Every number you have met so far lives on a number line, but not every point on that line is a fraction. Some numbers, like \(\sqrt{2}\) or \(\pi\), have decimal digits that go on forever with no pattern at all. In this chapter you will sort numbers into rational and irrational, turn repeating decimals into fractions, work with square roots and cube roots, and learn how to estimate, locate and compare irrational numbers like a pro.
1. Rational numbers
A rational number is a number that can be written as a fraction \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
All of these are rational: \(5 = \dfrac{5}{1}\), \(-\dfrac{3}{4}\), \(0.25 = \dfrac{1}{4}\), and \(2\dfrac{1}{3} = \dfrac{7}{3}\). Every integer, every whole number and every fraction is rational.
When you divide the top of a fraction by the bottom, one of two things happens. The decimal either terminates (it stops), or it repeats (a block of digits comes back again and again). That is true for every rational number, because in long division there are only so many possible remainders, so a remainder must eventually come back.
Write \(\dfrac{7}{8}\) and \(\dfrac{5}{11}\) as decimals.
\(\dfrac{7}{8} = 0.875\), which terminates.
\(\dfrac{5}{11} = 0.454545\ldots = 0.\overline{45}\), which repeats. The bar goes over the block of digits that repeats.
2. Converting repeating decimals to fractions
Terminating decimals are easy: read the place value. For instance, \(0.375 = \dfrac{375}{1000} = \dfrac{3}{8}\). Repeating decimals need a clever trick that uses subtraction to make the endless tail disappear.
- Name the number: \(x = \) the repeating decimal.
- Multiply by \(10\) if one digit repeats, by \(100\) if two digits repeat, by \(1000\) if three repeat, and so on.
- Subtract the original equation from the new one so the repeating tails cancel.
- Solve for \(x\) and simplify the fraction.
Write \(0.\overline{7}\) as a fraction.
Let \(x = 0.777\ldots\) Then \(10x = 7.777\ldots\) Subtract: \(10x - x = 7.777\ldots - 0.777\ldots\), so \(9x = 7\) and \(x = \dfrac{7}{9}\).
Write \(0.\overline{36}\) as a fraction.
Two digits repeat, so multiply by \(100\). Let \(x = 0.3636\ldots\) Then \(100x = 36.3636\ldots\) Subtract: \(99x = 36\), so \(x = \dfrac{36}{99} = \dfrac{4}{11}\).
Write \(0.4\overline{5} = 0.45555\ldots\) as a fraction.
Here only the \(5\) repeats, but a \(4\) sits in front of it. Make two equations with the same repeating tail: \(100x = 45.555\ldots\) and \(10x = 4.555\ldots\) Subtract: \(90x = 41\), so \(x = \dfrac{41}{90}\).
\(0.\overline{45}\) is not \(\dfrac{45}{100}\). A repeating decimal never equals the fraction you get by just reading the digits. Use the subtraction method: \(0.\overline{45} = \dfrac{45}{99} = \dfrac{5}{11}\).
3. Square roots and cube roots
The square root \(\sqrt{p}\) of a positive number \(p\) is the positive number whose square is \(p\). The cube root \(\sqrt[3]{p}\) is the number whose cube is \(p\).
For example, \(\sqrt{81} = 9\) because \(9^2 = 81\), and \(\sqrt[3]{125} = 5\) because \(5^3 = 125\). Numbers like \(81\) and \(125\) are called perfect squares and perfect cubes. Memorize these:
| \(n\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(n^2\) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
| \(n^3\) | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Two things to remember. First, the equation \(x^2 = p\) has two solutions when \(p > 0\): \(x = \sqrt{p}\) and \(x = -\sqrt{p}\). Second, a cube root can be negative: \(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\). A square root of a negative number does not exist among real numbers, since no real number squared gives a negative result.
Solve \(x^2 = \dfrac{64}{81}\) and \(x^3 = -\dfrac{1}{27}\).
\(x = \pm\sqrt{\dfrac{64}{81}} = \pm\dfrac{8}{9}\). For the cube, \(x = \sqrt[3]{-\dfrac{1}{27}} = -\dfrac{1}{3}\).
4. Irrational numbers
An irrational number is a real number that is not rational. It cannot be written as a fraction of two integers. Its decimal expansion never ends and never repeats.
If a positive integer is not a perfect square, its square root is irrational. So \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{11}\) are irrational, while \(\sqrt{49} = 7\) is rational. In the same way \(\sqrt[3]{10}\) is irrational because \(10\) is not a perfect cube. The number \(\pi\), the circumference of a circle divided by its diameter, is irrational too.
The picture shows where \(\sqrt{2}\) comes from: a tilted square inside a \(2 \times 2\) square has area \(4 - 4 \times \dfrac{1}{2} = 2\). Its side \(s\) satisfies \(s^2 = 2\), so \(s = \sqrt{2} = 1.41421356\ldots\) A real length, but not a fraction.
The sum of a rational number and an irrational number is irrational, and so is the product of a nonzero rational number and an irrational number. For instance, \(3 + \sqrt{5}\) and \(2\pi\) are irrational.
\(3.14\) and \(\dfrac{22}{7}\) are only rational approximations of \(\pi\). The true value \(\pi = 3.14159265\ldots\) never stops and never repeats. Also, a long decimal is not automatically irrational: \(0.\overline{142857} = \dfrac{1}{7}\) is rational.
5. Estimating irrational numbers
You do not need a calculator to find where an irrational number sits. Trap it between two perfect squares (or cubes), then squeeze it with decimals.
- Find the two consecutive perfect squares around \(n\).
- Take their square roots to get two consecutive integers.
- Test decimals between them by squaring, until you reach the accuracy you need.
Estimate \(\sqrt{30}\) to the nearest hundredth.
\(25 \lt 30 \lt 36\), so \(5 \lt \sqrt{30} \lt 6\). Test tenths: \(5.4^2 = 29.16\) and \(5.5^2 = 30.25\), so \(5.4 \lt \sqrt{30} \lt 5.5\). Test hundredths: \(5.47^2 = 29.9209\) and \(5.48^2 = 30.0304\), so \(5.47 \lt \sqrt{30} \lt 5.48\). Since \(30\) is much closer to \(30.0304\) than to \(29.9209\), \(\sqrt{30} \approx 5.48\).
On my home planet we always ask: is \(n\) closer to the lower or the higher perfect square? \(30\) is \(5\) away from \(25\) and \(6\) away from \(36\), so \(\sqrt{30}\) is a bit closer to \(5\) than to \(6\).
6. Locating irrational numbers on a number line
Once you have a good estimate, you can place the number on a number line. Here are four square roots located with tenths marked between the integers.
Locate \(\sqrt{7}\) and \(-\sqrt{7}\).
\(4 \lt 7 \lt 9\), so \(2 \lt \sqrt{7} \lt 3\). Since \(2.6^2 = 6.76\) and \(2.7^2 = 7.29\), we get \(2.6 \lt \sqrt{7} \lt 2.7\): plot it between the marks \(2.6\) and \(2.7\). The opposite \(-\sqrt{7}\) is the mirror image on the other side of \(0\), between \(-2.7\) and \(-2.6\).
7. Comparing and ordering real numbers
The real numbers are all the rational and irrational numbers together: every point of the number line. To compare two of them, you have two reliable tools.
- Decimals: write each number as a decimal with enough digits to see which is bigger.
- Squares or cubes: for positive numbers, \(a \lt b\) exactly when \(a^2 \lt b^2\). So you can compare \(\sqrt{21}\) with \(4.6\) by comparing \(21\) with \(4.6^2 = 21.16\).
Order from least to greatest: \(\sqrt{5}\), \(2.3\), \(\dfrac{9}{4}\), \(\sqrt[3]{12}\).
Decimals: \(\sqrt{5} \approx 2.236\), \(2.3\), \(\dfrac{9}{4} = 2.25\), \(\sqrt[3]{12} \approx 2.289\) (because \(2.29^3 \approx 12.009\)). The order is \[\sqrt{5} \lt \dfrac{9}{4} \lt \sqrt[3]{12} \lt 2.3.\]
The order flips for negatives: \(-\sqrt{10} \lt -3\) because \(\sqrt{10} \approx 3.16\) is farther from \(0\) than \(3\) is.
Key takeaways
- A rational number is a fraction \(\dfrac{a}{b}\) of integers (\(b \neq 0\)); its decimal terminates or repeats.
- To convert a repeating decimal: multiply by \(10\), \(100\), \(1000\)… to line up the tails, subtract, and solve. For example \(0.\overline{36} = \dfrac{4}{11}\).
- \(\sqrt{p}\) is the positive number whose square is \(p\); \(x^2 = p\) has two solutions \(\pm\sqrt{p}\). A cube root can be negative.
- If \(n\) is a positive integer that is not a perfect square, \(\sqrt{n}\) is irrational. \(\pi\) is irrational.
- Estimate an irrational number by trapping it between consecutive perfect squares or cubes, then squeezing it with tenths and hundredths.
- Rational and irrational numbers together make the real numbers, and each one has its place on the number line.
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