## A Simple Way to Introduce Complex Numbers

Complex numbers don’t make any sense. How can such weird numbers have any real use? The term “imaginary part” suggests that complex numbers are fake, cooked up by a bunch of crackpot mathematicians. That’s what I thought when I was in high school. But since my high school days, I’ve learned to use complex numbers to solve AC circuit problems, to do 2D vector math, to really appreciate the fundamental theorem of algebra, and to explore the famous Mandelbrot set. Complex numbers are now an essential tool in almost every branch of mathematics, science, and engineering.

In previous posts, I discussed how teachers can help students better understand and use the quadratic formula. But in order to have a complete understanding of the quadratic formula, it’s necessary to have a basic understanding of complex numbers.

I begin my introduction to complex numbers by asking my students to imagine that they are 3rd grade students who know the basic whole number addition and multiplication facts. I then have them consider how they, a 3rd grade student, would answer the six questions below.

After some discussion, my students agree that a 3rd grader would correctly answer questions 1, 2, and 5, but would not be able to answer questions 3, 4, and 6, because they don’t know about negative numbers and fractions. When those 3rd graders grow older and learn about fractions and negative numbers, they will be able to answer questions 3 and 4 correctly.

My students, not the 3rd graders, can correctly answer questions 1 – 5 but can’t correctly answer question 6, because they don’t know about the strange complex number i where i = √(-1) and i2 = -1. I explain that 7i * 7i = 49i2 = 49(-1) = -49. I tell students that all numbers after the counting numbers (1, 2, 3, . . .) are inventions of the human intellect and were invented to solve specific types of equations. It has been said, “God gave man the counting numbers, and man invented all the other numbers.”

In the next part of the lesson, I develop a list of the powers of the complex number i. The list of powers and the graph below enable students to easily see the circular pattern in the powers of i. (Note: i3 = i2 *i = (-1)i = -i and i4 = i2 * i2 = (-1)(-1) = 1)

After they learn about the powers of the complex number i, I show students how to plot a complex number and how to graph a complex number as a vector because all complex numbers have a magnitude and direction. Initially, students find it strange that complex numbers don’t have a negative property like some real numbers. Example: If the complex number z = 6 – 12i, then –z = -6 + 12i. I tell students that they should say, “the opposite of z,” for the symbol –z. The graph below shows complex number -9 + 6i and its conjugate -9 – 6i graphed as vectors. The other complex numbers in the graph below are graphed as a single point. Of course, 0 + 8i = 8i.

If time allows after the main lesson, I show students some interesting geometric patterns generated by the powers and roots of complex numbers. Students will learn how these pattern come about when they study De Moivre’s Theorem in a later course. It’s fun to make conjectures about the patterns. The left graph shows z, z2, z3, . . . , z20 where z = 1.15Cos(350) + 1.15Sin(350)i. The right graph shows the 12 12th roots of -4,096.

You can download the student and teacher versions of the free handout Introduction to Complex Numbers from www.mathteachersresource.com/instructional-content.html. This handout has two pages of exercises and student activities that I use to introduce my students to complex numbers. We usually work about a third of the problems together and the remaining exercises are left as homework. To make your presentations more dynamic, project graphs on a screen and use simple mouse control clicks to plot points and draw vectors.

Teaching Points: (Of course, teachers can modify the lesson to meet the needs of their class.)

• Read and study the free handout Introduction to Complex Numbers. As the lesson progresses, students should be taking notes and writing on a teacher provided student version of the handout.
• Some of the exercises involve calculating the absolute value of a complex number. Remind students that the absolute value of any number equals the positive distance of the number from zero, and therefore the theorem of Pythagoras can be used to calculate the absolute of a complex number. The absolute value of any nonzero number is always a positive real number, and i is never used to describe the absolute value of a complex number.
• Point out the geometric relationship between a complex number and its conjugate. After doing the exercises in the handout, many students see a way to use the conjugate to calculate the absolute value of a complex number.
• The handout Introduction to Complex Numbers covers all of the basic types of complex number arithmetic problems that an advanced algebra, trig, or precalculus student would be expected to handle. When appropriate, the polar form of a complex number can be explained at a later time.
• A geometric understanding of complex numbers is very important. Graphing complex numbers makes complex numbers more real to students. On homework and tests, have students graph various complex number expressions. Example: Let z = -8 + 4i. Graph and label each of the following as a vector: z, -z, 1.5z, -0.5z, and the conjugate of z.
• If time allows, show students interesting geometric patterns generated by the powers and roots of a complex number. It is interesting to see what pattern observations that students come up with. Tell students that they will learn the details of how these patterns come about in a later course. In most elementary math courses, students are never exposed to the really cool and interesting aspects of mathematics.
• Some students will claim that they can use their graphing calculators to get the answer in a matter of seconds. They are right. Remind them that they will not be allowed to use their graphing calculator on a test or quiz until they have demonstrated that they can do basic complex number arithmetic.

The above graphics were created with the program, Basic Trig Functions, which is offered by Math Teacher’s Resource. In addition to graphing x-y variable relations and polar functions, users can graph the powers or roots of a complex number, and view a list of the powers or roots which appears to the right of the graphic output. Segments or vectors can be drawn by left-clicking and dragging the mouse. The Edit/ Edit Graphics menu provides options for setting segment color, pen width, and head/tail parameters.

The user interface for all program modules is simple and intuitive. When graphing equations, users can select a sample equation which is automatically pasted into the active equation edit box. When appropriate, the program provides comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

## Better Way to Teach the Power of the Quadratic Formula

“This report doesn’t make any sense. It’s supposed to be a professional report, not a text message to a friend. Do it over.” That’s what a senior nuclear engineer, who happens to be my son, told a recently hired junior engineer. He was responding to a report submitted by the junior engineer. My son is a very good writer, and he tells me that he is appalled at the poor writing ability of some engineering graduates. He has learned that even if an engineering student has a high GPA, it does not necessarily follow that the student can write well. In fact, some math, science, engineering, and technical types go into a college major thinking that they will not be required to write papers and reports. Many years ago, one of my geometry students told me that he was going to be a minister, because ministers don’t have to write.

What does this have to do with teaching the quadratic formula? In my previous post, I discussed how teachers can help students understand the quadratic formula from both algebraic and geometric points of view. In this post, I’ll show you how teachers can use custom-made handouts created with computer technology not only to help students gain a deeper understanding of the quadratic formula but also learn how to be better writers. (When I use the term ‘teacher,’ I’m referring to anyone who teaches math, not just the traditional classroom teacher.) The ability to write well is a major goal of Common Core and STEM education. This lesson will not only help students better understand math, but will also help them become better communicators.

To best understand this discussion, download the student and teacher versions of the free handout Power of the Quadratic Formula from www.mathteachersresource.com/instructional-content.html. This handout provides seven ideas for teacher-guided quadratic formula learning/verification activities. Students should have basic competency using the quadratic formula. They will also need a graphing calculator, a ruler to box answers, and a teacher provided handout that gives structure to the lesson. During the lesson, students are expected to be actively engaged by calculating values of expressions, checking results, writing on the handout, and graphing equations on their graphing calculator. To make your presentations more dynamic, project graphs on a screen and use simple mouse control clicks to plot points and find x-intercepts of graphs.

The teacher version of the free handout Power of the Quadratic Formula contains all solutions to the student version of the handout. The free handout Observations About the Roots of a Polynomial gives a summary of important theorems about the roots of a polynomial. You can download it at the link mentioned above.

The graph of the equation y = x4 – 8x2 + 4 is shown below. The equation x4 – 8x2 + 4 = 0 is the first equation in the free handout Power of the Quadratic Formula. Because the ratio of the exponents is 2:1, the quadratic formula can be used to solve this equation. From the graph of the equation, we can see that there are four solutions. After solving the equation, students will see that the four solutions are irrational numbers that can be approximated to 9 decimal places. When a teacher shows them an efficient way to check solutions with their graphing calculator, some students are amazed that the value of the expression really equals zero or almost zero. Seeing the graph and checking solutions makes it more real to students.

The graph of the equation 2x2 – 3xy = 4y – 2 is shown below. This is the sixth equation in the free handout Power of the Quadratic Formula. Students will see how to use the quadratic formula to express x as an explicit function of y. By rearranging the equation, y can be expressed as a function of x, which makes it possible for students to use their graphing calculator to graph the equation. As the teacher version of the handout points out, different equation formats give us different insights about the graph of the equation. The program Basic Trig Functions, offered by Math Teacher’s Resource, can graph all three versions of the equation. I love to experiment with different equation formats. I’m still amazed that different equation formats always result in the same graph. (I must be getting old.)

Teaching Points: (Of course, teachers can modify the lesson to meet the needs of their class.)

• Teachers should read and study the handouts mentioned above.
• As the teacher derives the solution, students should carefully record the steps leading to the solution. This will give students a model of how solutions should be communicated in an organized manner to a friend, parent, in a homework assignment, or on a test.
• Answers should be expressed in a manner that reflects an understanding of the results found. Example: There are four irrational roots: x ≈ + 2.70828182 or x ≈ + 5.00968842.
• Solutions should be expressed in decimal format because exact radical format is meaningless to most students. Of course, for mathematically mature students, exact radical format is fine.
• Show students an efficient way to check a solution. All solutions should be checked. Have students use a check mark to certify that they have checked solutions.
• No more than a couple exercises of this type should be assigned in a homework assignment. Homework exercises of this type should be assigned periodically throughout the course.
• From time to time, a problem of this type should appear on a test or quiz.
• Some students will claim that they can get the answer in a matter of seconds. They are right. Remind them that this is not only about getting the right answer but also about learning how to communicate ideas to another human being and gaining a deeper understanding of a math concept.
• Remind students that learning to write well is hard work. It takes time and a great deal of effort to get it right. So what’s wrong with that? It’s worth it.
• When the occasion arises, teachers should explain how Descartes’ rule of signs can be used to predict the number of positive and negative real roots. There is no reason to wait until a later chapter in the book or the next math course.
• Explain to students that many calculator outputs have a rounding error. An output like 3.08 * 10-13 = 0.000 000 000 000 308 should be treated as equal to zero. Many beginning students don’t realize that calculator outputs like 6.18 * 10-10 essentially equal zero.

The above graphics, created with the program Basic Trig Functions, are offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a textbook. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities.

The user interface for all program modules is simple and intuitive and provides numerous sample equations along with comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

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photocredit: morguefile.com. Used by permission.

## A Different Way to Teach the Quadratic Formula

One of my core beliefs is that, whenever possible, math concepts should be understood from both an algebraic and geometric point of view. In previous posts, we looked at how René Descartes (1596 – 1650) gave us the synthesis of algebra and geometry. Now let’s look at how teachers can help students understand the quadratic formula from both an algebraic and geometric point of view by using custom made handouts created with computer technology.

To best understand this discussion, download the student and teacher versions of the free handout Quadratic Formula (teacher version) from http://www.mathteachersresource.com/instructional-content.html. This handout provides six ideas for teacher-guided quadratic formula discovery/verification activities. During the lesson, students are expected to be actively engaged calculating values of expressions and writing on the handout, so in addition to a handout, they will need a calculator and ruler. To make your presentations more dynamic, project graphs on a screen as you plot points and draw line segments with simple mouse control clicks.

The graph of the equation y = x2 + 5x – 8 is shown below. This quadratic equation is the first equation considered in the free handout Quadratic Formula. The added graphics are the graphics that students would be expected to add as the lesson progresses.

The graph of the equation h = -16t2 + 132t + 60 is shown below. This equation is the fourth equation in the free handout, Quadratic Formula (student version). The activity is about a toy rocket that is shot upward with an initial vertical velocity of 132 feet/second. The added graphics are the graphics students would be expected to add as the lesson progresses. The slope of the secant line through (6.5, 242) and (7, 200) tells us that the average vertical velocity of the toy rocket over the time interval [6.5, 7] equals -84 feet/second. The goal of this activity is to show students how mathematics can be used to extract useful information from an equation.

Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

• Students can be shown the derivation of the formula before the lesson or at a later time.
• Teachers will have to demonstrate how to enter an expression into a calculator. Students will probably make mistakes initially. Practice is the only way to improve.
• Teachers should have students use the graph to estimate answers before the actual calculation.
• Students should learn how to express answers in decimal format, because radical format is too abstract.
• Some answers require more than a simple numerical value. Teachers can dictate an English sentence that would be an appropriate way to answer the question. This is a good way for students to practice writing skills.
• The equation h = -16t2 + 132t + 60 in the toy rocket activity describes the relationship between t and h in the gravitational field in which we live. Students will learn how this equation comes about when they take a course in physics. ( -16 equals ½ of the gravitational constant for planet earth, 132 ft/sec = the initial vertical velocity, and 60 feet = the initial height above ground level.)
• Students and teachers can explore how gravity causes the average vertical velocity of an object to change over time.
• If students understand the basics of complex numbers, teachers can present activities five and six in the Quadratic Formula handout.

The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a textbook. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities.

The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition, relative minimum points, relative maximum points, x-intercepts and, intersection points can be found with simple mouse control clicks. A Help menu provides a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

## The Genius of René Descartes – Part 2 (The Parabola)

In my previous blog, I discussed how René Descartes (1596 – 1650) discovered a way to synthesize geometry and algebra, which resulted in a revolution in mathematics. This synthesis is the reason Descartes is credited as the father of analytic of geometry. Because of Descartes’s discovery, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on a conic curve.

In this blog, I will discuss how teachers can use modern computer graphing technology to help students gain a better understanding of the definition of a parabola and the equation of a parabola. In a future blog, I will discuss some of the magical properties and applications of the parabola. This discussion is intended to provide teachers with a general approach to teach the parabola. The specific approach is left to the discretion of the individual teacher. Teachers may want to provide a handout that students complete as the lesson progresses. Students should be required to use their calculators to make various calculations and verify specific facts during the lesson. When graphs are projected on a screen, presentations can be dynamic, especially with a moving trace mark and mouse drawn line segments.

Now for a quick review of the parabola. All parabolas are defined in terms of a fixed point, the focus of the parabola, and a fixed line, the directrix of the parabola. Point (x, y) is on the parabola if and only if the distance from (x, y) to the focus point equals the distance from (x, y) to the directrix line. All parabolas have an axis of symmetry and the directrix of the parabola is perpendicular to the axis of symmetry. The vertex and focus are on the axis of symmetry, and the vertex point is equidistant from the focus and directrix. Study the basic parabolic graph below.

Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

• Similar to the graph above, show the graph of a parabola and its directrix, in a handout and or on a projection screen.
• Students should be told something about a focus point and the directrix line but not the definition of a parabola. The definition of a parabola and the derivation of the equation will come at a later time.
• For at least four points (x, y) on the parabola, have students use the theorem of Pythagoras to calculate the distance from (x, y) to the focus and from (x, y) to the directrix line.
• Hopefully, most students will realize an amazing property of the parabola. For every point (x, y) on the parabola, the distance from (x, y) to the focus always equals the distance from (x, y) to the directrix. The class can experiment with other points on the parabola.
• Repeat the experiment with the equation y = 0.75x2. Does this new parabola have the same amazing property?
• Consider the parabola with equation y = kx2. How does changing k change the focus point? In general, how does changing k affect the shape of the graph? If we know the focus point of the parabola, can we find the equation of the parabola? Teachers and students can answer these questions by experimenting and just fiddling around with a computer graphing program.
• In another lesson, teachers can show students how the equation y = kx2 comes about. Since all parabolas are geometrically similar figures, the equation of most parabolas can be found by applying the standard equation transformations rules.
• Special equation transformation rotation rules:
Rotate graph 90 degrees counterclockwise about (0, 0): Replace x with –y and y with x.
Rotate graph 90 degrees clockwise about (0, 0): Replace x with y and y with –x.
Rotate graph 180 degrees about (0, 0): Replace x with -x and y with –y.
• Homework practice problems and exam questions.
Given the vertex and focus of a parabola, have students find the equation of the parabola and sketch the graph of the parabola and the directrix.
Given the graph of a parabola with key points, find the equation of the parabola, find the x-y coordinates of the focus and find the equation of the directrix line.

Teachers might consider allowing students to have some fun by having them do a project in which they are told to experiment and fiddle around with a computer graphing program to see what interesting relation graphs they can come up with. I can guarantee that an inquisitive student will come up with a graph that no human in the history of mankind as ever seen. Other than myself and some of my students, no human has ever seen the graph of the relation 2xSin(3x) + 2y = 3yCos(x + 2y) + 1.

The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a text book. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives users a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

## The Genius of René Descartes – Part 1

René Descartes (1596 – 1650), a French philosopher, mathematician and writer, discovered a way to synthesize geometry and algebra that resulted in a revolution in mathematics and science. Without Descartes’s brilliant insight, it would not have been possible to develop differential calculus, integral calculus, and many other branches of mathematics. What was revolutionary to Descartes’s contemporaries, now seems natural and almost intuitively obvious, a part of our culture. (Before Isaac Newton, the concept of gravity was unknown, and now all adults and most children know something about gravity.)

So what was Descartes’s world changing discovery all about? He first invented a right angle based coordinate system in which every point in the Euclidean plane is assigned a unique ordered pair of numbers, which represents the point’s location, denoted by (x, y) where both x and y are real numbers. He then demonstrated how to create algebraic equations or formulas to calculate the distance between two points, midpoint of a line segment, and the slope of a line. With these basics established, he showed how to find an x-y variable equation that describes the relationship between the x-coordinate and y-coordinate for every point on a curve and only those points on the curve. Once the equation of a curve is known, the equation can be algebraically manipulated to reveal important properties of the curve and solve a wide variety of application problems.

The diagrams below illustrates how Descartes’s great discovery is used to calculate the distance between points A and B, and the slope of the line that contains points A and B. The distance calculation is, of course, a direct application of the theorem of Pythagoras. If we let AB equal the distance from point A to point B and let m equal the slope of the line that contains points A and B, then AB = √( 82 + (-6)2 ) = √(100) = 10 units, and m = Δy / Δx = -6/8 = -3/4 or -0.75.

From the definition of a conic section and the theorem of Pythagoras, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on the curve. Study the graph of the circle and its equation. If you listen carefully, you will hear Pythagoras whisper from his grave, “x squared plus y squared equals 4 squared for every point (x, y) on the circle.” The graphs below are the graphs of various conic curves and a line. All equations are special cases of the general conic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are real number constants.

The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered as indicated to the right of the graphic. Example: The inequality x2 + 4y2 ≤ 64 is entered as x^2 + 4y^2 ≤ 64. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results.

The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives a quick summary of all the magical mouse control clicks. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

## Why is Division by Zero Forbidden?

What is 5/0? When I ask my beginning algebra students that question, the most popular incorrect answer they give me is 0. The next most popular incorrect answer is 5. After repeated reminders by their math teachers, students eventually learn that 5/0 is undefined, has no value, or is meaningless. (I once told a class of 9th grade algebra students that if they use their calculator to divide a number by zero, the calculator will explode in their face. One student looked at me and said, “Really?” I forgot how literal 9th graders can be. At least I got the student’s attention.) When I ask college algebra, trigonometry, statistics, technical math or calculus students why a number divided by zero is undefined, I either get an answer that begs the question or students say it’s simply a mathematical fact that they learned in a previous course.

So how do you explain division by zero? There are two ways. The first depends on a basic understanding of division of two numbers. It goes something like this: Students learn that a / b = c if and only if a = b*c. Therefore 986 / 58 = 17 because 58*17 = 986. Is 5 / 0 = 0? No, because 0 * 0 ≠ 5.   Is 5 / 0 = 5? No, because 0*5 ≠ 5. Since 0 times any number never equals 5, 5 / 0 is NOTHING or undefined. So what about 0 / 0? The problem here is that 0 times any number equals 0, and therefore 0 / 0 would have infinitely many answers, which in turn would be rather confusing. So we say that any number divided by zero is undefined.

The second explanation involves a deep mathematical insight from the 12th century Indian mathematician and astronomer, Bhāskara II, who developed the basic concepts of differential calculus. The 17th century European mathematicians, Newton and Leibniz, independently rediscovered differential calculus. This second explanation due to Bhāskara II goes something like this. Consider a single piece of fruit. If we divide 1 piece of fruit by ¼, we get 4 pieces of fruit. If we divide 1 piece of fruit by 1/10,000, we get 10,000 pieces of fruit. As 1 is divided by smaller and smaller numbers that approach zero, the number of pieces of fruit increases without bound. Therefore 1/0 = ∞ and, in general, n/0 = ±∞ if n does not equal 0.

Bhāskara II, Newton and Leibniz discovered the revolutionary concept of a limit of a function at a point, which enabled them to get around the problem of division by zero. Once that problem was solved, it was a relatively easy task to find methods to calculate a rate of change over a time interval of length zero, rate of change over a fleeting instant of time, or rate of change over a flux of time, as Newton would say. In The Ascent of Man, Dr. Bronowski tells the viewer, “In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.” Differential calculus is all about the mathematics of variable rates of change. I should mention that differential calculus students learn a slick technique for finding the limiting value of an x-variable expression as x approaches a constant k and the value of the expression when x = k is 0/0 or ∞/∞.

The graphic below shows the graphs of the functions y = 2Sin(x) and y = 2Csc(x) along with its vertical asymptotes. The graphs are color coded green, blue and red respectively. Because Csc(x) = 1 / Sin(x), the Csc(x) function is undefined at precisely those values of x where Sin(x) = 0. It’s interesting and fun to advance a trace mark cursor on the graphs of these functions. On both graphs, the horizontal velocity of the trace mark is constant, but the vertical velocity of the trace mark changes as the value of the x changes. As x approaches a vertical asymptote, the trace mark races towards ± ∞. Differential calculus gives us a complete understanding of the phenomena of the moving trace cursor.

The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. The equations entered into the program were: y = 2Sin(x), y = 2Csc(x), and Sin(x) = 0. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

Differential calculus is not only interesting and fun, but it can also be a stress reliever. At least it was for Omar Bradley, the famous American WWII general. He took a calculus book with him on battle campaigns, and when opportunity allowed, he worked differential calculus problems to relieve the stress of a battle campaign.

## Theorem of Pythagoras and “The Ascent of Man”

Everyone who has studied mathematics is aware of the theorem of Pythagoras. However, relatively few people are aware of the history of the theorem and how the theorem reveals some of the secrets of the universe. In the early 1970s, The Ascent of Man, a thirteen-part BBC series written and hosted by Dr. Jacob Bronowski, appeared on public television. (Both the DVD and book version can be purchased online.) Even to this day, I periodically read or view parts of the series in order to enjoy, ponder, and savor special moments that were epiphanies for me. In this blog, I would like to discuss some of the insights about the theorem of Pythagoras that Dr. Bronowski shared with his viewers.

Dr. Bronowski tells his viewers, “To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics.” To hear this statement from a tier-one mathematician and scientist is astonishing. I was awe struck by the profound insights he revealed as he explained history of mathematics and the proof of the theorem. As a high school geometry teacher, I was well aware of the importance of the theorem of Pythagoras, but did not understand or appreciate that Pythagoras established a fundamental characteristic of the space in which we move. His theorem describes the relationship between the lengths of the sides of a right triangle, and this relationship is true if and only if the triangle is a right triangle. The sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

The Egyptians used a set square with sides of 3 units, 4 units and 5 units to build the pyramids, and the Babylonians used set squares, beside the 3-4-5 set square, to build the Hanging Gardens. By 2000 BC, the Babylonians knew hundreds of Pythagorean triples. The fact that the Babylonians knew 3,367-3,456-4,825 is a Pythagorean triple is testimony that the Babylonians were very good at arithmetic. Listed below are all of the Pythagorean triples with sides less than 100 units and the lengths of sides are relatively prime. Many beginning geometry students are surprised when they learn that any multiple of a Pythagorean triple is another Pythagorean triple. Example: Since 3-4-5 is a Pythagorean triple, the triples 6-8-10, 9-12-15, 12-16-20, 15-20-25, and 120.75-161.00-201.25 are Pythagorean triples.

3-4-5,           5-12-13,     8-15-17,     7-24-25,     9-40-41,     11-60-61,   12-35-37,   13-84-85,   16-13-65,          20-21-29,   28-45-53,   33-56-65,   36-77-85,   39-80-89,   48-55-73,   65-72-97

About 550 BC, Pythagoras proved why the relationship for the sides of Egyptian and Babylonian set squares is true, and this relationship is true for any right triangle, not just the set squares of the ancient builders. Dr. Bronowski shows his viewers how Pythagoras probably proved the great theorem to his followers. Pythagoras first created the square pattern shown below and to the left. The area of this square = c2. Pythagoras then created the pattern to the right by rearranging the pattern on the left. The area of the square pattern on the left must be equal to area of the pattern on the right. Furthermore, the area of the pattern on the right equals the sum of the areas of two squares which equals a2 + b2. Therefore a2 + b2 = c2.

There are hundreds of proofs of the theorem of Pythagoras, but none are as elegant as the proof shown above. I can only begin to imagine how Pythagoras must have felt after he completed his magnificent proof. It is said that he offered a hundred oxen to the Muses in thanks for the great inspiration. Book 1, Proposition 47 of Euclid’s Elements, written in about 300 BC, gives a proof of the theorem of Pythagoras. In 1876, when James Garfield, the 20th President of the United States, was serving in the United States Congress as Representative of Ohio’s 19th District, constructed a proof of the great theorem.

Even if you don’t consider yourself to be a math or science type, I think you’ll find The Ascent of Man to be fascinating. Bronowski has a gift for explaining fundamental discoveries in a wide variety of human enterprises in a manner that makes perfect sense to the thoughtful reader or viewer. Musicians will learn that Pythagoras found a basic relationship between musical harmony and mathematics. Artists will learn about the geometric designs created by Arab artists-mathematicians which led to a complete understanding of the symmetries of space, which in turn explains why molecular structures can only have certain shapes.

Basic Trig Functions, offered by Math Teacher’s Resource, has a module that enables teachers to create a wide variety of trig-circle diagrams in which the sides of the right triangles can be multiples of any of the 16 Pythagorean triples listed above. These diagrams can be used to create handouts, homework assignments and test questions. Teachers can go to www.mathteachersresource.com/instructional-content.html and download the free handouts Trig Exercises # 1 and Trig Exercises # 2. These handouts will give teachers a good idea of the kinds of course materials they can create. Teachers may also want to download the free handout Basic Math Facts which is a compilation of some of the basic math facts that I want my beginning algebra students to understand and be able to apply when they finish my course. I even give this handout to my college algebra and technical math students. You can use our software to create all types of course materials, which I invite you to post and share on our web site. Besides handouts, teachers can use our software to create dynamic classroom presentations. Go to www.mathteachersresource.com to view numerous screen shots of different program modules.

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Photo of Pythagorus by Galilea at de.wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], from Wikimedia Commons

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## Functions and Relations: Teaching the Concepts

“Why are functions and relations important?” students ask. “How will I ever use functions and relations? Who cares?”

The answer is that the concepts of function and relation are core concepts in mathematics, and almost every course a student will take in the future will use these concepts. What’s more, we use functions and relations every day without realizing it.

All functions and relations have three components:

Domain Component: The domain of a function or relation can be any collection of objects. Elements in the domain can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The domain variable of a function or relation represents all possible input values and is said to be an independent variable.

Range Component: The range of a function or relation can be any collection of objects. Elements in the range can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The range variable of a function or relation represents all possible output values. Range variables are called dependent variables, because their values depend on domain input values.

Matching Rule Component: The matching rule component tells us how to match domain objects with range objects. Matching rules are usually in the form of an equation, table, list of ordered pairs, catalog, or set of directions. Every function is a relation, but not every relation is a function. If a relation is a function, every domain object is matched with only one range object. If a function is a one-to-one function, then any two different input values always have different output values. If a and b are domain input values, the symbols f(a) and f(b) represent the corresponding output values. If f(x) is a one-to-one function and f(a) = f(b), then a = b.

To introduce functions and relations to students, I often use the following examples:

Restaurant Menu Function: When a person is handed a menu in a restaurant, they instinctively start matching food items on the menu with prices. The domain of a restaurant menu equals all the food items for sale, and the range equals the prices of the food items. Every food item of a specific type and size is matched with exactly one price. Since different food items may have the same price, a restaurant menu is not a one-to-one function. When a customer sees chateaubriand for two on the menu with a price of \$150, the customer probably thinks that chateaubriand is out of his price range. The matching rule is usually described on a sheet of heavy paper, sign on a wall, or on a chalkboard mounted on a wall.

Student Report Card Function: The domain of a student’s report card equals all courses in which the student was enrolled. The range of a report card equals all grades the student received for the courses enrolled. Students receive one grade for each course taken. Because students can receive the same grade in different courses, most report cards are not one-to-one functions. The matching rule for a report card is usually described on a sheet of paper or in a database that can be accessed at the school’s web site.

People and Social Security Number Function: The domain equals all people with a valid social security number, and the range equals all valid social security numbers. Individuals can have only one valid social security number. The matching rule is determined and maintained by the Social Security Administration. This function is one-to-one because two different people who have valid social security numbers always have different social security numbers.

Cars and VIN Number Function: The domain equals all legally manufactured cars, and the range equals all vehicle identification numbers (VINs) assigned to the cars. Every car has only one VIN, and no two cars have the same VIN. The one-to-one matching rule is described in some computer database.

People and Phone Number Relation: The domain equals all people who have a phone, and the range equals the phone numbers of all people who have a phone. Since an individual can have more than one phone number, the matching rule is a relation, but not a function.

Musical Notes and Sound Function: The domain of a piece of music is the collection of note symbols that appear on a sheet of paper and the range equals the sounds that correspond to the notes on the sheet of paper. The matching rule is one-to-one because different note symbols always result in different sounds. The music matching rule is a set of rules developed by musicians over the course of many centuries.

Mathematical Functions and Relations: In beginning algebra courses, both the domain and range of a function or relation are subsets of the real number system. The matching rule is usually a relatively simple equation such as y = -x + 3, y = 3x^2 + 1, y = (2x – 5)/(x + 3) or x^2 + y^2 = 36.

Free Functions and Relations Handout:

Math teachers and students can go to www.mathteachersresource.com/instructional-content.html to download the free Functions and Relations Introduction Handout. I use this handout with my developmental algebra, technical math, and college algebra students. This handout also contains graphs of mathematical functions and relations. I recently used this handout to tutor a developmental algebra student in the math lab at the junior college where I teach. She said, “When you explain it that way, it makes perfect sense.” By the way, all the 2D graphs in this handout were created with the program Basic Trig Functions.

Math Teacher’s Resource offers the program, Basic Trig Functions, which allows students and teachers to graph a wide variety of mathematical functions and relations. Go to www.mathteachersresource.com to view numerous screen shots of the program’s capabilities. Click the <get all the details> button in the TRIG FUNCTIONS AND GRAPHING section to get to the screen shots. At the bottom of each screen shot are comments that teachers will find helpful. The polar graphs demo has a cool animation of the trace cursor as it races around the edges of the leaves of a five petal rose.

## All About π : Mystery, History, and Epiphany

The recent date 3-14-15 generated a few news stories about National Pi Day, but outside of a math class, most people rarely think about this fascinating irrational number. Here’s how I teach students about π.

I begin by explaining the basic difference between rational and irrational numbers. Don’t panic, stay with me. It will be okay. All rational numbers can be expressed as the ratio of two integers. The following numbers are examples of rational numbers:  0 = 0/2, 1.5 = 3/2, 53 = 53/1, 2.6666 . . . = 8/3 and 4.3187187187 . . . = 43,144/9,990.

At some point in the decimal expansion of any rational number, the digits start to repeat in a pattern. If a number is irrational, there is no point in the decimal expansion where the digits start to repeat in a pattern. In grad school, an epiphany came when I learned that there are infinitely many more irrational numbers than rational numbers and the number of rational numbers is an infinitely small fraction of the irrational numbers. The first 51 digits of the irrational number π are:

π = 3.14159  26535  89793  23846  26433  83279  50288  41971   69399  37510  . . .

Starting with the 762nd digit and ending with the 767th digit, the decimal expansion of π = 999999. This sequence of digits is called the Feynman Point, named after the brilliant physicist Richard Feynman. In one of this lectures, Feynman said he would like to memorize π to the 762nd digit so he could recite the digits to that point and then quip, “nine nine nine nine nine nine and so on.”

When I ask people the definition of π, most of them, including some college grads, can’t give me the correct definition. The constant π is defined to be the ratio of the circumference of a circle to the diameter of the circle. As I understand it, Euclid’s Elements did not explicitly state that the ratio of the circumference of a circle to the diameter of a circle is always the same number for any circle. The intuitive reason that this ratio is the same for all circles is that all circles are similar figures, and corresponding parts of similar figures are proportional. (There are some interesting discussions on the web about why this ratio is the same for any two circles.)

In the Bible, I Kings 7:23 indirectly gives π a value of 3. “Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference.” Apparently some people have found this approximation of 3 for π to be alarming, because it might give people the idea that certain passages in the Bible are not to be taken literally. The creationist author Theodore Rybka did a creative piece of mathematics to show that the Bible actually implies π = 3.14 which agrees with the modern version of π to two decimal places.

William Shanks (1812–1882), a British amateur mathematician, over a 15 year period calculated π to 707 decimal places. Shanks also calculated the digits of other mathematical constants such as e and the natural logarithm of 2, 3, 5 and 10 to 137 decimal places, and he published a table of the prime numbers up to 60,000. In 1944, D. F. Ferguson, using a mechanical calculator, showed that Shanks’s calculation was correct only up to the first 527 places.

Modern computers continue to set records for calculating digits of π. On January 7, 2010, using only a personal computer, Fabrice Bellard, a French computer scientist, set a new world record by calculating almost 2.7 trillion digits of π. The last time I checked, over 10 trillion digits of π have been calculated.

What about memorizing the digits of π? It’s not as difficult as you might think. When my son was in fifth grade, a friend of his came to our house to play. One of the boys asked me a question about π. I just couldn’t help myself. Within one hour, both boys could recite the first 50 digits of π. They didn’t know any better; to them it was just great fun. The last year I taught high school math, I recited 70 digits of π to the students in one of my classes. (The first 70 were posted on the wall above the chalkboard, and a student had challenged me to memorize them. Of course, I could not resist the challenge.)

Recently, one of my wife’s friends asked me if there will another Pi day. I said that if she lives to 3-14-2115, she’ll see another one. Then I added that there should be a Pi Day next year, because π rounded to 4 decimal places = 3.1416. This put a smile on her face and gave her enough reason to schedule a Pi party for next year.

No doubt about it, π is a fantastic number. Some people feel that it has mystical properties. How π is used to describe relationships in mathematics and nature is unbounded. Go to the web and find out the world record for memorizing the digits of π—you will be blown away. You can also explore the vast properties, mysteries, and rich history of π. Who knows? You may start a new hobby.

For more of my blogs and person insights, visit my website at http://www.mathteachersresource.com.

*Vector version of w:Image:Pi.eq.C.over.d.png from the English Wikipedia, Public Domain