Sorting a number into its sets

When you reach for it

The numbers met so far came in families, each built by adding to the one before: counting numbers, then zero, then the negative Integers, then fractions and decimals, and most recently numbers such as \displaystyle \sqrt{2} that are not fractions at all. So one number usually belongs to several families at once. 5 is a counting number, and also an integer, a rational number and a real number.

Sorting a number means listing every family it belongs to. You need it whenever a statement depends on the kind of number in play: a count of people must be a whole number, and “solve in the integers” throws out an answer such as \displaystyle \frac{7}{2}. It takes two things: knowing how the families sit inside one another, and seeing past the way a number is written to the number itself.

The procedure

The six sets

• The natural numbers, also called the counting numbers: \mathbb{N} = \{1, 2, 3, \ldots\}. The braces list the members, as in roster notation, and the dots mean the list goes on forever. In this lesson they start at 1.
• The whole numbers: 0, 1, 2, 3, \ldots, the natural numbers with 0 added. This set has no standard letter.
• The integers: \mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}, the whole numbers together with their opposites.
• The rational numbers, \mathbb{Q}: every number that can be written as \displaystyle \frac{a}{b} with a and b integers and b \neq 0. As What makes a number rational shows, these are exactly the numbers whose decimals either stop or repeat.
• The irrational numbers: the real numbers that are not rational. Their decimals never stop and never repeat. \displaystyle \sqrt{2} is one, and so is \pi.
• The real numbers, \mathbb{R}: the rational and irrational numbers together, which are exactly the numbers that label the points of the number line.

How the sets nest

Every natural number is whole, since the whole numbers are the natural numbers with one member added. Every whole number is an integer, since the integers add the opposites. Every integer n is rational, since \displaystyle n = \frac{n}{1}. And every rational number is real. Writing \subset for “sits inside”,

Each step adds something new: 0 is whole but not natural, -1 is an integer but not whole, \displaystyle \frac{1}{2} is rational but not an integer, and \displaystyle \sqrt{2} is real but not rational.

The irrational numbers are not in the chain. They are the real numbers that are not rational, so the two sets share no member, and together they make up all of \mathbb{R}. Every real number is rational or irrational, and none is both.

The smallest set decides the rest

A number in a region is also in every region drawn around it, and in no other. So once you know the smallest of the six sets a number belongs to, the full list follows. A real number can land in only five places, so there are only five possible answers:

The useful order of questions runs from the outside in. Is the number rational or irrational? If irrational, that is the last row and you are done. If rational, is it an integer, meaning that in lowest terms its denominator is 1? If it is an integer, its sign settles the rest: positive means natural, zero means whole but not natural, negative means an integer and no smaller set.

Find the value before you sort

Which sets a number belongs to is a fact about the number, not about how it is written. \displaystyle -\frac{8}{4} has a fraction bar, but it is the integer -2; \displaystyle \sqrt{49} has a root sign, but it is 7. So simplify exactly first, and sort second, with tools you already have.

• A fraction bar is division. If the top is a multiple of the bottom, the number is an integer.
• For a whole number n, \displaystyle \sqrt{n} is a whole number when n is a perfect square (0, 1, 4, 9, 16, \ldots), and irrational otherwise.
• For a \geq 0 and b > 0, \displaystyle \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} and \displaystyle \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} (operations with radicals). So \displaystyle \sqrt{2} \cdot \sqrt{32} = \sqrt{64} = 8, a natural number made from two irrational ones.
• A repeating decimal turns into a fraction, and a terminating one is a fraction over a power of 10.

The definition of rational asks only whether the number has some name \displaystyle \frac{a}{b} with integers a and b \neq 0. A name of another form, such as a fraction with roots in it, proves nothing either way.

Rational or irrational

A decimal known completely settles it: one that stops or repeats is rational, one that does neither is irrational, where “completely” means a rule for every digit. Beyond that, Irrational numbers gives three facts to use. The square root of a whole number that is not a perfect square is irrational, and so is \pi. If r is rational and x is irrational, then r + x and r - x are irrational, and so is r \cdot x when r \neq 0: if r + x were a rational s, then x = s - r would be rational. And two irrational numbers together follow no rule, since \displaystyle \sqrt{2} \cdot \sqrt{32} = 8 but \displaystyle \sqrt{2} \cdot \sqrt{5} = \sqrt{10}, so such a case has to be simplified.

All, some and none

“All A are B” is true exactly when A sits inside B, and one member of A outside B, a counterexample, makes it false. “All integers are rational” is true; “all rational numbers are integers” is false, and \displaystyle \frac{1}{2} alone shows it.

“Some A are B” says at least one number is in both, and one example proves it. In mathematics “some” does not mean “not all”: “some whole numbers are integers” is true, even though all of them are.

“No A is B” says the two sets share no member. “No irrational number is an integer” is true, because every integer is rational and nothing is both. The figure answers most such questions at a glance.

Worked examples

Boundaries

The form \displaystyle \frac{a}{b} counts only with integers a and b, and b \neq 0. \displaystyle \frac{\sqrt{35}}{5} has a fraction bar and is irrational. \displaystyle \frac{5}{0} is not a number at all, so it belongs to none of the six sets.

“Natural” does not mean the same thing in every book. Here the natural numbers start at 1, and the set with 0 added is the whole numbers. Some books, and most writing on computing, start the natural numbers at 0. Check which a text means.

The digits on show do not decide. A decimal written 1.41421356\ldots with no rule for the rest might be \displaystyle \sqrt{2}, or a rational number that shares those first digits and then repeats. A calculator display always stops, so it always shows a rational number, even when the exact answer is irrational.

Not every number is real. No real number squares to a negative number, so \displaystyle \sqrt{-9} is not real and belongs to none of the six sets. It is one of the complex numbers.

Where this goes next

Once you can tell where a number lands, you can ask which sets keep their answers inside. The integers are not closed under division, since \displaystyle 1 \div 2 = \frac{1}{2} falls outside them, while the rational numbers are, apart from division by zero. That is the closure property, decided by exactly this sorting.

The real number line shows the same sets as points, with the irrational numbers filling every gap the rational ones leave; placing one there starts by bracketing it between perfect squares. The same sorting works for cube roots, with a twist: a negative number has a real cube root, so \displaystyle \sqrt[3]{-125} = -5 is an integer. Pictures of sets inside sets are Venn diagrams, and beyond \mathbb{R} lie the complex numbers, built so that \displaystyle \sqrt{-9} has a meaning.

Exercises

Sort the number

List every set each number belongs to: natural, whole, integer, rational, irrational, real.

1. -15
2. \displaystyle \frac{21}{3}
3. -2.\overline{5}
4. \displaystyle \sqrt{40}
5. \displaystyle -\sqrt{64}
6. \displaystyle \frac{0}{9}

Simplify first

List every set each number belongs to.

7. \displaystyle \sqrt{12} \cdot \sqrt{3}
8. \displaystyle \frac{\pi}{2}
9. \displaystyle (\sqrt{19})^2 - 22
10. \displaystyle \frac{\sqrt{75}}{\sqrt{3}}
11. \displaystyle \frac{\sqrt{2} + \sqrt{2}}{\sqrt{2}}

Read the decimal

Decide whether each number is rational or irrational. Where it is rational, write it as a fraction.

12. 0.123456789101112\ldots, the natural numbers written one after another
13. 1.\overline{216}
14. -0.0125

All, some or none

Decide whether each statement is true or false, with a reason or a counterexample.

15. All natural numbers are whole numbers.
16. Some integers are not rational.
17. No whole number is negative.
18. Some irrational numbers are greater than 100.

Reasoning

19. Explain why no real number belongs to all six sets, and why none belongs to only one of them.
20. A real number belongs to exactly four of the six sets. Which number is it? Explain why it is the only one, by saying how many sets each of the other kinds of real number belongs to.