin if p > 0, to the left if p < 0) gives us the point representing p/3. We can replace 3 with any positive integer q by dividing the unit interval into q equal parts instead. This lets us match every rational number p/q to a point on the line—to the left of the origin if it's negative, to the right if it's positive.

The number line also gives us a concrete way to picture a number's decimal expansion. Take any positive number a. There's some integer N such that N ≤ a < N + 1; we call N the integral part of a. If N = a, we're done. If not, divide the interval between N and N + 1 into ten equal parts, and let d₁ be the number of these parts needed to reach a. This digit d₁ (a whole number from 0 to 9) is the tenths digit, and we write the result as N.d₁. If N.d₁ = a, we're finished. If not, we repeat the process: divide the interval between N.d₁ and N.(d₁+1) into ten equal parts to find d₂, the hundredths digit, giving us N.d₁d₂. Continuing this process generates a sequence of decimal approximations—N, N.d₁, N.d₁d₂, and so on—that get closer and closer to a. This gives us a reliable method for approximating a to any level of precision. As we saw in seventh grade, this process sometimes never ends (never reaches a exactly in a finite number of steps), as with 1/3 or 1/7.

Now let's flip this idea around: instead of turning a number into a decimal, let's turn a length on the number line into a decimal. Take any point a on the line (say, a positive one). The same process assigns a decimal expansion to a—an effective way of measuring the distance from 0 to a using tenths, then tenths of tenths, and so on, of the unit interval. We'll start by examining lengths that can be built using geometric construction, which will lead us to answer this question: are there lengths that cannot be represented by a rational number? To explore this, we need to extend our numerical system from the line to the plane.

Using the number line we just built, draw a perpendicular (vertical) line through the origin, applying the same unit-interval method. Now every pair of rational numbers (a, b) can be matched to a point in the plane: move along the horizontal line (the x-axis) to the point a, then move a distance b up or down along the vertical line through a. This gives us the point (a, b).

Example 1

In Figure 1, each unit length is half an inch. Using a ruler, estimate the lengths AB, AC, and BC to the nearest tenth (or hundredth) of an inch.

Solution. We measure with a ruler, keeping in mind that we may need to adjust for scale, since a "unit" in the figure may not exactly equal half an inch on our ruler. After accounting for this scale difference, we find that AC is approximately 3.05 inches.

This example shows that a ruler always lets us estimate a length using a fraction (a rational number)—and how precise that estimate is depends on how finely marked the ruler is. This raises an important question: can every length be described exactly using a rational number?

The coordinate system on the plane lets us assign precise lengths to line segments. Let's explore this further.

Example 2

In Figure 2, a tilted square (shown with dashed sides) is drawn inside a horizontal square. Each small square outlined in solid lines is a unit square (side length of 1), so the whole figure has an area of 2 × 2 = 4 square units. The tilted square takes up exactly half the area of each unit square it overlaps, since every triangle outside the tilted square matches a triangle inside it. This means the tilted square's area is 2 square units. Since a square's area equals the square of its side length, the side length of the tilted (dashed) square must be a number whose square is 2—written √2.

We use the symbol √A ("square root of A") to represent a number a whose square equals A: a² = A. Since the square of any nonzero number is always positive (and √0 = 0), √A only makes sense when A is zero or positive. Because 2² = 4, 3² = 9, 4² = 16, and 5² = 25, the numbers 4, 9, 16, and 25 all have whole-number square roots. A positive integer with a whole-number square root is called a perfect square. For other numbers—like 2, 3, 5, and 6—we still need a method to calculate their square roots. The tilted-square strategy gives us a way to construct lengths matching the square roots of whole numbers, as we'll now demonstrate.

Example 3

In Figure 3, the large square has a side length of 3 units, giving it an area of 9 square units. Each triangle outside the tilted square is a right triangle with legs of length 1 and 2, so each has an area of 1 square unit. Since there are four such triangles, the tilted square's area is 9 − 4 = 5 square units, meaning its side length is √5.

Using the same reasoning: in Figure 4, the large square measures 7 × 7, giving an area of 49 square units. Each outer triangle is a right triangle with legs of length 3 and 4, giving each an area of 6 square units. With four such triangles, that's 24 square units total, so the tilted square's area is 49 − 24 = 25 square units. Since 25 = 5², the tilted square's side length is 5—that is, √25 = 5.

Extension. We've used the "tilted square" method to turn square roots of whole numbers into actual, physical lengths—our long-term goal being to find a way to express these lengths numerically (in other words, to eventually generate the decimal expansion of a square root). Some students may notice that this construction works for any starting lengths a and b, not just whole numbers. They'd be right—that generalization is the Pythagorean theorem, which relates the side lengths of a right triangle. We'll return to this observation and explore the Pythagorean theorem further in Chapter 8. The figure below offers a preview of that discussion.