## Getting Comfortable with Negative Exponents This post shows how I use a reciprocal pairs approach to introduce zero and negative integer exponents to my students. Several exercises that I have used to make students more comfortable using zero and negative integer exponents are also included. It seems to me that students learn to tolerate negative exponents, but they are not really comfortable using negative exponents.

I begin my intuitive discussion of negative exponents by creating a table of the first 4 positive integer powers of an integer such as 6 for example. Refer to the table below. From this point, the discussion goes something like this:

• What do we do to 1,296 to get 216? What do we do to 216 to get 36?
• Some groups immediately notice that we divide by 6 to get the next number in the list, and other groups require a bit of coaching to see that we divide by 6 to get the next number.
• In order to maintain the pattern, it’s clear that the fifth row of the table must be 60 = 1.
• To get the remaining rows, I say, “Reduce the exponent by one and divide by whole number six to get the next fraction in the list.” (Some students see a rational number like 3/8 only as a single entity, not one whole number divided by another whole number. Also some groups need to be reminded how a fraction is divided by a whole number.)
• As shown in the diagram, I bracket the reciprocal pairs of numbers in the list.
• Now the important question; “What is the relationship between bracketed pairs of numbers?”
• Some groups immediately notice the reciprocal pair relationship, and other groups need a little coaching to see the relationship. It takes a lot of coaching to get them to say out loud, “They are reciprocal pairs because all products of a pair of bracketed numbers equals 1.”
• I point out that we could do the same process with any nonzero real number, but the math might be a little messy.

Shown below is a reciprocals pairs table for -2/5 and -5/2. If a teacher decides to show his/her class a reciprocal pairs table for two fractions, and fraction operation skills are fragile, I strongly advise that the implied multiplication and division of fractions in the table be explicitly demonstrated. It’s sufficient to demonstrate multiplication and division of fractions for one or two lines in the table. After exploring one or two tables of reciprocal number pairs, I give my students an informal summary of what can be deduced from the table. Of course, the concepts that I discuss depends on the class I’m teaching. Base b equals any nonzero real number, p is an integer, and x and y are real numbers. Students should use their calculators to verify the specific examples.

Concept Example
b0 = 1 3.140 = 1
1 / (x/y) = y/x 1 / (-7/2) = -2/7
All real numbers have an implied exponent of one. π = π1
Changing the sign of the exponent of a number gives us the reciprocal of the number. 4-1 = 1/4
If integer n > 0, bn means repeated multiplication of b by itself n times. 43 = 4•4•4 = 64
If integer n > 0, b-n means repeated multiplication of 1/b by itself n times. 4-3 = (1/4)•(1/4)•(1/4) = 1/64
b-p = (1/b)p (1.2)-5 = (5/6)5
bp = (1/b)-p (-2.25)3 = (-1/2.25)-3
xy = y / (1/x) 0.008x = x/125
If x > 0, Log(x) = -Log(1/x) because logarithms are exponents. Log(5) = -Log(0.2) ≈ 0.699
The graphs of y = a(bx) and y = a(1/b)-x are equal. See graph below.
The graphs of y = a(bx) and y = a(b-x) are reflection images of each other over the y-axis because the graphs of y = f(x) and y = f(-x) are always mirror images over the y-axis. See graph below. I will close this post by showing you some practice exercises that I have used to promote understanding of negative exponents, and to increase student comfort level in working with negative exponents. One may think that these exercises are nothing more than mental gymnastics, however, calculus students need to know how to rewrite expressions in terms of negative and fractional exponents. Readers can download my free handout, Properties of Exponents and Logarithms, by going to the algebra and pre-calculus tab in our instructional content page. This handout is filled with examples demonstrating the laws of exponents and logarithms.   Early in a beginning algebra course, students are taught how to add and subtract signed numbers. Addition of signed numbers is taught by giving students a set of rules that they can follow to get the right answer. Some number line diagrams are thrown in to illustrate addition of signed numbers. Subtraction of signed numbers is usually taught by the “add the opposite” rule. This rule sounds plausible, however, it does not give students any insight into what subtraction is all about from a geometric point of view. You might ask, “Isn’t learning how to apply a set of rules to get the answer sufficient?” In my view, it’s not. As I stated several times in previous posts, whenever possible, students should understand a math concept from both an algebraic and geometric point of view. The primary purpose of this post to show how I have used vector diagrams to illustrate addition and subtraction of real numbers and 2D vectors. I will also discuss geometric interpretations of expressions and relationships such as (a + b)/2, |x – 4| < 5, and |x – 5| > 7.

Based on my own experience, most kids quickly learn how to apply the rules for adding and subtracting positive and negative numbers. When simplifying polynomial expressions, rational polynomial expressions, and polynomial long division, most mistakes occur at the step where a negative number is subtracted. Students are generally pretty good at adding signed numbers, but a little weak when it comes to subtracting signed numbers. This is why my favorite high school math teacher, Vivian Jones, said, “When subtracting a number, just change the sign of the number and add”. This is what I tell my college algebra students when they do polynomial long division. As taken from a developmental algebra textbook, the rules for adding and subtracting signed numbers are as follows:

Adding Two Real Numbers a and b

• If a and b have the same sign, add their absolute values. Use the common sign of a and b as the sign of the answer.
• If a and b have different signs, subtract their absolute values. The sign of the answer is the sign of the number that has the largest absolute value.

Subtracting Two Real Numbers a and b

• ab = a + (-b)
• In other words, ab equals a plus the opposite of b
Vector diagrams similar to diagrams A, B and C below are used to illustrate how the addition and subtraction rules for signed numbers work. The bottom half of diagram A shows that two negatives do not necessarily make a positive.

Diagrams B and C illustrate the subtraction rule for signed numbers. The problem with the subtraction rule and the corresponding vector diagram is that they don’t give us any insight into what subtraction is all about from a geometric point of view.   Diagrams D and E below illustrate the idea that subtracting two real numbers gives us the directed distance between two reals numbers. The concept of directed distance is fundamental in understanding subtraction from a geometric point of view. For real numbers a and b, the directed distance from b to a = a – b, and the directed distance from a to b = ba. The positive distance or just the distance from a to b = |a – b| = |b – a|. The distance concept is related to many concepts in mathematics such as the amount of change in a variable, how much a data value deviates from some fixed constant, margin of error, etc. Notice that the subtraction problems in diagrams D and E are the same problems in diagrams B and C. After comparing diagrams B and C with diagrams D and E, it becomes clear that diagrams D and E give us a much better way to understand subtraction from a geometric point of view.  We will now take a look at 2D vectors which are essential in understanding a variety of concepts in math and physics. We can treat 2D vectors as line segments that have the properties of length and direction. Any two vectors are equivalent if and only if they have the same length and direction. In this post, 2D vectors will be denoted by bold face capital letters, and a pair of vertical absolute value bars will denote the length or magnitude of a vector. The arrow at one tip of a segment indicates the direction the vector. Geometric rays have infinite length, and therefore 2D vectors are not geometric rays. In passing, I will mention that 2D vectors are just a special case of an abstract mathematical object named vector. If you want to stretch your mind, take a course in infinite dimensional vector spaces. The student and teacher versions of my free handout, Introduction to Vectors, can be downloaded by going to the trigonometry section of our instructional content page.

The graph below shows how two 2D vectors are added. All vectors drawn in the same color are equal to each other because they have the same length and direction. All 2D vectors can be represented by a pair of real numbers of the form < x, y > where x and y equal the x-component and y-components of the vector. Knowing the x-y components of a vector, it’s easy, at least for trig students, to calculate the vector’s magnitude and direction. Note the following as you study the graph:

• A 2D vector can be expressed as the sum of its x-component and y-component. Example: Vector A = < -7, 0 > + < 0, 3 > = < -7, 3>.
• To find the x-y components of a vector, start at the tail of the vector and count the number of spaces left/right and the number of spaces up/down to the head of the vector.
• The x-y components of any two equivalent vectors are equal.
• The tail of the second vector in a vector sum is located at the tip of the first vector.
• Vector addition is commutative. In other words, it makes no difference in what order vectors are added.
• Any two equivalent vector pairs will always result in equivalent vector sums.
• The theorem of Pythagoras is used to calculate the length or magnitude of a vector.
• The inverse tangent function, Tan-1(x), is used to calculate the direction of a vector.
• Vectors A, B and S = A + B = B + A in the diagram have the following properties:
• A = < -7, 3 >, |A| = √(58) units, and direction of A ≈ 156.800
• B = < 3, 5 >, |B| = √(34) units, and direction of B ≈ 59.040
• S = < -4, 8 >, |S| = √(80) units, and direction of S ≈ 116.570
• |A| + |B| > |S| The graph below shows how two 2D vectors are subtracted. Vector subtraction gives us a vector that represents a difference vector between the tips of the two vectors. Vector AB has its tail at the tip of B and its head at the tip of A.  Vector BA has its tail at the tip of A and its head at the tip of B. All vectors drawn in the same color are equal to each other because they have the same length and direction. Note the following as you study the graph:

• The difference vector connects the tip of one vector to the tip of the other vector.
• Vector subtraction is not commutative. Vectors AB and BA have the same length, but they point in opposite directions; they are vector opposites.
• Vectors A, B, AB,  BA have the following properties:
• A = < 6, 2 >, |A| = √(40) units, and the direction of A ≈ 156.800
• B = < 2, 9 >, |B| = √(85) units, and the direction of B ≈ 59.040
• AB = < 4, -7 >, |A – B| = √(65), and the direction of A – B ≈ 299.750
• B – A = < -4, 7 >, |B – A| = √(65), and the direction of B – A ≈ 119.750

Once a student understands addition and subtraction from a geometric point of view, many math problems become much easier to solve. Consider the three routine math problems shown below.

Problem 1: Suppose the IQ score I of a person is in the normal range if the IQ score deviates from 100 by 10 points or less. What interval on a number line and inequality describes a normal IQ score?

Solution: The expression |I – 100| gives us the positive distance of the variable I from 100. Therefore the range of normal IQ scores is described by the inequality |I – 100| ≤10. Problem 2: A part will fail inspection if its diameter d deviates from 2.5 cm by more than 0.001 cm. What interval on a number line and inequality describes the rejection region?

Solution: A part will fail inspection if the positive distance from 2.5 to d is more than 0.001 cm. Therefore the rejection region can be described by the inequality |d – 2.5| > 0.001. Of course, we tacitly assume that there are practical restrictions on values of d. Problem 3:  The graph of the closed interval [0.84, 2.68] is shown below. Find the following:

• Length of the interval
• Coordinate of the midpoint M
• Write an inequality that describes the interval.

Solution: (Many of my elementary statistics students initially struggle with review problems like this.)

• Length of interval = 2.68 – 0.84 = 1.84
• Coordinate of midpoint M = (0.84 + 2.68)/2 = 1.76
• Radius of the interval = 1.84/2 = 2.68 – 1.76 = 1.76 – 0.84 = 0.92
• Inequality: |x – 1.76| ≤ 0.92

Miscellaneous facts I tell my students:

• If you subtract a smaller number from a bigger number, the answer is positive.
• If you subtract a bigger number from a smaller number, the answer is negative.
• If you subtract a positive number from n, the answer is smaller than n.
• If a positive influence is removed from your personal life, the quality of your personal life goes down.
• If you subtract a negative number from n, the answer is bigger than n.
• If you remove a negative influence from your person life, your personal life gets better.
• If you add a negative number to n, the answer is smaller than n.
• If a negative influence is introduced into your personal life, your personal life gets worse.
• To find out how far apart two numbers are, subtract the numbers.
• To find a number half way between two numbers, find the average of the numbers.

I will close this post with a true story about an epiphany I experienced early in my teaching career. The class was a regular high school geometry class. We were learning how to solve story problems involving complementary and supplementary angles. I could see that little Elmo (not his real name) was not getting the idea that if x is the measure of an acute angle, then 90 – x is the measure of the complement of the angle. So I asked Elmo a series of about 5 questions like: If an angle measures 200, what is the degree measure of the complement of the angle? Elmo got every one of my questions right. I then asked Elmo the following question: If an acute angle measures n degrees, what is the degree measure of the complement of the angle? All I got from Elmo was a blank stare. I’m thinking to myself, why doesn’t he get it? Then it hit me. When asked the degree measure of the complement of a 650, Elmo figured out how many degrees he needed to add to 650 to get 900. For me, this was an enlightening and humbling experience.

## Division, Fractions, Proportional Parts, and Other Neat Stuff After his freshman year in college, a former math student paid me a visit. Aaron (not his real name) indicated that his first year of college was a great experience. This was no surprise to me. I found Aaron to be a fairly perceptive individual who got along well with his high school classmates. Then Aaron said to me, “You know Mr. Johnson, after a year of college chemistry, I finally understand division.” He went on to explain how the concepts of division, fractions, decimals, ratios, rates, proportions and percent all fit together for him. In order to solve chemistry problems, Aaron was naturally motivated to learn and understand division and related concepts. Of course, I was surprised to hear Aaron make such a statement. He was an above average student who had attended a highly regarded local private school through junior high school. I then realized that if high school students like Aaron don’t really understand division, many of my students probably don’t really understand division either.

Based on numerous conversations with high school math teachers, in both public and private high schools, I have come to the conclusion that many high school students don’t really understand division and related concepts. It’s almost like a dirty little secret that high school math teachers don’t like to talk about in public. Many students have learned how to apply a set of rules to get the right answer, but they don’t really understand why a rule works or what the answer means. I see this all the time when I tutor developmental math students and above who are graduates of local area high schools.

Elementary, middle school, and high school teachers should NOT in any why interpret the above remarks to mean that they are not doing their job. In my view, teaching elementary and middle school students is far more difficult than teaching high school students. Of course, teaching high school math is not so easy either. If anyone thinks that they have a bullet proof method of teaching division and related concepts to a class of 30 typical middle school students, they should tell middle school teachers how to do it. I’m sure that middle school teachers would be ever grateful.

We constantly hear comments about how public education has deteriorated over the years. My now deceased father-in-law once complained to me that one of his office workers couldn’t figure out how many parts to ship to customers in proportion to the number of parts the customers ordered. (There were more parts ordered than parts in stock.). He said something like this, “Schools don’t teach kids the basics anymore. In my day, schools taught the fundamentals.” My father-in-law and many of his high school classmates learned basic math very well, but only 29% of the general population graduated from high school in the 1920’s. It’s my observation that today’s top students and athletes perform much better than the top students and athletes 40 or 50 years ago. As I see it, the perceived problem with modern education is due to the fact that public schools are attempting to reach a student population with a far greater range of abilities.

So what’s the purpose of this post? The purpose of this post is to share some of the methods that I have used in my classes to explain division and related concepts. Readers may find some of my examples and explanations to be ordinary, but other examples and explanations are different in that they are not found in textbooks. I invite readers to share some of their favorite methods that they use to explain division and related concepts to kids. I would love to do a post based on reader response. Readers should reply by sending an email to info@mathteachersresource.com. If your example has a graphic, please attach a file that contains the graphic. I will create the graphic for you if you give me a good description.

My first set of examples illustrates why the division rule for dividing two fractions gives us the correct answer. I assume that students already know how to apply the multiplication and division rules for fractions, and they know how to reduce a fraction. These examples reinforce the idea that dividing by a number is the same as multiplying by the reciprocal of the number.

• Diagram A below illustrates the solution of following problem: If we have 12 pounds of hamburger and a meat loaf recipe that calls for 2 pounds of ground beef, how many meat loafs can we make? Solution: 12/2 = 12*(1/2) = 6 meat loafs.

• Diagram B below illustrates the solution of following problem: If we have 12 pounds of hamburger, how many 1/4 pound hamburger patties can we make? Solution: 12/(1/4) =12*(4/1) = 12*4 = 48 hamburger patties.

• Diagram C below illustrates the solution of following problem: If we have 12 pounds of hamburger, how many 3/4 pound hamburger patties can we make? Solution: 12/(3/4) = 12*(4/3) = 48/3= 16 hamburger patties. Alternatively, 12 pounds of hamburgers divided into 16 patties = 12/16 = 3/4 pounds per patty.

When I introduce right triangle trigonometry to my students, I give them the sides of a right triangle, and then we find the trig ratios of the two acute angles α and β of the triangle. Students quickly notice that Sin(α) = Cos(β). I want my students to not only understand the definitions of trig ratios and understand that trig function values are ratios, but I also want them to see that trig function values are percentages in disguise. The diagram below shows a right triangle, accurately drawn to scale, with 150 and 750 acute angles. Knowing only the acute angles of the right triangle and a scientific calculator, we can deduce the facts below. When students study the diagram, they often say, “When you look at it this way, it makes more sense.”

• Cos(150) = b/c ≈ 0.966. This tells us the length of the leg adjacent to a 150 angle is always about 96.6% of the length of the hypotenuse.
• Sin(150) = a/c  ≈ 0.259. This tells us the length of the leg opposite a 150 angle is always about 25.9% of the length of the hypotenuse.
• Tan(150) = a/b  ≈ 0.268. This tells us the length of the leg opposite a 150 angle is always about 26.8% of the length of the leg adjacent to a 150 angle.
• Tan(750) = b/a ≈ 3.73. This tells us the length of the leg opposite a 750 angle is always about 373% of the length of the leg adjacent to a 750 angle.

In the next two problems, an intuitive approach is used to solve proportional parts problems. This would have made my father-in-law happy.

Problem 1: Find the three angles of a triangle if the angles are in a ratio of 3 : 4 : 5.

Solution: Refer to the diagram below. The problem boils down to dividing 1800 into a ratio of 3 : 4 : 5 parts. Because 3 + 4 + 5 = 12, we divide 180 into 12 equal parts with each part equal to 180/12 = 15. The smallest angle = 3 * 15 = 450. The next largest angle = 4* 15 = 600 and largest angle = 5 * 15 = 750. (I tell my students that I will only draw a diagram like the one below only one time because it takes too much time to draw the diagram. To my amazement, I once observed a student draw a diagram with well over 200 dots to solve a proportional parts homework problem. I’m not kidding.) Problem 2: Customers A, B, C, D, and E have placed orders for 5, 7, 10, 20, and 40 widgets respectively. Since the company has only 60 widgets in stock, it was decided to immediately fill the orders in proportion to the number of widgets ordered, and ship the remaining widgets when they become available. How many widgets should be shipped to each customer?

Solution: Because 5 + 7 + 10 + 20 + 40 = 82, we divide 60 into 82 equal parts with each part equal to 60/82 = 0.732.

• Customer A: Ship 5 x 0.732 = 3.66 or 4 widgets
• Customer B: Ship 7 x 0.732 = 5.124 or 5 widgets
• Customer C: Ship 10 * 0.732 = 7.32 or 7 widgets
• Customer D: Ship 20 x 0.732 = 14.64 or 15 widgets
• Customer E: Ship the remaining 29 widgets

The classic “working together” problems are difficult for beginning algebra students to understand. Before I show students the standard algebraic technique for solving a working together problem, I show them how to use ratios to solve a working together problem. The problem below is a typical working together problem.

Problem: It would take homeowner Bob 7 days to roof his garage and professional roofer Clyde could roof Bob’s garage in 3 days. If Bob and Clyde cooperate with each other, how many days should it take them to roof Bob’s garage if they work together? Assume that a work day equals 8 hours. The text box below shows the solution of the problem. I suppose Bob and Clyde would work over time the second day so that they could finish the job in 2 days. (There is always someone who will make an initial guess of 5 days!) Now let’s take a look at a couple of percent problems from a slightly different point of view.

Problem 1: A sofa that normally sells for \$2,799.95 is on sale at 20% off. Local sales tax rate equals 9%. Find the sale price of the sofa and the cost of the sofa with sales tax. The solutions are given in the text box below. Problem 2: Brad and his lovely wife Angelina dined at an upscale restaurant to celebrate their 20th wedding anniversary. The couple was fortunate to have a 15% discount coupon to help cover the \$76.80 cost of their meal plus 9% sales tax. Since Brad was a high school math teacher, he thought that he had two options as to how he should apply the 15% discount. With option 1, he could take a 15% reduction of the cost of the meal plus sales tax. With option 2, he could first take a 15% discount on the cost of meal, and then pay sales tax on the discounted meal. With either option, Brad will leave a 20% tip. What option should Brad choose? (If Brad had stopped to really think about his options for a minute, he would have realized that it makes no difference what option he chooses.)

• Option 1: Total cost = 0.85(1.09 x 76.80) = \$71.16
• Option 2: Total cost = 1.09(0.85 x 76.80) = \$71.16
• Total cost with tip = 1.20 x 71.16 = \$85.39 or \$85.00
I will close this post with some sample exercises that I have used to give students practice drawing valid conclusions and hopefully promote their mathematical reasoning ability. Each exercise involves a conditional relationship which is true or false depending on the values of the variables in the conditional statement. From a list of about 25 possible conclusions, student are to select all conclusions that are necessarily true. Given the long list of possible conclusions, most students don’t find these exercises so easy on first exposure. Each of the text boxes below shows a conditional statement and all valid conclusions which are contained in a list of possible conclusions.

## Why is the Product of Two Negative Numbers a Positive Number?

When I ask most adults or high school students what is -5 times -10, I usually get the correct response of positive 50 or just 50. When I ask them if they can give me simple explanation or an example to show why the product or two negative numbers is a positive number, they can’t.  Students seem to remember the rule “two negatives make a positive”, but some forget that this rule applies to the product or quotient of two real numbers, but not to the sum or difference of two real numbers. I believe it’s fair to say that many lower level math textbooks and math courses treat this rule as just a simple mathematical fact of life, and rarely if ever give an intuitive explanation of why this rule is true.

The purpose of this post is to give examples of intuitive explanations as to why the product of two negative real numbers is a positive real number. It’s important that mathematics makes sense to students, and math should not be just a bunch of somewhat arbitrary rules that can be used to get the right answer. I have used and still use many of these examples when I explain why the product of two negative numbers is positive number. Some of these examples are probably familiar to many readers of this post. The first three examples are my favorites, and I use them exclusively. In all of the examples, it is assumed that students understand multiplication is repeated addition, and students have a basic understanding of the addition rules for positive and negative numbers. Of course, this is a big assumption.

Before I show you my examples, I will give you a brief description of the state of math education on the 1960’s and 1970’s in the United States. The Russian launch of the satellite Sputnik on October 4, 1957 caused a national panic which led to in complete revision of math and science curriculums. The revised math curriculum was called “new math” which was based on the advice of university research mathematicians and other professional mathematicians; not on the advice of wise and experienced math educators. New math placed emphasis on set theory, the fundamental properties of real numbers, functions, relations and the symbols of modern abstract mathematics. The new math approached mathematics from a more rigorous and abstract point of view as opposed to an intuitive and practical point of view. Students and parents found the new math strange and mystifying. Like most new waves in education, the new math was eventually replaced by another new wave. I clearly remember one nationally renowned math educator at a NCTM national convention in the 1980’s state (almost word for word), “Our research shows that a more rigorous abstract approach of teaching math only works with very bright students.” Of course, every experienced math teacher in the audience already knew this. The new math failed because you can’t make an abstraction (identify common core properties of different objects/systems) if you don’t have a base of knowledge and experience from which to make an abstraction. Since elementary, middle school, high school students, and normal adults don’t have this crucial base of knowledge and experience, it should be no surprise that the new math failed. By the early 1980’s, math education started to move in a new direction.

My first example is taken from the book Why Johnny Can’t Add: the Failure of the New Math written by Morris Kline (1908-1992) who was a scientist and professor of mathematics at NYU. One of Professor Kline’s core beliefs is that math concepts should be explained by using concrete examples that students can relate to. The example below is similar to the example Kline used to explain the product rule for positive and negative numbers.

Suppose homeowner Bob hires neighbor boy Bill to do general yard work at \$10.00/hour. We have four situations to consider; two from Bob’s point of view and two from Bill’s point of view. Bill is gaining \$10 every hour he works and Bob is losing \$10 every hour every hour Bill works.

• 4 hours in the future, Bill will be \$40 richer. (+4 * +10 = +40)
• 4 hours ago, Bill was \$40 poorer. (-4 * (+10) = -40)
• 4 hours in the future, Bob will be \$40 poorer. (+4 * (-10) = -40)
• 4 hours ago, Bob was \$40 richer. (-4 * (-10) = +40)

The next example involves filming a person walking forward at the rate of 4 ft/sec for 10 seconds, and then filming the same person walking backwards at the rate of 4 ft/sec for 10 seconds. By running the two films forwards and backwards in a film projector, we have four cases to consider. Let film A show the person walking forward, and film B show the person walking backwards. (It’s fun to pace back and forth across the room to illustrate this example.)

• If film A is run forward in the projector for 10 seconds, we will see the person walk forward 40 ft. (+4 * +10 = +40)
• If film A is run backwards in the projector for 10 seconds, we will see the person walk backwards 40 ft. (4 * (-10)) = -40)
• If film B is run forward in the projector for 10 seconds, we will see the person walk backwards 40 ft. (-4 * 10 = -40)
• If film B is run backwards in the projector for 10 seconds, we will see the person walk forward 40 ft. (-4 * (-10)) = 40)

The next example is probably familiar to many readers. From my personal experience, a few students find the two previous examples somewhat confusing, but the pattern approach illustrated in the text box below seems to make the most sense to students. The first 4 rows of the table follow from the fact that the product of a positive number and a negative number is a negative number which makes perfect sense to most students. As the value of numbers in column A decrease by 1, the product of A and B gets bigger by 5. When A decreases from 0 to -1, I tell by students they can’t change horses in midstream; so the pattern must be maintained by increasing the product by 5 when A is decreased by 1. The next example hinges on the idea that multiplication is repeated addition under the following rules: (Rules are easier to understand if m and n are integers.)

• If m is positive, then m * n equals n added to itself m times.
• If m is negative, then m * n equals the opposite of n added to itself |m| times.

The text box below illustrates how these rules work then n = ±4 and m = ±6. My last example uses the properties of real numbers and mathematical reasoning to demonstrate  (-3)(-5) equals (3)(5) = 15. The demonstration hinges on the following properties of real numbers:

• The distribute property
• A negative number times a positive number is negative. (Previously established)
• m + n = 0 if and only if m and n are opposites of each other.

Because this demonstration requires a higher level of mathematical maturity, I advise against showing this demonstration to younger learners. I will close this post with a discussion of the concept of positive and negative numbers by looking at two different number systems. You may be surprised to learn that in some number systems, the concept of positive and negative numbers does not exist. My post, A Simple Way to Introduce Complex Numbers, discusses the basics of complex numbers.

The set of real numbers:

• For every real number x: x = 0, x < 0, or x > 0.
• Every real number x not equal to zero has a unique opposite which is denoted by the symbol –x.
• The opposite of a real number is the same as the additive inverse of a real number.
• Real numbers x and y are opposites of each other if and only if x + y = 0.
• If x > 0, then x is a positive number and –x is a negative number.
• If x < 0, then x is a negative number and –x is a positive number.
• The angular direction of all positive numbers is to the right or 00.
• The angular direction of all negative numbers is to the left or 1800.
• For all real numbers x and y, x*y = (-x)(-y). Note: This is true for any pair of real numbers. The expression (-x)*(-y) does not indicate we are multiplying two negative real numbers.
• The symbol –x means the opposite of x; not negative x.

The set of complex numbers:

• All complex number z can be expressed in the form z = a + bi where a and b are real numbers and i is the unit imaginary number such that i2 = -1.
• Every complex number z not equal to zero has a unique opposite which is denoted by –z.
• If z = a + bi, then –z = -a – bi.
• The opposite of a complex number is the same as the additive inverse of a complex number.
• Complex numbers w and z are opposites of each other if and only if w + z = 0.
• The angular direction of complex number z can range from 00 to 3600.
• In general, complex numbers are neither positive or negative because the angular direction of a complex number can range from 00 to 3600; not just 00 or 1800.
• For all complex numbers w and z, w*z = (-w)(-z). Note: This is true for any pair of complex numbers. The expression (-w)*(-z) does not indicate we are multiplying two negative complex numbers because complex numbers in general don’t have a positive or negative property.
• The symbol –z means the opposite of z; not negative z.

## Student Use of Technology in the Classroom I strongly support the use of technology in the classroom, but of course with some caveats.  In the December 2015/January 2016 issue of MATHEMATICS teacher, published by the National Council of Teachers of Mathematics, NCTM, there were several excellent articles about teaching in a world with technology. The authors of the focus article, Corey Webel and Samuel Otten, discussed using the application PhotoMath in the classroom. PhotoMath is a free application from Apple that works by holding a phone or tablet up to a math equation in a textbook. If the equation is not too complicated, PhotoMath will solve the equation and show the steps to solve the equation. Without going into the pros and cons of each option, I will list the three options the authors considered.

Option 3: Consider a Different Division of Labor

Option 3 is about how teachers can treat technology as a tool and how tech tools should be used to engage students in ways that promote mathematical reasoning. If applications like PhotoMath are only used to get the right numerical answer, students are being cheated because they are not learning anything of real value. The authors suggested an activity in which students compare PhotoMath’s solution of a linear equation with an intuitive approach to solve the equation.

The purpose of this post is to share some of my thoughts regarding the use of technology in the modern classroom. I make no claim that computer software technology is the grand elixir for solving or fixing the problems of math education. My intention is to provide some ideas as to how software technology should and can be used to enhance math education. Of course, good visuals and physical manipulatives are just as important as ever. I’m looking forward to hearing what readers of this post think about using technology to enhance math education. Please feel free to provide your feedback by replying to this post, commenting on Facebook, or emailing me at info@mathteachersresource.com.

I fondly remember when electronic calculators became widely available in the 1970’s, and later micro computers in the 1980’s. In those early years, it was generally assumed that students and teachers should learn how to program a computer. Eventually it was realized that students and teachers should learn how to use application programs written by professional programmers. I also mention that in the 1980’s in most high schools, student computer access was limited to one or two computer labs and a relatively small number of computers scattered about in a few classrooms. Projection and computer technology is now an important feature of modern classrooms.

My core belief is that all high school math students should eventually understand and memorize a basic set of math facts and relationships before using software technology. Of course, software technology is a wonderful aid for teaching basic math facts. A high school student who needs a calculator to do problems like 8 + 39 or 7 * 9 has been cheated in his/her prior education. A beginning algebra student who needs a calculator to do a simple problem like 27 – 13 is wasting time and energy, and possibly losing some focus on the original problem. In my opinion, only after students have learned a basic set of math facts and relationships is the use of software technology appropriate. Readers can download my free handouts Basic Math Facts and Slope and Equation of Line Summary to use as resource when teaching some of these basic math facts. Some possible inclusions in the basic set of math facts and skills might be the following items:

• Addition, subtraction, multiplication and division facts
• Squares of the first 20 counting numbers
• Cubes of the first 10 counting numbers
• Powers of 2 such as 25 = 32, 210 = 1,024 and 2-6 = 1/64
• Decimal equivalent of common fractions such as 5/8 = 0.625
• Common Pythagorean triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41 and 11-60-61
• Properties of real numbers such as the commutative and associative properties
• The order of operation rules
• Common polynomial factoring patterns
• Slope of a line and the different forms of the equation of a linear relationship

After students understand and have memorized a set of basic facts, they should be taught how do a variety of mental math (no paper or calculator) problems. This is important not only to solve common math problems that routinely arise in daily living, but also to develop good estimation skills to see if calculator output is reasonable. Also by learning how to do these types of problems, students will develop a feel for how numbers work. All experienced teachers have stories of students who have given absurd answers that make absolutely no sense. I remember the nationally renowned math educator, Dr. Lola J. May (1923 – 2007), tell her audience, “You can’t teach math to an empty head.” The following examples illustrate some of the types of mental math problems, without hints of course, that I believe most students should eventually be able to handle:

• 25 * 89 * 4 = ? (Sample problem given in a talk by Dr. Lola May)
• 57 + 40 + 13 + 8 = ? (Hint: Consider the fact 13 = 3 + 10)
• 5*\$89.95 = ?  (Hint: Consider 5 times (\$90 – 1 nickel))
• Estimate a 15% tip if the bill is \$63.58 (Hint: What is 10% of \$64.00?)
• What is the average of 81, 82 and 87? (Hint: What is the average of 1, 2 and 7?)
• What is the length of the shortest side of a right triangle if the other sides have lengths of 30 and 34?  (Hint: Consider the Pythagorean triple 8 – 15 – 17)
• Estimate the casualty rate suffered by a regiment if only 47 of 262 survived the battle. (Hint: 50/250 = 1/5 = 20% that were not a casualty)
• 43*37 = ?  ( Hint: Consider (40 + 3)(40 – 3) = 402 – 32 )
• The fraction 17/20 is equivalent to what percent? (Hint: How many nickels make a dollar?)
• Estimate the decimal value of π/12. (Hint: π is close to what integer?)

In order to effectively use technology to teach math in a modern classroom, it’s imperative that all students have ready classroom access to a variety of software tools and physical manipulatives at any time in the classroom. No software tool or physical manipulative alone can teach math, however, teacher guided activities that require student engagement can naturally result positive student benefits. Creating the appropriate teacher guided activity is the hard part. The list below gives the tools that I have used or wished that I had when I taught high school math:

• Graphing calculator such as the TI 84
• Logo programming language (Great tool for teaching kids how to think.)
• Cuisenaire rods (Fantastic physical manipulative for teaching children core math concepts.)
• Easy to use, yet comprehensive graphing program
• Easy to use, yet comprehensive probability/statistics simulation program (I can’t image teaching probability theory without a good probability simulation program.)
• Program to calculate common statistics, confidence intervals, and hypothesis test results
• Word processing program
• Presentation software such as PowerPoint

Some Personal Thoughts Regarding the Use of Technology:

• Incorporate more history of mathematics and science into math courses. There are abundant online resources that students can use to create interesting PowerPoint presentations relating to a specific history of mathematics and/or science topic. This would be a natural way for students to improve writing skills. (I found the story about Kepler’s laws of planetary motion, and the relationship between Kepler and Tyco Brahe fascinating.)
• From time to time, have students graph the equations involved in a problem situation, and then check their paper and pencil solution against the graph on the computer screen. When I was tutoring a student several years ago, she asked me how she could show that the graphs of two relations are orthogonal. This involved finding the points of intersection, and then showing that the tangent lines at the points of intersection are perpendicular. I then graphed the relations and tangent lines so that she could better understand what the problem was about; and she did.
• I envision that the mathematics classroom of the future will be more like a science lab where students have a safe environment to ask questions, use software tools and physical manipulatives to conduct math experiments, and just play around with mathematics. I fully realize that most students are not free spirits in search of truth and are only taking a math course to meet some required course credit. With appropriate teacher guided structured activities, I believe it’s possible to gradually move math education in this direction. Creating the appropriate teacher guided structured activities will take a lot of hard work and sharing of ideas. Will math textbooks of the future primarily contain structured activities for learning math?
• It’s not necessary to understand all of the mathematics underlying a particular software tool. What’s important is to learn when and how to use the tool and correctly interpret the output generated by the software tool. A master mechanic has a set of tools that he uses to repair cars. The mechanic does not understand the underlying chemistry, physics, mathematics and design principles that allowed the creation of a particular tool, but he knows when and how to use the tool.

I will close this post with a true story about by daughter, Liz. When Liz was a sophomore in college, she asked me if she should take a Mathematica based calculus course. Mathematica was and still is the premier math program in the world. If an engineer, scientist or mathematician has some serious number crunching, any type of symbolic manipulation or data analysis to do, Mathematica is the tool of choice. Mathematica’s ability to create an endless variety of stunning graphic output is mind boggling. I thought that the course would be really cool, so I suggested that she take the course. Liz’s instructor was a knowledgeable user of Mathematica who was a grad student working on a PhD in mathematics. Liz thought that the instructor was a pleasant and likeable person. Students were given Mathematica files relating to the course and were essentially told, “go to it tiger.” Very little explanation was provided as to what concepts the students were supposed to learn by using Mathematica. Liz got an A in the course, but later she told me, “Dad, I feel that the course cheated me.” One student later told me that a Mathematica based calculus course is great if you already understand the course concepts. The point of this story is that no piece of software can teach math without proper and careful teacher guidance.

## When 0/0 or ∞/∞ Can Have a Value That Makes Sense In a previous post, Why Division by Zero is Forbidden, I explained why a nonzero number divided by zero is undefined and why zero divided by zero gives us infinitely many answers.

Several comments from math teachers indicate that it is not easy to get this concept across to younger students so that they have a real understanding of the concept. I fully agree. From personal experience, I have found that students initially attain a rudimentary understanding of a math concept, but only time, practice working with the concept, and increased mental maturity can students gain a deeper understanding of a concept.

One reader, Ali-Carmen Houssney, wondered why an expression of the form 0 / 0 is called “indeterminate.” The purpose of this post is to discuss the indeterminate forms of the type 0 / 0 and ∞ / ∞. There are other indeterminate forms which you can look up online, but the forms 0 / 0 and ∞ / ∞ are the two major indeterminate forms that appear in calculus text books.

To get started, I need to discuss the concept of a limiting value of a function f(x) at some specific point x = k. The limit concept is a core concept in differential calculus. The limiting value of a function when x = k is simply a number or value that f(x) gets arbitrarily close to when x gets arbitrarily close to k or +∞.  The definition of a limit of a function only requires that the input variable x gets arbitrarily close to some constant k and not necessarily equal to k. In most cases the limiting value of a function as x gets arbitrarily close to k is just f(k) itself. In other cases when f(k) results in the indeterminate form 0 / 0 or ∞ / ∞, the limiting value of f(x) when x gets arbitrarily close to k may exist or it may not exist. If the limiting value does exist, the limiting value will be a single finite real number or + ∞ which means that the function is increasing/decreasing without any upper/lower boundary. The text boxes and companion graphs below illustrate the limit concept for the functions y = f(x) = Sin(x) / x, y = g(x) = x/2x and y = h(x) = Floor(x). All function outputs in the examples are rounded to 9 decimal places.

What’s important to notice in the examples below, is that values of x very close to k were selected to test whether or not the corresponding function values are very close to some fixed constant L, the limit of the function at x = k. A mathematically rigorous demonstration of the existence of the limit at x = k would show that there is always an open interval (a, b) about k in which all x in (a, b), except x = k, so that the corresponding function values are arbitrarily close to L, the limiting value of the function at x = k.      From a geometric point of view, it’s interesting to see why the limiting value of Sin(θ) / θ = 1 as θ approaches 0. Refer to graph D below. If central angle θ is in radians, the length of the arc on the unit circle that subtends angle θ equals θ units. Now imagine what happens as θ approaches 0; central angle θ approaches 0, Sin(θ) and length of the arc get closer and closer to each other. Hence, Sin(θ) / θ approaches 1 as θ approaches 0. This fundamental limit is used to derive the formulas for the derivative of all trigonometric functions. There is another interesting fact. If θ is converted to degrees, the limiting value of Sin(θ) /θ equals π/180 ≈ 0.017453293. You can easily convince yourself that this is true by setting the angle mode of your calculator to degrees and entering values of θ in the expression Sin(θ) /θ that are very close to 0. In closing this post, I will mention L’Hôpital’s rule which is a very useful theorem for finding the limiting value of a function f(x) at x = k when f(k) results in the indeterminate form 0 / 0, ∞ / ∞, (-∞) / ∞, ∞ / (-∞) or (-∞) / (-∞).  When f(k) results in indeterminate forms such as 0 * ∞, 1, 00, or ∞ – ∞, calculus students learn techniques for rewriting the expression for f(x) so that L’Hôpital’s rule can be applied.  In order to use L’Hôpital’s rule, one needs to know how to find the derivative of a function.

## The Famous Cantor Ternary Set Many mathematicians consider the Mandelbrot set to be the most complicated and interesting set in all of mathematics. In contrast, the Cantor set is deceptively simple, but it has properties that are just as counter-intuitive and astonishing as some of the properties of the Mandelbrot set. Henry John Stephen Smith discovered the Cantor set in 1874, and Cantor introduced it to the world in 1883. As mentioned in a previous post, Cantor’s work helped lay the foundation upon which much of modern mathematical analysis rests. After studying some of Cantor’s mathematics in grad school, I gained a deeper understanding of fundamental concepts in calculus. When you think that you understand a concept fairly well and later learn that your understanding is a bit lacking, it can be humbling. If you have not read my first two Cantor posts, Infinity Does Not Necessarily Equal Infinity, and Why There Are Infinity Many Different Levels of Infinity, I strongly suggest that you read them before continuing. The concepts presented in this post heavily depend on an understanding of the concepts presented in those posts. As you continue to read this post, please slow down and read carefully. My intention is to be as clear and precise as possible with very limited use of the symbols of abstract mathematics. Cantor is very deep, and therefore this post will not be easy reading for many. Don’t get frustrated and give up if you don’t get it on the first or second read. If you are persistent, you will get it. You will find that the struggle was well worth your time and effort.

As I see it, part of the difficulty in understanding Cantor is that there is a vast difference between mathematical objects and physical objects. There is a theorem in mathematics that tells us that no matter how close one real number is to another real number, say 10-100 apart, there are infinitely many rational numbers between them.  Mathematicians elegantly describe this theorem as follows: The rational numbers are dense in the real numbers. This theorem is certainly not obvious to a normal person, and it seems to be counter-intuitive nonsense. We forget that mathematical points/numbers and curves have no thickness. The set of real numbers is far more complicated and mysterious than we realize. A pure mathematician can cut any interval on a real number line into infinitely many pieces without blinking an eye, but he or she can only cut a piece of lumber into a finite number of pieces. If two atoms are sufficiently close to each other, there is no room to fit another atom between them because atoms have a thickness property. As far as I know, no physical object contains infinitely many atoms. The speed of everything is finite. Pure mathematical knowledge is obtained through a mental activity of logical reasoning from a set of postulates or axioms. Knowledge in physics is obtained through a process of observation, inductive logic, deductive logic and experimentation with physical objects. To say that two real numbers are “just a little bit apart” is imprecise nonsense to a pure mathematician. For a pure mathematician, two reals are equal or they are not equal, but never just a little bit apart. A math professor once told me: “A woman is pregnant or she is not pregnant, but she is never just a little bit pregnant.” Cantor’s work reflects the mind of a pure mathematician who deals with mathematical objects that exist only in the mind of man and God. If God directly communicates math concepts to humans, as Cantor believed, God must be a jokester who is roaring with laughter as He watches us struggle to understand Cantor. If you understand and accept Cantor’s definition of equal cardinality of two sets, his counter-intuitive and absurd theorems are not so counter-intuitive and absurd after all. My three Cantor posts reflect my struggle to understand Cantor.

The purpose of this post is to give the reader a description of the Cantor set and some of its basic properties which seem counter-intuitive, preposterous, absurd, and astonishing. I will avoid the language of formal abstract mathematics as much as possible, and provide numerous explicit examples that illustrate the concepts presented.  Keep in mind the following concepts, definitions, and facts as the construction of the Cantor set is explained.

• Open interval (p, q) equals all real numbers between p and q, but not including p or q.
• Closed interval [p, q] equals all real numbers between p and q, and including both p and q.
• Two sets are disjoint if and only if the intersection of the sets equals the empty set.
• Numbers expressed in binary or base 2 format have only the digits 0 or 1.
• Numbers expressed in ternary or base 3 format have only the digits 0, 1, or 2.
• Numbers expressed in decimal or base 10 format have only digits 0 through 9.
• The terms “all”, “each”, and “every” mean without exception.
• “If and only if ” means whenever the first statement is true, the second statement is true and vice versa.
• A real number is a Cantor number if and only if it’s in the Cantor set.

To get started, let’s see how the Cantor ternary set is constructed. Like the Mandelbrot set, the Cantor ternary set is a fractal because it’s created by an infinite iterative procedure that determines what numbers are in the set. Numbers in the Mandelbrot set are complex numbers (a + bi) in a region of the complex coordinate plane. All Cantor numbers are real numbers contained in [0, 1]. Like the Mandelbrot set, construction of the Cantor set can only take place in the human intellect.

Construction of the Cantor set starts with the closed interval [0, 1] on the real number line with nothing removed. This interval is represented by the top solid bar in the graph below. At each step in the construction, the open middle 1/3 of each closed interval is removed to produce a new set of closed intervals. The Cantor set equals the set of points/numbers remaining in the closed interval [0, 1] after infinitely many iterations. The graph below depicts the closed intervals remaining after each of the first six iterations in the construction. Every horizontal line of the graph depicts the union of a set of closed intervals, and this union of closed intervals contains all of the Cantor numbers. It might appear that the Cantor set is empty, but you will later see that there are as many numbers in the Cantor set as there are real numbers in [0,1]. We end up with the same number of points we started with! In other words, the cardinal number of the Cantor set equals the cardinal number of the set of the real numbers in [0,1]. What remains in [0, 1] is fractal dust. So the Cantor set is nothing more than fractal dust that has the same cardinality as the set of reals in [0, 1].  (That is absurd! How can that be possible?) The text box below lists the closed intervals remaining after each of the first 5 iterations in the construction of the Cantor set. Note that the endpoints of the closed intervals are Cantor numbers, and the number of endpoints doubles on each iteration. After the 5th iteration, we have found at least 64 Cantor numbers. Later you will see that there are numbers between the endpoints of closed intervals that are also in the Cantor set. Furthermore, all Cantor numbers are contained somewhere inside the union of the disjoint closed intervals at every step in the construction. Example: 12/13 is contained in [8/9, 9/9], 12/13 is in the Cantor set, and 12/13 is not an endpoint of any of the closed intervals. Later you will see why 12/13 is a Cantor number. As an exercise, you could explicitly list all 64 closed intervals remaining after the 6th iteration. This may help you better understand how the Cantor set is constructed. Before I go any further, I need to discuss a theorem that tells us how to calculate the value of an infinite geometric sum. In this post, the infinite geometric sum formula is used to calculate the value of a binary or ternary expansion of a number with infinitely many digits. Infinite geometric sums are calculated follows:

• Let a equal the value of first term of the sum.
• Let r equal the constant multiplier of the terms where |r| < 1.
• Sum = a + ar + ar2 + ar3 + . . . = a(1/(1 – r))

The text box below shows how to apply the infinite geometric sum formula to calculate the binary or ternary expansion of a real number that has infinitely many digits. We can now see why the sum of the lengths of all open intervals removed from the interval [0, 1] equals 1. This is astonishing and leads to counter-intuitive conclusions. The text box below lists all of the disjoint open intervals removed from [0, 1] in the first 5 iterations. None of these open intervals contains a Cantor number. The sum of all removed open intervals = 1/3 + 2/9 + 4/27 + 8/81 + . . . = 1/3(1 / (1 – 2/3)) = 1. The length of the interval [0, 1] = 1, and the total of the lengths of all of the infinitely many open disjoint intervals removed equals 1. Therefore after infinity many iterations, the sum of the lengths of the closed disjoint intervals that contains the Cantor set must equal 0. All that remains is fractal dust. Mathematicians say that the Cantor set has Lebesgue measure zero. Later you will see that the cardinal number of [0, 1] equals the cardinal number of the Cantor set. (How is it possible that the cardinal number of fractal dust equals the cardinal number of all reals in [0, 1]? That is absurd!) It turns out that we can easily determine whether or not a real number x is a Cantor number if we know the ternary expansion of x. An important theorem about Cantor numbers states that every real number x in [0, 1] is a Cantor number if and only if there exists a ternary expansion of x that uses only digits 0 and 2. The proof of Cantor’s theorem hinges on this theorem. We will accept this theorem without a proof. The text box below shows the ternary expansion of various rational numbers in the Cantor set. Notice that some Cantor numbers like 1/27 and 1/3 have two equivalent ternary expansions. What’s important to understand is that the ternary expansion of all Cantor numbers, rational or irrational, can be uniquely expressed using only ternary digits 0 and 2. The ternary expansion of 1/2 = (0.111 . . .)3, and 1/2 is not in the Cantor set. It’s not important to know how to convert an arbitrary number to ternary format. Note that the ternary expansion of 12/13 = (0.220220220 . . .)3. If you are bored and want to add a little spice to your life, find the first 100 digits of the ternary expansion of 1/π or 1/e. Before we can get to the proof of Cantor’s theorem, we need to understand one more important idea. Every real number, rational or irrational, in the closed interval [0, 1] can be expressed as a unique binary coded number of the form (0.b1b2b3 . . .)2 where each binary digit bi equals 0 or 1. Some examples: 0 = (0.0)2, 1 = 1/2 + 1/4 + 1/8 + 1/16 + . . . = (0.1111 . . . )2, 2/3 = (0.10101010 . . . )2 and 0.875 = (0.111)2. What’s important to understand is that there is a unique binary expansion of the form just described for every real number in [0, 1]. How to find the binary expansion of an arbitrary number is not important; we just need to know that it can be done.

Cantor’s theorem states that the cardinal number of the set Cantor numbers equals the cardinal number of the set  reals in  [0, 1]. In other words, the number of Cantor numbers equals the number of reals in [0, 1]. A proof of Cantor’s remarkable theorem can now be given and it goes something like this:

• Let C equal the set of ternary expansions, using only the digits 0 and 2, of all reals in [0, 1]. Therefore C equals the set of Cantor numbers and C is a proper subset of the reals in [0, 1]. C is the fractal dust that is contained in the closed interval [0, 1].
• Let R equal the set of all binary expansions of the reals in [0, 1]. Therefore R equals the set of all reals in [0,1].
• Construct a one-to-one function f(x) with domain C and range R that matches all elements of C with all elements of R as follows: (This construction is so simple.) Let x equal any element of C. If the nth ternary digit of x = 0, then set the nth binary digit of f(x) = 0. If the nth ternary digit of x = 2, then set the nth binary digit of f(x) = 1.

Examples: f((0.20022202)3) = (0.10011101)2  and f((0.020220222)3) = (0.010110111)2

• For every element y in R, there is an element x in C such that f(x) = y.

Example: If y = (0.11000101)2, then x = (0.22000202)3.

• From the results discussed above and the definition function f, Cantor’s theorem easily follows. Since the cardinal number of the reals in [0, 1] equals the cardinal number of the set of all real numbers, it follows that the cardinal number of the Cantor set equals the cardinal number of the set of all real numbers.

How to construct the inverse function of f(x) is obvious. I don’t know what the graph of f(x) looks like, and I really don’t care. It’s only important to know that f(x) is a one-to-one function that pair wise maps set C to set R. Perhaps it’s a bit too dramatic and somewhat misleading to say “The number of Cantor numbers equals the number of reals in [0, 1].” It’s probably better to just say “There is a one-to-one function that pair wise matches the set of Cantor numbers with the real numbers in [0, 1].” On the other hand, how else can we compare the number of elements in two sets? If you understand and accept Cantor’s definition of equal cardinality, Cantor’s work makes more sense. Note that the above proof did not use the technique of proof by contradiction.

There is also a diagonal proof of Cantor’s theorem which uses the technique of proof by contradiction. My post Infinity Does Not Necessarily Equal Infinity gives Cantor’s famous diagonal proof which states that the cardinality of the set of real numbers is strictly greater than the cardinality of the set of counting numbers. The diagonal proof can be easily modified to show that the cardinality of the Cantor set is strictly greater than the cardinality of the set of counting numbers. This is easily accomplished by just replacing the strings of binary digits with strings of ternary digits consisting of 0 or 2 only. Therefore the cardinality of the Cantor set and the cardinality of the set of real numbers is strictly greater than the cardinality of the set of counting numbers. The continuum hypothesis states that there is no cardinal number between the cardinal of the counting numbers and cardinal number of all real numbers. If we accept the continuum hypothesis, it follows that the cardinality of the Cantor set equals the cardinality of the set of all real numbers; not just the reals in [0, 1].

I will close this post with a short discussion of the Cantor ternary function. Warning! This is not the function that was defined in the proof of Cantor’s theorem above. Basic properties of the ternary function and its graph are shown below. After a student studies the Cantor ternary function in a graduate level math course, he or she gains a deeper understanding of concepts learned in undergraduate level math courses. Functions are no longer just some formula like f(x) = 3x2 – 2x + 1 or g(x) = 3Cos(x) – 5. The first derivative of the ternary function can’t be found by applying the standard differentiation rules because there is no explicit formula for it. Wikipedia has an excellent article on the Cantor function.  ## Why There Are Infinitely Many Different Levels of Infinity

Because a recent post about the German mathematician Georg Cantor (1845-1918) generated a great deal of interest, I decided to do two more posts about Cantor’s contributions to mathematics. Of course, I find Cantor’s mathematics fascinating, and apparently many readers found the content of that post intellectually stimulating as well. You might think that Cantor’s work amounts to a bunch of clever and interesting mathematical mind games, but this is not the case. His work helped lay the foundation upon which much of modern mathematical analysis rests. If you have not read my first Cantor post, Infinity Does Not Necessarily Equal Infinity, I strongly suggest that you read it before continuing. The concepts presented in this post heavily depend on an understanding of the concepts presented in that post.

The purpose of this post is to show how Cantor proved that there are infinitely many levels of infinity or there are infinitely many different infinite cardinal numbers! The purpose of the next Cantor post will be to give readers a general description of the Cantor ternary set which is counter-intuitive, preposterous, absurd, and astonishing. I will not delve deeply into formal abstract mathematics, because my understanding of Cantor’s work only scratches the surface of his deep mathematics.

To get started, I will do a quick review of some basic definitions and concepts in set theory.

• A set can be any collection of objects such as numbers, character symbols, cars, people, cats, etc.
• Set A is a subset of set B if and only if every element of set A is an element of set B.
• Let A equal any subset of B. A is a proper subset of B if and only if there is an element in B that is not in A.
• The null or empty set is a set that contains no elements. The symbols { } or Ø denote the empty set.
• Sets {0}, {Ø}, and {{ }} are not the empty set because each of the three sets contains an element.
• The null set is both a subset and proper subset of every set.
• Set A equals set B if and only if the sets are subsets of each other.
• In set theory and Boolean algebra, the word “or” means “one or the other and possibly both.” In contrast, when a parent uses the word “or” with a child, the parent means “one or the other, but not both.”
• In set theory and Boolean algebra, the word “and” means “both are in the set” or “both are true.”
• The union of sets A and B, denoted by AB, is the set of elements that are in A or B.
• The intersection of sets A and B, denoted by A ∩ B, is the set of elements that are in A and B.
The text box below uses sets of numbers to illustrate the set definitions above.

The power set of set A, denoted by P(A), equals the set of all possible distinct subsets of A. In other words, P(A) is just another set that contains all of the distinct subsets of set A. To get a better idea of what P(A) means, the text box below gives P(A) for different finite sets of counting numbers. Note that if set A has n elements, then P(A) has 2n elements.

To see why increasing the number of elements in a set by one causes the number of elements in the power set to double, consider how you could go about creating a list of all 32 subsets of the set {1, 2, 3, 4, 5}. The first 16 subsets of {1, 2, 3, 4, 5} are given by the power set of {1, 2, 3, 4}. The other 16 subsets of {1, 2, 3, 4, 5} can be obtained by forming the union of {5} with each of the 16 subsets of {1, 2, 3, 4}. You should later go ahead and list all 32 subsets of {1, 2, 3, 4, 5} and then all 64 subsets of {1, 2, 3, 4, 5, 6}. I’m very serious about this suggestion because it will help you learn to think in a different way and help you better understand the fundamental counting principle, permutations, and combinations. You may find this task tedious and boring, but you will be rewarded with a better understanding of fundamental counting concepts. I can now explain the proof Cantor’s theorem which states that the cardinal number of P(A) is strictly greater than the cardinal number of A where A is any finite or infinite set. Cantor’s theorem can be used to show that there are infinitely many different infinite cardinal numbers. Recall from my first Cantor post that the cardinal number of a set equals the number of elements the set contains. Therefore the cardinal number of a google of water molecules equals 10100, and the cardinal number of a MLB active roster equals 25.  Also recall from my first Cantor post that the symbol for the cardinal number of the counting numbers is ℵ0, the symbol for the cardinal number of the real numbers is ℵ1, and ℵ0 < ℵ1.

For finite sets, Cantor’s theorem is obvious. If the cardinal number of finite set A equals n, then the cardinal number of P(A) equals 2n. For infinite sets Cantor’s theorem might seem obvious, but it’s much more difficult to prove. To make Cantor’s proof more comprehensible for infinite sets, I will first give a proof that shows that the cardinal number of P(C) is strictly greater than the cardinal number of C where C equals the set of counting numbers. Like many deep abstract mathematical proofs, Cantor’s proof uses the sophisticated technique of proof by contradiction. For set C, his proof goes something like this:

• Assume that there is a one-to-one function f with domain C and range P(C) that matches the counting numbers in C with all of the elements of P(C). The text box below shows one of the infinitely many possible ways that we could create a matching rule for f. The order in which domain and range elements are listed makes no difference. The important point is that there exists a one-to-one function f such that the domain of f equals set C and the range of the function, all function values, equals the set P(C).
• Construct a special subset M of the counting numbers as follows: (See the text box above.) Let set M equal all counting numbers n such that n is not contained in f(n). M is not the empty set because counting number j, such that f(j) = {   }, must be an element of M by definition.
• Set M raises a contradiction as follows: There must be a unique counting number k such that f(k) = M. Either k is contained in M or k is not contained in M. If k is contained in M, then by the definition of set M, k is not an element of M. If k is not contained in M, they by the definition of set M, k is an element of M. Therefore the function f can’t exist. Hence there is no counting number k that matches with M, and the cardinal number of set P(C) is greater than the cardinal number of set C. (The contradiction is somewhat like damned if you do and damned if you don’t.)

Now let’s see how we can prove that the cardinal number of the set of real numbers is strictly less than the cardinal number of the power set of real numbers. We need only to modify the last proof a little bit to give us our proof. The following algorithm describes how to create the modified proof:

• Let R equal the set of real numbers and the variable x equal any real number or element of R.
• Assume that there is a one-to-one function f with domain R and range P(R) that matches the real numbers in R with all of the elements of P(R).
• Construct a special subset M of the real numbers as follows:
1. Let set M equal all real numbers x such that x is not contained in f(x).
2. M is not the empty set because real number y, such that f(y) = {   }, must be an element of M by definition.
• Set M raises a contradiction as follows:
1. There must be a unique real number k such that f(k) = M. Either k is an element of M or k is not an element of M.
2. If k is an element of M, then by definition of set M, k is not an element of M.
3. If k is not an element of M, they by definition of set M, k is an element of M.
4. Therefore function f can’t exist. Hence there is no real number k that matches with M, and the cardinal number of set P(R) is greater than the cardinal number of set R.
Using the proofs described above as a model, it’s relatively easy to prove that the cardinal number of P(A) is strictly greater than the cardinal number of set A where A is any infinite set. By letting our imaginations run wild and considering set expressions such as P(P(A)) and P(P(P(P(A)))), we can create as many different infinite cardinal numbers we wish.

1. How will I ever apply set theory and cardinal numbers in my daily life? A: Probably never.
2. Do many engineers and scientists use set theory and cardinal numbers? A: Very few.
3. Name a math class that uses set theory? A: Venn diagrams to model probabilities of events in statistics.
4. Who uses set theory on a regular basis? A: People who design computer logic circuits use Boolean algebra.
5.  Is the study of set theory and cardinal numbers really just a mind game played by crackpot mathematicians? A: No – Cantor’s work helped create the foundation upon which much of modern mathematics rests.
6. Should very bright and curious high school math students be exposed to some of Cantor’s ideas? A: Yes.
I will close this post with a bit of personal information about Cantor. Cantor was a devout Lutheran who acknowledged the Absolute infinity of God. He believed that his theories about different levels of infinity were communicated to him by God. Some contemporary Christian theologians viewed Cantor’s work as a direct challenge to the idea that there is a unique infinity that only resides in God. For about the last 35 years of his life, Cantor suffered recurring bouts of depression. Most likely, the numerous vicious attacks on his work by many of his contemporaries contributed to his bouts of depression. Eventually Cantor’s work received praise and accolades from prominent contemporaries. In 1904, the Royal Academy awarded Cantor the Sylvester Medal which was the highest honor in mathematics. The brilliant mathematician David Hilbert said “No one can expel us from the Paradise that Cantor has created.” He spent the last year of his life in a sanatorium where he died on January 16, 1918.

## Infinity Does Not Necessarily Equal Infinity

A light year is about 6 trillion miles and the U.S. national debt reached 18 trillion dollars in 2015. Numbers of this magnitude are almost impossible to comprehend, but compared to infinity they are rather small. The German mathematician Georg Cantor (1845-1918) invented set theory and the mathematics of infinite numbers which in Cantor’s time was considered counter-intuitive, utter nonsense, and simply wrong. Many of Cantor’s contemporaries considered him to be nothing more than a charlatan. Set theory and the mathematics of infinite numbers are now part of mainstream mathematics.

To understand why Cantor upset so many mathematicians, I need to explain a basic concept of Cantor’s set theory. The cardinal number of a set or collection of objects equals the number of objects the set contains. Therefore the cardinal number of a gross of pencils equals 144 and the cardinal number of a mole of atoms is about 6.023 x 1023. For finite sets, the concept of cardinality is simple and straight forward, but for infinite sets the concept of cardinality can be counter-intuitive and utter nonsense. Cantor proved that the cardinal number of one infinite set can be greater than the cardinal number of another infinite set; infinity no longer necessarily equals infinity. Cantor also proved that there are infinitely many levels of infinity. In other words, there are infinitely many different infinite cardinal numbers!

What does Cantor mean when he says that two so seemly different infinite sets can have the same cardinality? Why are there as many real numbers between 0 and 1 as there are from –∞ to +∞? Why are there as many counting numbers as there are rational numbers? Why is the cardinality of the set of rational numbers less than the cardinality of the set of real numbers?  The purpose of this post is to provide answers to these questions without delving into formal abstract mathematics. If you want a mathematically rigorous discussion of cardinal numbers and set theory, take a graduate level course in set theory or point-set topology. My previous post Relations, Functions, and One-to-One Functions discussed concepts that will be used in this post and therefore readers may find it helpful.

To get started, I will do a quick review of the different types of real numbers. All real numbers are either rational or irrational. The set of rational numbers is composed of counting numbers, whole numbers, integers, and numbers that can be expressed as the ratio of two integers. The decimal expansion of all rational numbers starts to repeat in a pattern of fixed finite length at some point. The decimal expansion of an irrational number never starts to repeat in a pattern of fixed finite length. The text boxes below give examples of the different types of real numbers. Note that there is a pattern in the decimal expansion of n, but the length of the pattern increases. The term “real number” is unfortunate because it suggests that some numbers are valid and other numbers like the imaginary numbers are fake numbers.  Cantor uses the concept of cardinality to define when two sets have the same cardinality. Set A has the same cardinality as set B if and only if there is a one-to-one function that matches elements of A with elements of B such that the domain of the function is set A and the range of the function is set B. When both sets have a finite number of elements, this definition makes perfect intuitive sense. Example: When two bags of golf balls contain an equal number of golf balls, it’s easy to see how we can match the golf balls one-to-one, to show that the two bags of golf balls have the same cardinality. When both sets A and B have infinitely many elements, Cantor’s definition leads to a new and profound understanding of the nature of infinity. The matching rule for the one-to-one function in Cantor’s definition may be described by an equation or general algorithm that tells us how to match domain elements with range elements.

Using Cantor’s definition, let’s see why it makes sense to say that the set of real numbers between 0 and 1 has the same cardinality as the set of real numbers greater than 1. Initially, this seems preposterous. Two numbers are reciprocals if and only if the product of the two numbers equals 1. It’s a mathematical fact that every nonzero real number has a unique reciprocal and the reciprocals of two numbers are different if the numbers are different. If 0 < n < 1, then 1/n > 1. If n > 1, then 0 < 1/n < 1. Therefore the one-to-one function y = 1/x with domain equal to the open interval (0, 1) and range equal to the open interval (1, ∞) leads us to the conclusion that the open intervals (0, 1) and (1, ∞) have the same cardinality. Graph A below shows the graph of this function. If you think about it, the only practical way to show that two sets have the same cardinality is to show that there exists a one-to-one function that pair wise matches the elements of the two sets. I will now demonstrate that the open interval (p, q) has the same cardinally as the open interval (-∞, +∞) for any pair of real numbers p and q such that q > p. Let the width of the interval w = q – p and midpoint of the interval m = (p + q)/2. The one-to-one function y = Tan(π/w(x – m) with domain (p, q) has a range equal to (-∞, +∞). From Cantor’s definition, it follows that the cardinality of the open interval (p, q) equals the cardinality of the open interval to (-∞, +∞). Graph B below illustrates that the open interval (0.75, 1.25) has the same cardinality as the open interval (-∞, +∞). The next part of this post will demonstrate that the cardinality of the counting numbers equals the cardinality of the positive rational numbers. This will be accomplished by showing that there is a one-to-one function, f, that matches the counting numbers with the positive rational numbers. The matching rule of the one-to-one function is an algorithm that describes how we can systematically go about matching every counting number with a unique positive rational number in such a manner that every positive rational number gets matched with a counting number. Mathematicians say that the rational numbers are countable.

The algorithm for the matching rule of the one-to-one function is as follows:

1) Organize the positive rational numbers in a rectangular grid as shown below.

2) Start in the upper left corner of the grid. Set the counting number n = 1 and let f(n) = 1/1.

3) Continue moving from grid element to grid element forever as indicated in the diagram. If grid element p/q is not equivalent to a previous function value, then increase n by 1 and let f(n) = p/q. If grid element p/q is equivalent to a previous function value, then skip the grid element and go to the next grid element. (Note that skipped grid elements in the diagram are crossed out.) Now for Cantor’s famous diagonal proof that the real numbers are not countable. His proof used the sophisticated technique of proof by contradiction which is commonly used by mathematicians to prove a theorem. The diagonal proof goes something like this.

• Assume that there is a one-to-one function f(n) that matches the counting numbers with all of the real numbers. The box below shows the start of one of the infinitely many possible matching rules for f(n) that matches the counting numbers with all of the real numbers. The real numbers in the range of the function are represented as strings of base 2 real number digits or binary digits (i.e. consisting only of zeros and ones).
• Now construct a real number p as follows: Let n equal any counting number and f(n) equal the corresponding function value.
• If the nth binary digit of f(n) = 0, then set the nth binary digit of p = 1.
• If the nth binary digit of f(n) = 1, then set the nth binary digit of p = 0. (See the text box below.) It’s clear that the real number p is not in the range of f(n) which in turn contradicts the original assumption about f(n). Therefore the cardinality of the real numbers is greater than and the cardinality of the counting numbers and the real numbers are not countable.

Some Comments Regarding Cardinal Numbers and Real Numbers:

• The symbol for the cardinal number of the counting numbers is ℵ0. (aleph naught)
• The symbol for the cardinal number of the real numbers is ℵ1.
• The continuum hypothesis states that there is no cardinal number between ℵ0 and ℵ1.
• If you can prove the continuum hypothesis, you will become world famous overnight.
• There are infinitely many rational numbers between any two real numbers.
• The rational numbers are an infinitely small fraction of the real numbers.
• To work with irrational numbers in practical applications, we use rational numbers to approximate irrational numbers. (3.1416 ≠ π)
• The points, lines and curves that we draw on a chalkboard or computer screen are just crude approximations of true mathematical points, lines, and curves.
• True mathematical points and curves are infinitely thin, and therefore they can’t reflect light which in turn tells us that we really can’t see true mathematical points and curves in the physical sense.
• Mathematical objects only exist in the mind of man and God.

## Applying the Order of Operation Rules to Solve an Equation Experienced teachers know that some students seem to have a natural feel of how to solve an equation. They just know how and when an operation should be applied to both sides of the equation. Capable math students may not be fully aware of why they are using a particular strategy to solve an equation, but they know how to apply the strategy. Students who struggle with solving basic equations ask questions like the following: How do you know whether to add or subtract the same number to both sides of the equation?  How do you know whether to multiply or divide both sides of the equation by the same number? How do you know in what order various operations need to be applied to both sides of an equation? What does “find x” or “what is x” really mean? The purpose of this post is to provide answers to these types of questions. Readers who have read the posts Inverses of Relations and Functions and Inverse of a Matrix will immediately see the connection between those posts and this post.

To get started, you can download the free handouts Basic Equation Solving Strategies, Strategies for Solving Exponential and Logarithmic Equations, and Basics of Solving Inequality Relations. These free handouts (and many more) can also be accessed by visiting mathteachersresource.com/instruction-content. Depending on the course I’m teaching, I give one or more of these handouts to my students. Basic Equation Solving Strategies breaks down the basic algebraic equations that students will encounter in a lower level math course into six equation types which I call “case 1” through “case 6”. When solving different types of equations, I routinely ask my students to identify the type of equation being solved. When students can identify the type of equation being solved and know the basic algorithm to solve that type of equation, they can quickly and efficiently solve the equation; no wasted time. In application problems, solving an equation is usually just one small step in finding a solution of a problem.

This post will focus on how to solve a case 1 type of equation by the “work backwards” method. Case 1 equations are equations in which the variable appears only once in the equation. The work backwards technique of solving an equation is well known, but the manner in which it’s presented in this post is different. I use the following analogy to explain the basic reasoning behind the work backwards method: Suppose that you have established a base camp on a camping trip and take a day hike in the woods. You can always get back to your base camp by simplify reversing your steps.

Sample case 1 equations are shown in the text box below. I certainly would not have a beginning algebra student initially solve equations of this complexity; however, it’s my hope that they will eventually learn how to solve equations of this complexity. It takes a while for students to remember that equations involving the squaring or absolute value operations can have two solutions or no solutions. From personal experience, the work backwards method as presented in this post is effective with both low and high ability students. I remember one of my students saying to a classmate, “The work backwards method really works, but you need to know the order of operation rules.” Is the work backwards method effective with all students? Of course not!

The text box below gives five examples of case 1 equation types. When solving for a variable in terms of other variables, think of the other variables as constants. To use the work backwards method, it’s necessary to understand which operation reverses a given operation. After looking at specific examples of each type of operation, most students quickly develop an understanding of what reversing an operation means. The text box below gives a summary of common operations and the corresponding inverse operation. Note that the reciprocal operation is its own reverse or inverse operation. There are three major steps in solving a case 1 type of equation.

Step1) Recognize that an equation describes a process that starts with an unknown value of a variable, then performs a series of back-to-back mathematical operations, according to the order of operation rules. This process is described in what I call “the equation solve plan” which may or may not be explicitly stated by the student. Equation solve plans are road maps that lead to the solution of an equation. The student now knows what operation needs to be applied to both sides of the equation and when the operation needs to be applied. Initially, I require beginning students to explicitly state the equation solve plan as shown in the examples below where the solve plan is described in a box to the left of the list of equations. The equation solve plan is the series of back-to-back operations in the equation that are performed, according to the order of operation rules, on the variable being solved for.

Step 2) Follow the equation solve plan in reverse order from the last step to the first step. At each step, apply the reverse or inverse operation to both sides of the equation. Continue working backwards until the solution appears. This process is like peeling back an onion layer by layer.

Step 3) Check numerical solutions by plugging the solutions into the original equation. This is very important because it helps students better understand the equation and catch errors. I have no sympathy for a student who gives an incorrect solution and has not bothered to check the solution. With modern calculators, there is no reason that solutions can’t be checked.

The next six text boxes illustrate how the work backwards method works. The text in the box to the left of the list of equations is the equation solve plan. The operation symbol to the right of a step indicates what operation was applied to both sides of the equation. The check mark certifies that the solution was checked. Notice that the solution of the sixth equation is a somewhat different approach of solving the equation.      The handouts previously mentioned in this post are intended to provide a set of efficient algorithms that students can use to solve the types of equations and inequalities found in lower level math courses. Of course, some students will find a faster way to solve an equation or inequality in a special situation. However, I have observed students who seem to have a knack for making easy problems difficult. One time I saw a student rewrite the equation and then use the quadratic formula to solve the equation. The solution could have been obtained in 3 or 4 steps by using the work backwards method!

Some Personal Observations:

• Equation solving is not an end in itself, but a small step in a larger application problem.
• Practice solving equations is really nothing more than good mental gymnastics.
• Equation solving is an essential skill, but not creative mathematics.
• Discovering an equation that models a law of nature is creative mathematics.
• Discovering a new algorithm to solve a math problem is creative mathematics.
• Students primarily take algebra to start learning to how to reason abstractly with symbols; not how to learn to manipulate polynomial expressions, graph equations and solve equations. Of course, they don’t realize this.
• Professionals such as engineers, scientists, writers, artists, musicians, educators, company managers, business executives, military planners, etc. routinely think and reason abstractly with symbols, but they never or seldom factor a polynomial or use the quadratic formula.
• Beginning students should be required to express solutions in decimal format because an expression like 3 + √(29) has no real meaning for them.
• It’s necessary to remind beginning students that dividing by a number is the same as multiplying by the reciprocal of the number and vice versa.
• When the solution of an equation is an algebraic expression, imagine replacing the single variable that was solved for with the expression. If you carefully study the resulting equation, you will see the equation magically transformed itself into an identity. It’s amazing.