1.1 Real Numbers

Learning Objectives

By the end of this section, you will:

• know the definitions of different types of numbers (natural, rational, real, positive, negative)
• visualize real numbers on a number line and understand greater than/less than relationships
• know the arithmetic operations and in what order to perform them

Numbers! Can't live with 'em, can't live without 'em, or something like that? Well, they're everywhere at any rate, so we're going to learn how to use them. Let's start out with counting. (A-one...a-two-whoo...a-three...)

We're familiar with the counting numbers, aka whole numbers, aka natural numbers denoted by the symbol \( \mathbb{N} \), which are \( 0, 1, 2, 3, ..., 17, ..., 10001, ... \) etc. They go on forever getting bigger, and you can tick them off on a number line like the one below.

Why are there extra ticks and an arrow indicating the number line also goes off forever to the left? Glad you asked. Those are for the rest of the integers (denoted by \( \mathbb{Z} \)), which include negative whole numbers as well as the (positive) natural numbers. Check it out.

Addition and subtraction are two arithmetic operations we can use on numbers. If you add two integers, the result is their sum, and it will also be an integer. You can think of subtraction as adding a negative number, and the resulting difference will also be an integer. Addition and subtraction are inverse operations, in the sense that they undo each other. Also, it doesn't matter what order you add in, because addition is a commutative operation, but order does matter with subtraction! Consider the examples below illustrating these properties.

Example \(\PageIndex{1}\)

1. \( 3 + 2 = 2 + 3 = 5 \)
2. \( 4000 + 101 = 4101 \)
3. \( -3 + (-4) = -7 \)
4. \( 8 - 6 = 8 + (-6) = 2 \)
5. \( 300 - 10 = 290 \), but \( 10 - 300 = -290 \)
6. \( 2 + 4 - 4 = 2 \)

Exercise \(\PageIndex{1}\)

1. Think of two integers whose sum is 13.
2. Think of two integers whose difference is 5.
3. Think of an integer such that subtracting 7 will result in -2.

Answer

1. \(8\) and \(5\) will work, and so will \(10\) and \(3\), or even \(20 + (-7)\).
2. \(20\) and \(15\) (in the order \(20 - 15\)) will work, and so will \(47\) and \(42\), etc.
3. For this one, only positive 5 will work. That's \(5 - 7 = -2\).

By the way, you can think of addition (and subtraction) as hopping forwards (or backwards) a certain amount of times along the number line. For example, you can represent \( -2 + 3 = 1 \) with the picture below.

The next arithmetic operations to look at are multiplication and division. We multiply two numbers (sometimes called factors) using a dot symbol, and the result is called their product. Multiplying is code for adding a bunch of copies of a number, and multiplication is also a commutative operation. For example, \( 5 \cdot 3 = 5 + 5 + 5 = 15 \) and equivalently \( 3 \cdot 5 = 3 + 3 + 3 + 3 + 3 = 15 \). This interpretation of multiplication should make clear why...

• any number times 0 is zero!
• any number times 1 is just itself!

However, you do NOT want to be calculating products by writing out a gazillion additions, so you must learn your "times tables" if you don't already have them memorized. You can find multiplication drill sheets in the appendix! Set a timer and speed run those until they're second nature. You should be able to do things like the list below using mental math pretty quickly.

Exercise \(\PageIndex{2}\)

1. \( 8 \cdot 4 = \)
2. \( 10 \cdot 3 = \)
3. \( 7 \cdot 9 = \)
4. \( 13 \cdot 2 = \)
5. \( 200 \cdot 400 = \)
6. Think of two integers whose product is 18.

Answer

1. 32
2. 30
3. 63
4. 26
5. 80000 (Any time you're multiplying things with zeros on the end, just look at the non-zero parts on the front, and then tack on the total number of zeros in the numbers. Two zeros came from the 200 and two more came from the 400.)
6. 6 and 3 will work, or 9 and 2, or 18 and 1.

We use the dot symbol \( \cdot \) so we don't mix up \( \times \) and \( x\) later, but if you see things in parentheses right next to each other, that also means multiplication, such as \( (2)(4) = 8 \), or just \( 2(6) = 12 \).

One thing to watch out for is when negative numbers are multiplied. Sometimes the signs will change, and sometimes they won't. Here's the rundown:

Ingredients	Product	Example
positive times positive	positive	\( (7)(3) = 21 \)
positive times negative	negative	\( (-2)(3) = -6 \)
negative times negative	positive	\( (-11)(-2) = 22 \)
\((-1)\) times a negative, or "minus a minus"	positive	\( -(-4) = (-1)(-4) = 4 \)

Try using these rules when you practice in the Exercises section.

Division is the inverse operation to multiplication. Intuitively, it's taking a number or quantity and splitting it into a certain number of equal pieces. Like \(\dfrac{12}{3} = 4 \) because splitting 12 candy bars among 3 children will result in each child getting 4 candy bars. Order matters in division; that is, \(\dfrac{12}{3} \neq \dfrac{3}{12}\). You might also see division written with the symbol \( \div \), but that's gross and I won't use it often. You'll want to know how to do simple divisions in your head as well, so if that's rusty, check out the drill sheets in the appendix.

Exercise \(\PageIndex{3}\)

1. Divide 50 by 2.
2. Compute \(\dfrac{9}{3} \).
3. Compute \(\dfrac{24}{6} \).
4. Take the number 4 and multiply it by 8. Then divide the result by 8. What happened?

Answer

1. 25
2. 3
3. 4
4. \( 4 \cdot 8 = 32 \) and then \( \dfrac{32}{8} = 4 \). We got back to the same number. This is because multiplication and division are inverse operations.

The number you start with is called the dividend, and the number you divide BY is the divisor, and the result is called the quotient. You'll also want to watch out for negative numbers involved in division, just like with multiplication above.

Ingredients	Quotient	Example
positive divided by positive	positive	\( \frac{25}{5} = 5 \)
positive divided by negative or vice versa	negative	\( \frac{-12}{3} = -4 \) or \( \frac{36}{-12} = -3 \)
negative divided by negative	positive	\( \frac{-16}{-8} = 2 \)

Try using these rules when you practice in the Exercises section.

Did you notice that \( \dfrac{3}{12} \) up above?? I can't split 3 candy bars among 12 children without cutting the candy bars into pieces! In fact, I need to cut them into four pieces each to have enough... So you see, introducing the division operation is leading us to talk about fractions. Goodbye integers and hello rational numbers! A rational number is one that can be written as a ratio of two integers, and the set of all such numbers is denoted by the symbol \( \mathbb{Q} \). Since any whole number can be written as itself over 1, the rational numbers include all the integers. We're talking about things like

\[ \frac{3}{2}, \quad \frac{4}{1},\quad \frac{37}{9},\quad \frac{-2}{5}, \quad\frac{1}{100}, \quad ... \notag \]

and so on and so forth. You can visualize rational numbers on the number line between the whole number ticks.

Rational numbers can be written as decimals instead of fractions, and these decimals will always "end" somewhere, like how \( \dfrac{3}{2} = 1.5 \) or \( \dfrac{1}{8} = 0.125 \) end after a certain number of decimal places. We will work with fractions and decimals in more detail in later sections, so for now let's get a move on toward those real numbers we're supposed to be learning about. If a decimal representation of a number never ends and never settles into a repeating pattern, like the special number \( \pi = 3.141592653589793... \), it's called an irrational number, meaning it canNOT be written as a ratio of integers. Other examples we will meet later are things like \(\sqrt{2} = 1.41421356237... \) or the special number \( e = 2.718281828... \). We could still toss \( \pi \) on our number line to visualize him using his approximation, though, right thereabouts...

Finally, the set of numbers that includes all integers, all rational numbers, AND all irrational numbers is called the set of real numbers.

The real numbers, denoted by \( \mathbb{R} \), contain all integers (positive and negative whole numbers), all rational numbers (numbers that can be written as a ratio of two integers or a decimal that ends or settles into a repeating pattern eventually), and all irrational numbers (numbers whose decimal representation never ends and doesn't repeat, like \( \pi, e, \) or \( \sqrt{2} \)).

Now let's learn how to compare numbers. I want to start representing numbers using letters so it's easier to talk about general ideas. I'll say things like, "Let \( m \) be an integer," which is code for, "I have some whole number nicknamed \( m \), maybe he's 2, maybe he's 17, maybe he's -4000, it doesn't matter." Try to see what I mean by the following description of possible situations:

Given two real numbers \( a \) and \( b \), we fall into one of the following three cases...

1. \( a > b \). We say \( a \) is greater than \(b \), aka \(a\) is further to the right on the number line than \( b \), aka the difference \(a - b \) is a positive number.
2. \( a < b \). We say \( a \) is less than \(b\), aka \(a\) is further to the left on the number line than \( b \), aka the difference \(a - b \) is a negative number.
3. \( a = b \). We say \( a \) and \(b \) are equal, aka they are the same number, aka the difference \( a - b\) is zero.

The symbols involved here are called inequality symbols, and if you picture them as an alligator opening his mouth, he's always opening his mouth toward the bigger number. Because he's greedy. Here's the whole list:

• \( a > b \) means \(a\) is strictly greater than \( b\), Case 1 above. To be clear, "strict" means \(a\) cannot equal \(b\). For example, \( 5 > 2 \).
• \( a < b \) means \(a \) is strictly less than \( b \). For example, \( -2 < 9 \)
• \( a \geq b \) means \(a\) is either greater than or equal to \( b\). For example, if \( a \geq 7 \), \(a\) could be 7, or 8, or 32.99, or anything as long as it is at least 7.
• \( a \leq b \) means \(a \) is either less than or equal to \( b\). If \( a \leq 4\), then \(a\) could be 4, or 3, or -100, as long as it's not more than 4. You get the idea.

Exercise \(\PageIndex{4}\)

True or False?

1. \( 8 < 12 \)
2. \( -5 > -2 \)
3. \( 1.8 \geq 1.5 \)
4. \( \pi \leq 3 \)

Answer

1. True, 8 is smaller than 12.
2. False! Note that \(-5\) is further to the left on the number line than \(-2\), which makes it "more negative," which makes it technically a smaller number! You can check this way: \( -5 - (-2) = -5 + 2 = -3 \). Yep, the difference is negative.
3. True, 1.8 is greater than 1.5, which satisfies one of the options for the \( \geq \) symbol. The difference \( 1.8 - 1.5 = 0.3 \) is positive.
4. False, we know \( \pi \approx 3.14159... \) which is actually a bit bigger than 3.

Order of Operations

All those arithmetic operations we covered above can be done one after another all in a row, to make a big combination of operations. For example, if I say, "Take 2 and add 4, then subtract 8, then multiply by 3, then add 9, then divide by 3," all that garbage translates to

\[ \textcolor{red}{2} \quad \longrightarrow \quad \textcolor{red}{2}+4 = 6 \quad \longrightarrow \quad \textcolor{red}{6} - 8 = -2 \quad \longrightarrow \quad (\textcolor{red}{-2})(3) = -6 \quad \longrightarrow \quad \textcolor{red}{-6} + 9 = 3 \quad \longrightarrow \quad \frac{\textcolor{red}{3}}{3} = 1. \notag \]

But good lord, I don't want to have to type that stuff ever again! The good news is we can write the whole process using math notation as long as we're careful about what order we're doing things in. See if you can understand why the expression below is the equivalent to the long-winded instruction I gave above.

\[ \frac{3((2+4)-8) + 9}{3} \notag \]

All of those parentheses are doing some heavy lifting there to indicate the order. We always look at parts in parentheses first, and we always work from the inside out, so the first step is that inner \( (2+4)\). Once that's done, we look at the next layer of parentheses, which include the subtraction, so that happens next. Outside all of that, we see the multiplication by 3. Moving along to the right, we see the addition of 9 step. And all of that is finally being divided by the 3 downstairs. Woof.

Computations follow the conventional Order of Operations: PEMDAS

1. Parentheses: Start with the innermost parentheses, ignore everything else going on outside, and proceed. Each time you finish working out the contents of some parentheses, move to the next innermost layer and repeat.
2. Exponents: This is when things are raised to powers, but we're going to deal with powers and radicals in detail later. If these were around, you would do them next.
3. Multiplication/Division: Perform any multiplications and divisions, moving from left to right (which may not matter, but it's convention). Note that if a bunch of stuff is happening in the numerator of a fraction, it's as if there are parentheses around the numerator, and you should do all of it before performing the division. That is,
   \[ \frac{a+b}{c} = \frac{ (a+b)}{c} \notag \]
4. Addition/Subtraction: Perform any additions and subtractions, again moving from left to right.

These rules allow us to know for sure whether \( 2 + 3 \cdot 5 \) is supposed to equal \( 25\) or \(17 \). There are no parentheses, so we look for multiplications and do those first, getting \( 2 + 15 \). We now finish by adding to get \( 17\). Give the practice problems below a shot, and then check your answers.

Exercise \(\PageIndex{5}\)

Compute:

1. \( 7 - 2 \cdot 4 \)
2. \( \dfrac{2+7}{3} - 2 \)
3. \( 4(15 + (2-3)) - 6 \)
4. \( 2(3-(-3)) + 2\cdot 8 - \dfrac{10}{5} \)

Answer

1. \( 7 - 2 \cdot 4 = 7 - 8 = -1 \)
2. \( \dfrac{2+7}{3} - 2 = \dfrac{9}{3} - 2 = 3 - 2 = 1 \)
3. \( 4(15 + (2-3)) - 6 = 4(15 + (-1)) - 6 = 4(14) - 6 = 56 - 6 = 50 \)
4. \( 2(3-(-3)) + 2\cdot 8 - \dfrac{10}{5} = 2(6) + 2\cdot 8 - \dfrac{10}{5} = 12 + 16 - 2 = 26 \)

Order of operations is important, because if you mess it up, you could get a totally different answer!

Exercise \(\PageIndex{6}\)

Which is the correct math translation of the instructions, "Take the number 6 and subtract 2, then divide by 4, then add 9, then multiply by 5," and what is the correct answer?

\[ \frac{6-2}{4} + 9 \cdot 5 \quad \text{ or } \quad 5\left(\frac{6-2}{4} + 9 \right) \notag \]

Answer

The first one is wrong, because if you follow the order of operations, you will multiply the 9 and 5 too early, and only later perform the addition! If we want to add 9 first and then multiply by 5, we must use the second option, which performs the addition inside the parentheses, and multiplies by 5 last of all. The result of the whole fiasco should be 50. (The wrong expression gives an answer of 46.)

Okay, that's all, folks. Scroll back up to the Learning Objectives and make sure you feel good about the skills you were aiming to pick up from this section! Then move on to trying the exercises in the next section.