ORCCA Set Notation and Types of Numbers

Section A.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.

Figure A.6.1. Alternative Video Lesson

Subsection A.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 A.6.2. Set Notation.

Practice using (and not using) set notation.

According to Google, the three most common error codes from visiting a web site are 403, 404, and 500.

(a)

Without knowing which error code is most common, express this set mathematically.

Explanation.

Since we only have to describe a collection of three numbers and their order doesn’t matter, we can write {403,404,500}.

(b)

Error code 500 is the most common. Error code 403 is the least common of these three. And that leaves 404 in the middle. Express the error codes in a mathematical way that appreciates how frequently they happen, from most often to least often.

Explanation.

Now we must describe the same three numbers and we want readers to know that the order we are writing the numbers matters. We can write (500,404,403).

Subsection A.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 A.6.3. Types of Numbers

Warning A.6.4. 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 A.6.5. 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 exclude the negative integers.

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

Checkpoint A.6.6.

Give an example of a whole number that is not an integer.
Give an example of an integer that is not a whole number.
Give an example of a rational number that is not an integer.
Give an example of a irrational number.
Give an example of a irrational number that is also an integer.

Explanation.

Since all whole numbers belong to integers, we cannot write any whole number which is not an integer. Type DNE (does not exist) for this question.
Any negative integer, like $-1\text{,}$ is not a whole number, but is an integer.
Any terminating decimal, like $1.2\text{,}$ is a rational number, but is not an integer.
$\pi$ is the easiest number to remember as an irrational number. Another constant worth knowing is $e\approx2.718\text{.}$ Finally, the square root of most integers are irrational, like $\sqrt{2}$ and $\sqrt{3}\text{.}$
All irrational numbers are non-repeating and non-terminating decimals. No irrational numbers are integers.

Checkpoint A.6.7.

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?

(a)

The number of people in a math class that play the ukulele.

This number is best considered as a

natural number
whole number
integer
rational number
irrational number
real number

Explanation.

The number of people who play the ukulele could be $0,1,2,\dots\text{,}$ so the whole numbers are the appropriate set.

(b)

The hypotenuse’s length in a given right triangle.

This number is best considered as a

natural number
whole number
integer
rational number
irrational number
real number

Explanation.

A hypotenuse’s length could be $1\text{,}$ $1.2\text{,}$ $\sqrt{2}$ (which is irrational), or any other positive number. So the real numbers are the appropriate set.

(c)

The proportion of people in a math class that have a cat.

This number is best considered as a

natural number
whole number
integer
rational number
irrational number
real number

Explanation.

This proportion will be a ratio of integers, as both the total number of people in the class and the number of people who have a cat are integers. So the rational numbers are the appropriate set.

(d)

The number of people in the room with you who have the same birthday as you.

This number is best considered as a

natural number
whole number
integer
rational number
irrational number
real number

Explanation.

We know that the number of people must be a counting number, and since you are in the room with yourself, there is at least one person in that room with your birthday. So the natural numbers are the appropriate set.

(e)

The total revenue (in dollars) generated for ticket sales at a Timbers soccer game.

This number is best considered as a

natural number
whole number
integer
rational number
irrational number
real number

Explanation.

The total revenue will be some number of dollars and cents, such as $\$631{,}897.15\text{,}$ which is a terminating decimal and thus a rational number. So the rational numbers are the appropriate set.

Subsection A.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 dividing a number by a number from $\{9,99,999,\ldots\}$ with the same number of digits will create a repeating decimal that starts as “$0.$” and then repeats the numerator. We can use this observation to reverse engineer some fractions from repeating decimals.

Checkpoint A.6.8.

(a)

Write the rational number $0.772772772\ldots$ as a fraction.

Explanation.

The three-digit number $772$ repeats after the decimal. So we will make use of the three-digit denominator $999\text{.}$ And we have $\frac{772}{999}\text{.}$

(b)

Write the rational number $0.69696969\ldots$ as a fraction.

Explanation.

The two-digit number $69$ repeats after the decimal. So we will make use of the two-digit denominator $99\text{.}$ And we have $\frac{69}{99}\text{.}$ But this fraction can be reduced to $\frac{23}{33}\text{.}$

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 A.6.9.

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, $9.134343434\ldots\text{.}$