ORCCA Set Notation and Types of Numbers

Section 1.6 Set Notation and Types of Numbers

When we talk about how many or how much of something we have, it often makes sense to use different types of numbers. For example, if we are counting dogs in a shelter, the possibilities are only \(0,1,2,\ldots\text{.}\) (It would be difficult to have \(\frac{1}{2}\) of a dog.) On the other hand if you were weighing a dog in pounds, it doesn't make sense to only allow yourself to work with whole numbers. The dog might weigh something like \(28.35\) pounds. These examples highlight how certain kinds of numbers are appropriate for certain situations. We'll classify various types of numbers in this section.

Subsection 1.6.1 Set Notation

What is the mathematical difference between these three “lists?”

\begin{equation*} 28, 31, 30\qquad\{28, 31, 30\}\qquad(28, 31, 30) \end{equation*}

To a mathematician, the last one, \((28, 31, 30)\) is an ordered triple. What matters is not merely the three numbers, but also the order in which they come. The ordered triple \((28, 31, 30)\) is not the same as \((30, 31, 28)\text{;}\) they have the same numbers in them, but the order has changed. For some context, February has \(28\) days; then March has \(31\) days; then April has \(30\) days. The order of the three numbers is meaningful in that context.

With curly braces and \(\{28, 31, 30\}\text{,}\) a mathematician sees a collection of numbers and does not particularly care in which order they are written. Such a collection is called a set. All that matters is that these numbers are part of a collection. They've been written in some particular order because that's necessary to write them down. But you might as well have put the three numbers in a bag and shaken up the bag. For some context, maybe your favorite three NBA players have jersey numbers \(30\text{,}\) \(31\text{,}\) and \(28\text{,}\) and you like them all equally well. It doesn't really matter what order you use to list them.

So we can say:

\begin{align*} \{28, 31, 30\}\amp=\{30, 31, 28\}\amp(28, 31, 30)\amp\neq(30, 31, 28) \end{align*}

What about just writing \(28, 31, 30\text{?}\) This list of three numbers is ambiguous. Without the curly braces or parentheses, it's unclear to a reader if the order is important. Set notation is the use of curly braces to surround a list/collection of numbers, and we will use set notation frequently in this section.

Checkpoint 1.6.1. Set Notation.

Practice using (and not using) set notation.

Subsection 1.6.2 Different Number Sets

In the introduction, we mentioned how different sets of numbers are appropriate for different situations. Here are the basic sets of numbers that are used in basic algebra.

Natural Numbers

When we count, we begin: \(1, 2, 3, \dots\) and continue on in that pattern. These numbers are known as natural numbers.

\(\mathbb{N}=\{1,2,3,\dots \}\)

Whole Numbers

If we include zero, then we have the set of whole numbers.

\(\{0,1,2,3,\dots \}\) has no standard symbol, but some options are \(\mathbb{N}_0\text{,}\) \(\mathbb{N}\cup\{0\}\text{,}\) and \(\mathbb{Z}_{\geq0}\text{.}\)

Integers

If we include the negatives of whole numbers, then we have the set of integers.

\(\mathbb{Z}=\{\dots,-3,-2,-1,0,1,2,3,\dots \}\text{.}\)

A \(\mathbb{Z}\) is used because one word in German for “numbers” is “Zahlen.”

Rational Numbers

A rational number is any number that can be written as a fraction of integers, where the denominator is nonzero. Alternatively, a rational number is any number that can be written with a decimal that terminates or that repeats.

\(\mathbb{Q}=\left\{0,1,-1,2,\frac{1}{2},-\frac{1}{2},-2,3,\frac{1}{3},-\frac{1}{3},-3,\frac{3}{2},\frac{2}{3}\ldots\right\}\)

\(\mathbb{Q}=\left\{0,1,-1,2,0.5,-0.5,-2,3,0.\overline{3},-0.\overline{3},-3,1.5,0.\overline{6}\ldots\right\}\)

A \(\mathbb{Q}\) is used because fractions are quotients of integers.

Irrational Numbers

Any number that cannot be written as a fraction of integers belongs to the set of irrational numbers. Another way to say this is that any number whose decimal places goes on forever without repeating is an irrational number. Some examples include \(\pi\approx3.1415926\ldots\text{,}\) \(\sqrt{15}\approx3.87298\ldots\text{,}\) \(e\approx2.71828\ldots\)

There is no standard symbol for the set of irrational numbers.

Real Numbers

Any number that can be marked somewhere on a number line is a real number. Real numbers might be the only numbers you are familiar with. For a number to not be real, you have to start considering things called complex numbers, which are not our concern right now.

The set of real numbers can be denoted with \(\mathbb{R}\) for short.

a disc represents all real numbers; the disc is separated into two areas: one for irrational numbers with examples like pi, e, sqrt(15), and 1.010010001..., and the other for rational numbers with examples like 3/17, 1.25, and 4.3 repeating; within the rational area, there is a disc representing integers, with -42 as an example; within the integer area there is a region representing whole numbers with 0 as an example; within the whole numbers area there is a region representing natural numbers, with 23 as an example — Figure 1.6.2. Types of Numbers

Warning 1.6.3. Rational Numbers in Other Forms.

Any number that can be written as a ratio of integers is rational, even if it's not written that way at first. For example, these numbers might not look rational to you at first glance: \(-4\text{,}\) \(\sqrt{9}\text{,}\) \(0\pi\text{,}\) and \(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\text{.}\) But they are all rational, because they can respectively be written as \(\frac{-4}{1}\text{,}\) \(\frac{3}{1}\text{,}\) \(\frac{0}{1}\text{,}\) and \(\frac{1}{1}\text{.}\)

Example 1.6.4. Determine If Numbers Are This Type or That Type.

Determine which numbers from the set \(\left\{-102, -7.25, 0, \frac{\pi}{4}, 2, \frac{10}{3}, \sqrt{19}, \sqrt{25}, 10.\overline{7} \right\}\) are natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Explanation

All of these numbers are real numbers, because all of these numbers can be positioned on the real number line.

Each real number is either rational or irrational, and not both. \(-102\text{,}\) \(-7.25\text{,}\) \(0\text{,}\) and \(2\) are rational because we can see directly that their decimal expressions terminate. \(10.\overline{7}\) is also rational, because its decimal expression repeats. \(\frac{10}{3}\) is rational because it is a ratio of integers. And last but not least, \(\sqrt{25}\) is rational, because that's the same thing as \(5\text{.}\)

This leaves only \(\frac{\pi}{4}\) and \(\sqrt{19}\) as irrational numbers. Their decimal expressions go on forever without entering a repetitive cycle.

Only \(-102\text{,}\) \(0\text{,}\) \(2\text{,}\) and \(\sqrt{25}\) (which is really \(5\)) are integers.

Of these, only \(0\text{,}\) \(2\text{,}\) and \(\sqrt{25}\) are whole numbers, because whole numbers excludes the negative integers.

Of these, only \(2\) and \(\sqrt{25}\) are natural numbers, because the natural numbers exclude \(0\text{.}\)

Checkpoint 1.6.5.

Checkpoint 1.6.6.

In the introduction, we mentioned that the different types of numbers are appropriate in different situation. Which number set do you think is most appropriate in each of the following situations?

Subsection 1.6.3 Converting Repeating Decimals to Fractions

We have learned that a terminating decimal number is a rational number. It's easy to convert a terminating decimal number into a fraction of integers: you just need to multiply and divide by one of the numbers in the set \(\{10,100,1000,\ldots\}\text{.}\) For example, when we say the number \(0.123\) out loud, we say “one hundred and twenty-three thousandths.” While that's a lot to say, it makes it obvious that this number can be written as a ratio:

\begin{equation*} 0.123=\frac{123}{1000}\text{.} \end{equation*}

Similarly,

\begin{equation*} 21.28=\frac{2128}{100}=\frac{532\cdot4}{25\cdot4}=\frac{532}{25}\text{,} \end{equation*}

demonstrating how any terminating decimal can be written as a fraction.

Repeating decimals can also be written as a fraction. To understand how, use a calculator to find the decimal for, say, \(\frac{73}{99}\) and \(\frac{189}{999}\) You will find that

\begin{equation*} \frac{73}{99}=0.73737373\ldots=0.\overline{73}\qquad\frac{189}{999}=0.189189189\ldots=0.\overline{189}\text{.} \end{equation*}

The pattern is that dividng a number by a number from \(\{9,99,999,\ldots\}\) with the same number of digits will create a repeating decimal that starts as “\(0\text{.}\)” and then repeats the numerator. We can use this observation to reverse engineer some fractions from repeating decimals.

Checkpoint 1.6.7.

Converting a repeating decimal to a fraction is not always quite this straightforward. There are complications if the number takes a few digits before it begins repeating. For your interest, here is one example on how to do that.

Example 1.6.8.

Can we convert the repeating decimal \(9.134343434\ldots=9.1\overline{34}\) to a fraction? The trick is to separate its terminating part from its repeating part, like this:

\begin{equation*} 9.1+0.034343434\ldots\text{.} \end{equation*}

Now note that the terminating part is \(\frac{91}{10}\text{,}\) and the repeating part is almost like our earlier examples, except it has an extra \(0\) right after the decimal. So we have:

\begin{equation*} \frac{91}{10}+\frac{1}{10}\cdot0.34343434\ldots\text{.} \end{equation*}

With what we learned in the earlier examples and basic fraction arithmetic, we can continue:

\begin{align*} 9.134343434\ldots\amp=\frac{91}{10}+\frac{1}{10}\cdot0.34343434\ldots\\ \amp=\frac{91}{10}+\frac{1}{10}\cdot\frac{34}{99}\\ \amp=\frac{91}{10}+\frac{34}{990}\\ \amp=\frac{91\multiplyright{99}}{10\multiplyright{99}}+\frac{34}{990}\\ \amp=\frac{9009}{990}+\frac{34}{990}=\frac{9043}{990} \end{align*}

Check that this is right by entering \(\frac{9043}{990}\) into a calculator and seeing if it returns the decimal we started with, \(9.134343434\ldots\text{.}\)