Chapter 7: Rational and Irrational Numbers

In this chapter, we will look again at the real line, which is a way of showing numbers as points on a line. You learned about this in seventh grade. First, we'll remember how whole numbers (integers) and fractions (rational numbers) are placed as points on the line.

Since every rational number can be shown as a point on the line, we can ask an interesting question: does every point on the line stand for a rational number?

Remember, a point on the line matches 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, the length is negative.

By building special shapes called "tilted squares," we will discover something the Pythagoreans found about 2,500 years ago: some lengths — like the diagonal of a square with side length 1 — do not match any rational number. This special construction gives us numbers whose squares are whole numbers. That leads us to use the symbol √A to mean "a number whose square is A." We will also use the symbol ³√V to mean "the side length of a cube whose volume is V."

Building tilted squares also lets us discover the famous 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 an area of 2, and show that its side length (√2) can never be written as a fraction. This means √2 is not a rational number — it is called an irrational number. The same idea works for √5 and other lengths made using tilted squares. In fact, here's a rule: if N is a whole number, either N is a perfect square (the square of an integer), or √N is irrational — meaning it cannot be written as a fraction.

After that, we'll ask: can we write down lengths that aren't fractions, using numbers in some other way? The ancient Greeks couldn't do this, mostly because they didn't have a good number system for measuring lengths. Luckily, we do! It's called the decimal system (which you studied in seventh grade).

Remember from grade 7: a rational number can be written as a decimal that stops (terminates) only if its denominator (the bottom number of the fraction) is made up of only 2s and 5s multiplied together. That means many rational numbers — like 1/3, 1/7, and 1/12 — do not stop as decimals. Instead, they repeat forever. And any decimal that repeats also stands for a rational number.

We can think of a decimal expansion as a step-by-step method for getting closer and closer to a point on the line, by dividing things into tenths again and again. In fact, every decimal expansion stands for some point on the line — some number. But if the decimal doesn't stop or repeat, that number is irrational.

So now the big question is: can we write every possible length as a decimal? We'll start with square roots and use a method called Newton's method to estimate them. Here's how it works: start with a reasonable guess, then use this formula to get a better guess:

a_new = ½ (a_old + N ÷ a_old)

Through examples, we'll see that this method lets us find the decimal expansion of the square root of any number N, as accurately as we want.

Finally, we'll point out something important: when doing math with irrational numbers (as well as rational ones), we must be careful. To get a certain level of accuracy in our final answer, we may need to start with even more accurate numbers.

Section 7.1: Representing Numbers Geometrically

Let's remember how the rational numbers are shown as points on a line. Using a straight edge, draw a horizontal line. If you pick any two points, a and b, on the line, we say a < b ("a is less than b") when a is to the left of b. The piece of the line between a and b is called the interval between them. For any two different points, either a < b or b < a must be true. Also, if a < b, we can also write b > a.

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

We can also find points for numbers with halves. For example, the midpoint 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 matches 1/3, and the first two parts together match 2/3. In fact, for any whole number p, we can find the point for p/3 by laying end-to-end copies of one-third of the unit interval — to the right of 0 if p is positive, and to the left if p is negative. We can do the same thing with any positive whole number q instead of 3, by making a length that is one-qth of the unit interval. This way, every rational number p/q gets its own point on the line — to the left of 0 if it's negative, and to the right if it's positive.

The number line also helps us picture the decimal expansion of a number. Let's say a is a positive number. There is some whole number N so 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. Let d₁ be how many of these parts fit between N and a. This digit d₁ (a number from 0 to 9) is the tenths digit, and we write the number as N.d₁. If this equals a, we're finished. If not, we repeat: divide the interval between N.d₁ and N.(d₁+1) into ten equal parts, and let d₂ count how many parts fit between N.d₁ and a. This gives us the hundredths digit, written N.d₁d₂. If we keep doing this, we get a list of decimal numbers that get closer and closer to a. This gives us a real method for estimating a very closely. As we learned in grade 7, sometimes this process never stops exactly — like when a = 1/3 or 1/7.

Instead of just thinking of this as a way to turn a number into a decimal, let's think of it as a way to turn a length on the number line into a decimal. Take any point a on the number line (let's say it's positive). The same steps let us find a decimal expansion for a — a step-by-step way of measuring the length from 0 to a, using smaller and smaller tenths of the unit interval.

We'll start by looking at lengths we can actually build using shapes. This will help us answer the question: are there lengths that can't be written as rational numbers? To explore this, we need to move from numbers on a line to numbers on a flat surface (a plane).

Using the number line we made, draw a line straight up and down (perpendicular) through the origin, using the same unit length. Now we can match every pair of rational numbers (a, b) to a point on the plane: move along the horizontal line (called the x-axis) to point a, then move up or down a distance of b. This gives us the point (a, b).

Example 1

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

Solution. We use a ruler to measure. Sometimes the scale on our ruler won't match exactly — a "box" that should be half an inch might measure differently. After adjusting for scale, we find that AC is about 3.05 inches long.

By using a ruler, we can always estimate the length of a line segment using a fraction (a rational number). How accurate our estimate is depends on how precise our ruler is. But here's the big question: can every length be described exactly using a rational number?

The coordinate system on the plane helps us assign lengths to line segments. Let's look at a few more examples.

Example 2

In Figure 2, we've drawn a tilted square (shown with dashed lines) inside a square that sits straight (with horizontal and vertical sides). Each small square with solid lines is a unit square — its sides are each one unit long. The whole figure is a 2-by-2 square, so its area is 2 × 2 = 4 square units.

The tilted square takes up exactly half the area 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 the area of a square equals the side length multiplied by itself (squared), the side length of the tilted square is a number whose square is 2. We write this as √2.

We use the symbol √A (called a square root) to mean "a number a whose square equals A" (so 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.

Since 2² = 4, 3² = 9, 4² = 16, and 5² = 25, the numbers 4, 9, 16, and 25 have whole numbers as their square roots. A positive whole number whose square root is also a whole number is called a perfect square. But for other numbers, like 2, 3, 5, or 6, we still need a way to find their square roots. The tilted square method gives us a way to build lengths that match the square roots of whole numbers, as we'll see next.

Example 3

In Figure 3, the big square has sides of length 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, 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. So, the side length of the tilted square is √5.

Using the same idea: in Figure 4, the big square has sides of length 7, so its area is 49 square units. Each outside 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 side length of the tilted square is 5 units. That means √25 = 5.

Going further. We used the "tilted square" method to find exact lengths for square roots of whole numbers. Our goal is to eventually find a way to write these lengths as decimals. Some students might notice that this method works for any starting lengths, a and b — not just whole numbers. That's exactly right! This bigger idea is called the Pythagorean theorem, which connects the side lengths of a right triangle. We'll explore this further in Chapter 8. The next part gives a preview of that idea.