Chapter 7: Rational and Irrational Numbers

Getting Started

In this chapter, we look again at the number line. You learned about this in seventh grade. We will remember how whole numbers and fractions (called rational numbers) match up with points on a line.

Every rational number has a spot on the line. But here's a question: does every point on the line match a rational number?

A point's distance from the start (called the origin) tells us its length. If the point is to the left of the origin, we call that length negative.

Long ago, a group called the Pythagoreans discovered something surprising. Some lengths — like the diagonal (corner-to-corner line) of a square with sides of length 1 — are not rational numbers! We found this out by drawing special "tilted" squares.

This gives us two new number symbols:

• √A (square root of A) means a number that, when multiplied by itself, gives you A.
• ∛V (cube root of V) means the side length of a cube whose volume is V.

Tilted squares also help us see the Pythagorean theorem. It says: for a right triangle, if a and b are the two short sides (legs), and c is the longest side (the hypotenuse), then:

a² + b² = c²

Numbers That Aren't Fractions

Next, we build a square with an area of 2. Its side length is √2. We can prove that √2 is not a fraction. A number that isn't a fraction is called an irrational number.

The same is true for √5 and other lengths made with tilted squares. Here's a rule: if a whole number N is a perfect square (like 4, 9, or 16), then √N is a whole number. But if N is not a perfect square, then √N is irrational.

Can We Write These Numbers Down?

So — can we write irrational lengths as numbers? The ancient Greeks couldn't figure out how. They didn't have the right number system yet.

Today, we have decimals. That's our tool for writing these numbers down (you learned about decimals in seventh grade).

Remember: a fraction can be written as a decimal that stops (like 0.5) only if its denominator (bottom number) is made only of 2s and 5s multiplied together. Many fractions — like 1/3, 1/7, and 1/12 — don't stop. Instead, their decimals repeat forever. And any decimal that repeats forever is a rational number.

We can think of a decimal as a set of directions. It tells us how to get closer and closer to a point on the line by splitting things into tenths, again and again. Every decimal — even ones that go on forever — points to some spot on the line. But if a decimal doesn't stop and doesn't repeat, that number is irrational.

Finding Square Roots with Decimals

Can we write every length as a decimal? Let's start with square roots. We can use Newton's method to get closer and closer to a square root:

1. Start with a good guess.
2. Use this rule to get a better guess:

new guess = ½ × (old guess + N ÷ old guess)

If we keep repeating this, we get the decimal for √N as accurately as we want!

One more important note: when we do math (like adding or multiplying) with irrational numbers, we must be careful. To get a very accurate answer, we may need to start with numbers that are even more accurate.

Section 7.1: Showing Numbers as Points

Let's remember how we show rational numbers as points on a line.

Draw a straight line. Pick any two points, a and b. If a is to the left of b, we say a < b. The piece of the line between them is called an interval. For any two different points, one is always smaller — either a < b or b < a.

Pick a point and call it 0 (the origin). Use a ruler. Put its left end at 0. Pick a point to the right — call it 1. The distance from 0 to 1 is called one unit. Keep marking equal distances to the right: 2, 3, 4, and so on. Do the same to the left of 0, marking −1, −2, −3, and so on. Now all the whole numbers (integers) have a spot on the line!

We can also find halfway points. The midpoint between 3 and 4 is 3.5. The midpoint between −7 and −6 is −6.5.

If we split the space between 0 and 1 into three equal parts, the first part matches 1/3. The first two parts together match 2/3. This works for any fraction — we can split the unit length into any number of equal parts (let's call that number q), and match up fractions like p/q with points on the line. If p/q is negative, its point is to the left of 0. If positive, it's to the right.

Turning Points into Decimals

Here's how a number line connects to decimals. Take a positive number a. There's some whole number N that is just below or equal to a (written N ≤ a < N+1). We call N the integer part of a.

If N equals a exactly, we're done. If not, split the space between N and N+1 into ten equal parts. Count how many parts fit between N and a — call that count d₁. This is a single digit (0 through 9), and it's the tenths place. We write this as N.d₁.

If that's still not exactly a, we repeat: split that space into ten more parts, and find digit d₂, the hundredths place. We keep going: N.d₁d₂d₃... gets closer and closer to a. This never-ending process is exactly how we approximate numbers like 1/3 or 1/7, which never stop.

This same idea works for any point on the line, not just numbers we already know. Picking a point on the line, we can build its decimal step-by-step, getting closer and closer using tenths, then hundredths, and so on.

Now here's our big question: are there lengths that can't be matched with any rational number? To answer this, we need to move from a number line to a number plane (a flat grid with two directions).

Draw a line straight up through the origin (this is the vertical line), just like the horizontal line we already made. Now every pair of numbers (a, b) can point to a spot on the plane: go right to point a on the horizontal line, then go up (or down) a distance of b. That spot is called (a, b).

Example 1

Problem: In Figure 1, each unit is half an inch. Estimate the lengths of AB, AC, and BC to the nearest tenth of an inch.

Answer: Using a ruler, we measure carefully, adjusting for the half-inch scale. We find that AC is about 3.05 inches long.

Rulers can only give us fraction (rational) answers — and how exact those answers are depends on how fine our ruler's marks are. So here's our real question: can every length be named using a rational number?

Example 2

Look at Figure 2. There's a tilted square (dashed lines) drawn inside a straight square. The big square is made of unit squares (each side = 1 unit), so its whole area is 2 × 2 = 4 square units.

The tilted square takes up exactly half of each unit square — because each triangle outside the tilted square matches a triangle inside it. So the tilted square's area is 2 square units.

Since a square's area equals its side length multiplied by itself, the side of this tilted square is a number whose square is 2. We call this √2.

We use the symbol √A for a number whose square equals A. Since squaring a number always gives zero or a positive result, √A only makes sense when A is zero or positive.

Some numbers, like 4, 9, 16, and 25, have whole numbers as their square roots (2, 3, 4, and 5). These are called perfect squares. But numbers like 2, 3, 5, and 6 aren't perfect squares — we still need a way to find their square roots. The tilted square trick helps us do exactly that!

Example 3

Figure 3: This big square has sides of 3 units, so its area is 9 square units. Each triangle outside the tilted square is a right triangle with legs 1 and 2, so each has an area of 1. There are 4 such triangles, taking up 4 square units total. So the tilted square's area is 9 − 4 = 5 square units. That means its side length is √5.

Figure 4: This big square has sides of 7 units, so its area is 49 square units. Each outside triangle is a right triangle with legs 3 and 4, giving an area of 6 square units. There are 4 such triangles, using up 24 square units. So the tilted square's area is 49 − 24 = 25 square units. Since 25 = 5 × 5, the tilted square's side length is 5 — meaning √25 = 5.

A Little Extra: We used tilted squares to find square roots as real lengths. This is the first step toward writing these lengths as decimals. This idea can actually work for any two starting lengths, a and b — and that bigger idea is the Pythagorean theorem! We'll explore this more in Chapter 8.