Chapter 7

Rational and Irrational Numbers

In this chapter, we start by reviewing the real line model for numbers, something you saw in Chapter 2 of seventh grade. We'll remind ourselves how integers and then rational numbers (numbers that can be written as fractions) are matched up with points on a line. Once every rational number has a matching point on the line, we can ask an interesting question: 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 starting point (called the origin) to that point. If the point is to the left of the origin, its length is negative. Using a special construction with "tilted" squares, we'll rediscover something the ancient Greek mathematicians (the Pythagoreans) noticed 2,500 years ago: some lengths — like the diagonal of a square with side length 1 — cannot be written as a rational number. This construction gives us numbers whose squares are whole numbers, which leads us to introduce the symbol √A, meaning "a number whose square is A." We'll also introduce the cube root, written ³√V, which stands for the side length of a cube whose volume is V. Along the way, tilted squares will help us rediscover the Pythagorean theorem: a² + b² = c², where a and b are the two shorter sides (legs) of a right triangle, and c is the longest side (the hypotenuse).

Next, we'll go back to building a square with area 2, and prove that its side length (√2) cannot be written as a fraction. This means its length is not a rational number — we call it an irrational number. The same reasoning works for √5 and other lengths built from tilted squares. In fact, here's a useful rule: for any whole number N, either N is a perfect square (the square of a whole number), or √N is irrational — meaning it can never be written as one whole number divided by another.

This raises a bigger question: how can we represent lengths that are not fractions using numbers? The ancient Greeks struggled with this because they didn't have a good number system for expressing such lengths. Today, we have exactly the right tool: decimal representation, which you reviewed in Chapter 1 of seventh grade.

From seventh grade, remember that a rational number has a terminating decimal (one that ends) only if its denominator is made up of just 2s and 5s multiplied together. This means many rational numbers — like 1/3, 1/7, and 1/12 — don't end when written as decimals. Instead, they become repeating decimals. The reverse is true too: any repeating decimal represents a rational number. We can think of a number's decimal expansion as a step-by-step method for getting closer and closer to its exact point on the line, by repeatedly dividing sections into ten equal parts. In fact, every decimal expansion represents some point on the line — some number. But if that decimal never ends and never repeats, the number is irrational.

So now the question becomes: can every length be written as a decimal? We'll start with square roots and explore Newton's method, a technique for approximating square roots. You begin with a reasonable guess and then repeatedly improve it using this formula:

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

Through examples, we'll see that this method can produce the decimal expansion of the square root of N as accurately as we want. Finally, we'll point out that when doing math with irrational numbers (and even with rational numbers), we need to be careful — getting a certain level of accuracy in our answer often requires even more accuracy in the numbers we start with.

Section 7.1: Representing Numbers Geometrically

Let's begin by recalling how the rational numbers are 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 is to the left of b. The piece of the line between a and b is called an interval. For any two different points, either a < b or b < a must be true — there's no other option. (If a < b, we can also write this as b > a.)

Pick a point on the line, mark it, and call it the origin — labeled 0. Place a ruler with its left end at 0. Choose a point to the right of 0 (maybe the 1 cm or 1 inch mark) and call it 1. The distance from 0 to 1 is called one unit. Now mark off the same distance to the right of 1 and call that point 2. Keep going, and you can place every positive whole number on the line. Then mark off equally spaced points to the left of 0, calling them −1, −2, −3, and so on. Now every integer has a place on the line.

We can also find points for numbers in between. The midpoint of the interval between 3 and 4 is 3.5, and the midpoint between −7 and −6 is −6.5. If we split 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 whole number p, laying p copies of one-third of a unit end to end (to the right of 0 if p is positive, to the left if negative) gives us the point for p/3. We can do the same thing with any positive whole number q in place of 3, by splitting the unit interval into q equal parts. This lets us place every rational number p/q on the line — to the left of the origin if it's negative, and to the right if it's positive.

The number line also helps us understand decimal expansions visually. Take a positive number a. There's some whole number N such that N ≤ a < N + 1 — this N is called the integer part of a. If N equals a exactly, we're done. If not, we divide the interval between N and N + 1 into ten equal parts, and count how many of these parts fit between N and a. That count is a digit between 0 and 9, called d₁ — the tenths digit. We write this as N.d₁. If that matches a exactly, we're finished. If not, we repeat: divide the interval between N.d₁ and the next tenth into ten equal parts to find d₂, the hundredths digit, giving us N.d₁d₂. Continuing this process generates a growing string of digits that gets closer and closer to a — this is how we approximate numbers using decimals. As you learned in seventh grade, this process sometimes never finishes (never terminates), as with 1/3 or 1/7.

Instead of thinking of this only as a way to turn a number into a decimal, we can think of it as a way to turn any length on the number line into a decimal. Given any point a on the line (say, a positive one), this same step-by-step process gives us its decimal expansion — an effective way to measure the length from 0 to a using tenths, then tenths of tenths, and so on. We'll start by looking at lengths that can be built using geometric constructions, which will help answer this key question: are there lengths that can't be written as rational numbers? To explore this, we need to extend our number line idea from one dimension (a line) to two dimensions (a plane).

Using the number line we built above, draw a second, vertical line through the origin, perpendicular to the first, using the same unit length. 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 straight up or down from there. This gives us the point (a, b).

Example 1.
In Figure 1, each unit length equals half an inch. Estimate the lengths of AB, AC, and BC to the nearest tenth (or hundredth) of an inch.

Solution. We use a ruler to measure. Since our ruler's markings might not exactly match half-inch units, we need to account for the scale. After adjusting for scale, we find that AC is about 3.05 inches.

Using a ruler, we can always estimate the length of a line segment with a fraction (a rational number) — how accurate that estimate is depends on how finely marked our ruler is. This raises an important question: can every length be described using a rational number?

The coordinate plane helps us assign exact lengths to line segments, as the next examples show.

Example 2.
In Figure 2, a tilted square (shown with dashed lines) is drawn inside a larger square made of solid lines. Each small square has a side length of 1 unit, so the whole figure is a 2 × 2 square with an area of 4 square units. The tilted square takes up exactly half the area of each unit square, since every triangle outside the tilted square matches a triangle inside it. So the tilted square has an area of 2 square units. Since the area of a square equals the side length squared, the side length of the tilted square must be a number whose square is 2 — written as √2.

We use the symbol √A (the square root symbol) to represent a number a whose square equals A — meaning a² = A. Since the square of any nonzero number is always positive (and √0 = 0), the expression √A only makes sense when A is zero or positive. Since 2² = 4, 3² = 9, 4² = 16, and 5² = 25, we know that 4, 9, 16, and 25 all have whole numbers as square roots. A positive whole number whose square root is also a whole number is called a perfect square. But for numbers like 2, 3, 5, or 6, we still need a way to calculate their square roots. The tilted-square strategy gives us a way to construct lengths equal to the square roots of any whole number, as shown next.

Example 3.
In Figure 3, the large square has a side length of 3 units, so its area is 9 square units. Each triangle outside the tilted square is a right triangle with legs of length 1 and 2, giving it an area of 1 square unit; there are four such triangles, totaling 4 square units. So 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 is 7 × 7, giving an area of 49 square units. Each outside triangle is a right triangle with legs 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 units — in other words, √25 = 5.

Extension. We've been using the "tilted square" construction to turn square roots of whole numbers into real, physical lengths — with the ultimate goal of finding a way to express those lengths as decimals. (In other words, we're working toward finding a method to calculate the decimal expansion of square roots.) Some students may notice that this construction works for any starting lengths a and b, not just whole numbers. That's a sharp observation — it leads directly to the Pythagorean Theorem, which relates the side lengths of a right triangle. We'll explore this fully in Chapter 8. The example below gives a preview of that idea.