Sets of Numbers
Problem with Zero and Division
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.
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?
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 . 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 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:
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.
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:
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.
Repeating Decimals Written as Fractions
Consider the fraction This is a rational number because it is the ratio of two integers, 1 and 3. Yet, the decimal approximation of one-third is (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 .
We don’t know what number, as a fraction is , so we will write the unknown x.
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.)
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.
Divide both sides by 99 to solve for x.
Recall that x was originally defined as the fractional equivalent of the repeating decimal.
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.
We can try that again with but 1/3 is always larger.
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 . But in base ten this is . 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 . 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:
Then, solving for x: .
2.5 = x
But, 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:
Let us try the same process with 1/3.
Then, solving for x: .
As you can see, we will keep getting ten divided by three, forever.
1. Change the following into rational numbers:
is the following called undefined: ?
3. List all of the sets of numbers to which the following numbers belong:
0 b. 9
c. -5 d. e.
a rational number also be a whole number?
number is whole but not natural?
many seconds is 0.48 of an hour?
many inches is 0.25 of a foot?
two examples of how the Commutative Property does not work for subtraction.
9. Add parenthesis to this expression in a way the value does not change: 3 4 + 7·5·2
A Sideways Approach to Learning Associate Property and PEMDAS
I have heard, Math was okay until they threw the alphabet in it, so many times from so many adults, who I wish were trying to be witty, but are in fact completely serious. And these are reasonable, intelligent people. Why does the abstraction of a variable or an unknown become so confusing that it creates a huge disconnect?
I’d like to share with you an assignment that I believe will help students transition from the concrete application and properties of Real Numbers to the abstractions we deal with in Algebra and mathematics beyond.
The desired outcomes of this assignment are:
- Improve mathematical literacy by encouraging students to read the mathematical meanings created by the spatial arrangement of numbers and symbols
- Improve their understanding of the Order of Operations and to help student realize that the Order of Operations is not its own topic, compartmentalized, but rather an over-arching understanding of how math is performed.
- Promote abstract thinking about numbers and their properties
- Introduce some concepts that will come into play later in Algebra, like finding the x-intercepts of a quadratic once it is factored
How to Introduce the Assignment
Students must be aware that they will be dealing with abstract ideas and that there are sometimes more than one right answer. Also, a student can be right, but not completely right, they could also be wrong, but not always. By fostering a healthy discussion about these problems you can introduce the idea that in order for something to be mathematically true, it must always be true. If a single circumstance is untrue, then the statement is untrue.
Consider the problem: Given that a and b are real numbers, and the following is true, what do you know about the numbers a and b? a×b=0
A student might say, In this case a and b are both zero.
That is correct in one case, but there are many cases where that is not true. It is true that one of them must be zero.
The First Prompt
Given that a and b are real numbers, and the following statement is true, what can you conclude at the numbers a and b?
a – b = 0
At first have them think and write on their own. Make sure they’re all working, not avoiding this uncomfortable notion.
After a given amount of time (short, maybe one minute), instruct them to talk with two different people. Be clear that the expectation is that they take turns, one person shares, the other listens and responds.
After the time is up, have a whole-class discussion, but avoid being the authority until the discussion is winding down. Only be the authority on the subject to help summarize.
The Second Prompt
With the same properties of the numbers and the statement being true, provide them with the equation:
(a + 5)(b –7) = 0
Conduct conversations as you did with the first, maybe allowing more time for them to talk together as this is a more complicated situation.
The Third Act
Switching gears from properties of variables to applying some properties of real numbers will promote their understanding of when the associative and commutative properties as well as challenge their understanding of how math is written. We are really trying to promote their ability to read and write math and their fact that the spatial arrangement has meaning in math.
Have the students try and add parenthesis, as many as they like, to the following equation, so that it will be true. Have them try it on their own first, then provide a short amount of time for peer discussion.
3•2 – 72 + 5 = 80
When you conduct a whole-class discussion, make sure it’s student lead, your role is as a mediator, not a disseminator of facts.
Instruct students to create a similar problem by making a statement they know is true and removing the parenthesis. For example, they might make up:
8(5 – 3) + 11 = 27
But would only write
8•5 – 3 + 11 = 27
When they’re completed their problem, have them show you. Once everybody has a problem, hand out 3x5 cards. On the front of the card the student will write their problem, without the parenthesis. On the back, they’ll write their name. Have them pass the cards forward and you can distribute them at to another class the following day.
Last Thing for the Lesson:
The idea here is the same as with the previous activity, but we are accessing their knowledge from a different angle. They will be given an expression with parenthesis and be asked if the parenthesis can be removed without changing the value of the expression.
For example: (5 + 4) – 2, or 11 – (4 – 9).
Introduce these problems in the same fashion, with quiet thinking first, then small group discussions, then whole-class discussion.
The homework is critical here because it will challenge students to think and examine how the way in which math is written changes the meaning. It will also force them to think about numbers in a general fashion.
The last thing I’d like to mention is that this could easily be done over two separate days depending on the aptitude of the class you’re teaching.
1.8 Fundamentals of Algebra
1.10 Units, Ratios and Rates
For a PDF of solutions to the first set of practice problems, click here.
For a video of how I did the first set of practice problems, click here.
For a PowerPoint you can use in your classroom, click here.
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