## Properties of Real Numbers

1.1 Real Numbers

We will begin with the various types of numbers called Real Numbers.  Together, these numbers can be ordered and create a solid line, without gaps.   Real Numbers can be ordered in respect to size, and this is how the number line can be created.

Ø  Natural Numbers:  These are counting numbers, the smallest of which is 1.  There is not a largest Natural Number.

Ø  Whole Numbers: All of the natural numbers and zero.  Zero is the only number that is a Whole Number but not a Natural Number.

Ø  Integers:  The integers are all of the Whole Numbers and their opposites.  For example, the opposite of 11 is -11.

Ø  Rational Numbers:  A Rational Number is a ratio of two integers. All of the integers, whole and natural numbers are rational.

o   Decimals that terminate or repeat (have patterns) are rational as they can be written as a ratio of integers.

Ø  Irrational Numbers: A number that cannot be written as a ratio of two integers is irrational.  Famous examples are π, and the square root of a prime number (which will be discussed next).

Together these make up the Real Numbers.  The name, Real, is a misnomer, leading people to conclude that the word real in this context has the same definition as used in daily language.  That misconception is only strengthened when the Imaginary numbers are introduced, as the word imaginary here harkens back to a day when the nature of these numbers, and their practical use, was unknown.

Infinity

Suppose there is a largest natural number. Now, multiply that number by itself and you have discovered another, larger, natural number.  Therefore, it is impossible to have a largest natural number.  The set of natural numbers has no bound, “it keeps going as long as you keep looking.”  This is what infinity is.  Infinity is not a number, but rather an idea.  Whether infinity exists with tangible items is an interesting question.

The integers are also infinite.  Are there the same quantity of integers and natural numbers, or does one set contain more than twice as much as the other?

Two Properties

There are many properties of Real Numbers, but only two we will discuss here.  The first says that you can change the order of addition without changing the value.  An example would be 5 + 3 = 3 + 5.  This also works for multiplication but not subtraction or division.  This is called the Commutative Property.

One place the Commutative Property comes into play is with reducing Algebraic Fractions.

The second property is called the Associative Property.  Their names are not of particular importance, but the ideas behind each is highly valuable.  The Associative Property says the way in which you group repeated addition does not change the sum.  It works also for multiplication but not division or subtraction.

5 $–$ 3 = 2, but 3 $–$ 5 = -2

Is zero rational?

A rational number is a number that is the ratio of two integers.  Before we tackle the issues that arise from zero, let’s reframe how we think about rational numbers (fractions) and develop a different language for these to promote greater proficiency in Algebra and allow for greater ease in understanding how zero causes real problems with rational numbers.  (If you understand the nature of what follows you do not have to memorize or remember the tricks, you just understand.)

Consider the fraction 82$\frac{8}{2}$ .  You were likely taught to think of this fraction as division and would also likely be taught to ask the question, “How many times does two go into eight?”  That is sufficient for this level of mathematics, but the Algebra ahead is seemingly more complicated, but by simply rephrasing the language we use to talk about fractions, we can expose the seemingly more complex as being the same level of difficulty.

Instead of asking, “How many times does two go into eight,” the better question is, “Two times what is eight?”

It is true that 82=4,$\frac{8}{2}=4,$ because two times four is eight.  Simply answer the question “Two times what is eight,” and you’ve found the answer.

This will come into play with Algebra when we begin reducing Algebraic Fractions (also called Rational Expressions) like:

9x23x$\frac{9{x}^{2}}{3x}$.

If you ask the question, “How many times does three x going into nine x squared,” you’ll likely be stuck, especially when the expressions become more complicated.

But asking, “three x times what is nine x squared,” is a little easier to answer.

9x23x=3x,$\frac{9{x}^{2}}{3x}=3x,$ because 3x3x=9x2$3x\cdot 3x=9{x}^{2}$.

There will be much more on reducing Algebraic Expressions later in this chapter.  Let’s turn our attention to zero and how it “behaves” in with rational numbers.

Zero is an integer, and again, a rational number is a ratio of two integers. Consider the following:

5005$\frac{5}{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{0}{5}$

The first expression asks, “Zero times what is five?”

The second expressions asks, “Five times what is zero?”  (Again, phrase the question in this fashion to provide easier insight into the math.)

The product of zero and any number is zero.  So, the answer to, “zero times what is five,” is … well, there is no answer.  There is no number times zero that is five.  There is not a number times zero that equals anything except zero.  We say this is undefined, meaning, there is no definition for such a thing.

The second expression, “five times what is zero,” is zero.  Five times zero is zero.

One of these two expressions is rational, the other is not a number at all.  It does not just fail to fit within the Real Numbers, it fails to fit in with any number.

50 Not a Number05Rational

Repeating Decimals Written as Fractions

Consider the fraction 13.$\frac{1}{3}.$ This is a rational number because it is the ratio of two integers, 1 and 3.  Yet, the decimal approximation of one-third is 0.3¯$0.\overline{3}$ (the bar above the three means it is repeating infinitely).

Here is how to express a repeating decimal as a fraction.  Let us begin with the number 0.27¯¯¯¯$0.\overline{27}$ .

 We don’t know what number, as a fraction is 0.27¯¯¯¯$0.\overline{27}$, so we will write the unknown x. x=0.27¯¯¯¯$x=0.\overline{27}$ Since 0.27¯¯¯¯$0.\overline{27}$ is repeating after the hundredths place, we will multiply both sides of the equation by 100. (note, for 0.333333… we would multiply by 10, since the decimal repeats after the 10ths place, but we would multiply 0.457457457457…by 1,000 since it repeats after the thousandths place.) 100×x=0.27¯¯¯¯×100$100×x=0.\overline{27}×100$   100x=27.27¯¯¯¯$100x=27.\overline{27}$ The following step is done by a procedure learned with solving systems of equations, which will be covered later. (In fact, this procedure would be a great topic to review when systems of equations is learned.) Subtract the first equation from the second.       Note:  27.27¯¯¯¯−0.27¯¯¯¯=27$27.\overline{27}-0.\overline{27}=27$ 100x=27.27¯¯¯¯−(x=     0.27¯¯¯¯)−−−−−−−−−−−−    99x=27$\begin{array}{l}\underset{_}{\begin{array}{l}100x=27.\overline{27}\\ -\left(x=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.\overline{27}\right)\end{array}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}99x=27\end{array}$ Divide both sides by 99 to solve for x.   Recall that x was originally defined as the fractional equivalent of the repeating decimal. x=2799$x=\frac{27}{99}$

Why Are Some Rational Numbers Non-Terminating Decimals?

This question can lead into greater understanding of why we must use care when using calculators.  The issue with repeating decimals being rational numbers is related to our base-ten numbering system we use for decimals.  Consider the following information and how it could be discussed with students.

The fraction 1/3 = 0.3333333333333333…  And yet, we are told rational numbers include decimals that can be written as a fraction (the ratio of two integers).

How it works is sometimes very clear and clean.  For example, 0.7 is said, “Seven tenths.” And “Seven tenths,” can also be written as the ratio of seven and ten.  And the number seven tenths is of course equal to itself, regardless of how it is written.  The number 0.27 is said, “twenty seven hundredths,” which can easily be written as the ratio of twenty seven, for the numerator, and one hundred, as the denominator.  And this can continue so long as the decimal terminates.  But try the same thing with a repeating decimal and you do not end up with things that are equal.

Fact 1:  13=0.3¯=0.3333333….$\frac{1}{3}=0.\overline{3}=0.3333333\dots .$

Fact 2: 0.3=310,0.333=3331,000,and so on

Fact 3:  If a=b, then ab=0

0.3¯310>0because0.333333333…0.30.0333333….$\begin{array}{l}0.\overline{3}-\frac{3}{10}>0\\ \text{because}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.333333333\dots \\ \underset{_}{-0.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}0.0333333\dots .\end{array}$

We can try that again with 33100, or 3331,000, but 1/3 is always larger.  13 is non-terminating.

But that does not address why a rational number would be a non-terminating decimal.

The simple reason that some rational numbers cannot be expressed as terminating decimals has to do with our numbering system.  We use base 10 numbers.  Our decimal system provides us an easy way to write fractions with denominators that are powers of 10.

That means that we have ten numbers, including zero, that fill up one column, like a car’s odometer.  When you travel 9 miles the odometer will read 000009.  When you travel the tenth mile the odometer will read 000010.

Not all of our number systems are base ten. While the metric system is base 10, or at least translates into powers of ten, the Imperial system is base 12 from inches to feet, but 5,280 feet to the next unit of miles, and so on.

Time is another great example of bases other than ten.  Seconds and minutes are base sixty.  You need sixty seconds before you have an hour, not ten.  But hours are base 24 because 24 hours are needed to make one of the next category, which is days.

In time, 25 minutes of an hour is the ratio 25 minutes60 minutes in an hour.  But in base ten this is 0.416¯$0.41\overline{6}$ .  But this does not account for the ratio of minutes to an hour.  In the context of time the ratio of 25 to 60 is not 0.416¯$0.41\overline{6}$.  A typical mistake would be to say that 25 minutes is 0.25 of an hour.

Back to our original example of 1/3.  Not all numbers can be cleanly divided into groups of ten, like 3.  If we had a base 3 numbering system, where after the third number we moved, then 1/3 would just be 0.1.  But in our numbering system, 0.1 is one tenth.

Other numbers, like four, translate into ten more easily.  Consider the following:

14=x10$\frac{1}{4}=\frac{x}{10}$

Then, solving for x: 104=x$\frac{10}{4}=x$.

42.510.0820200$\begin{array}{l}4\begin{array}{c}\hfill 2.5\\ \hfill \overline{)10.0}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{\text{\hspace{0.17em}}-8\text{\hspace{0.17em}}\text{\hspace{0.17em}}↓}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{\text{\hspace{0.17em}}-20}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}$

2.5 = x

Then 14=2.510$\frac{1}{4}=\frac{2.5}{10}$

But, 2.510$\frac{2.5}{10}$ is not a rational number because 2.5 is not an integer and a rational number is a ratio of two integers.  But this can be resolved:

2.5101010=25100$\frac{2.5}{10}\cdot \frac{10}{10}=\frac{25}{100}$

So, 14=25100$\frac{1}{4}=\frac{25}{100}$

Let us try the same process with 1/3.

13=x10$\frac{1}{3}=\frac{x}{10}$

Then, solving for x: 103=x$\frac{10}{3}=x$.

33.310.091091$\begin{array}{l}3\begin{array}{c}\hfill 3.3\\ \hfill \overline{)10.0}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{\text{\hspace{0.17em}}-9\text{\hspace{0.17em}}\text{\hspace{0.17em}}↓}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}10\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\end{array}$

As you can see, we will keep getting ten divided by three, forever.

## Lesson: Real Numbers and Their Properties

Note:  The purpose of this to develop the mathematical thinking of students.  That goal is supported by learning the properties of Real Numbers.  During the delivery of the lesson, look for opportunities to ask probing questions that will uncover misconceptions and deepen understanding.

Before the lesson you will see a Big Idea, which is the concept you’re teaching, a Key Knowledge, which is the facts students must know, and a Pro-Tip, which can be shared with students to promote recall or efficiency.

This will be a two-day lesson.

### Big Idea

• There are Real Numbers and Imaginary Numbers. In Algebra I we deal with Real Numbers, but see Imaginary Numbers occasionally.
• Not every numeric expression is a number.
• Irrational Numbers fill in the gaps between the Rational Numbers, creating a continuous number line.
• Arithmetic properties between rational and irrational numbers.

### Key Knowledge

• The subsets of Real Numbers
• Key: A rational number is a number that can be written as a ratio of integers.
• Properties of Real Numbers
• What infinity means
• Why you cannot divide by zero
• Why some fractions are non-terminating when written as decimals

### Pro-Tip(for students)

• This lesson is more about ideas than procedures. To promote learning, make a list of questions and use those as flash cards to study.

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• Lesson Guide
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• Quiz and Key (Real Numbers and Primes)

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