Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks)

by Banks, Robert B.

  A good many of the documents in the work deal with strictly historical studies. Thus, considerable attention is given to studies carried out long ago in Egypt, Greece, India, China, and medieval Islam. And finally, a number of presentations are somewhat whimsical or even amusing selections. These include a 402-word mnemonic for π, constructed in the format of a circle, and a display of the numerous documents presented to the Indiana legislature in 1897 to decree the legal value of π.

  The Second Most Famous Number: e

  Although nearly everyone knows about the number π, its nearest rival in fame and importance, the number e, is virtually unknown outside of mathematics, science, and engineering. The main reason for this is that e, which has the approximate value e = 2.71828, is not really encountered or utilized—except in natural logarithms—until we get involved in calculus and other areas of more advanced mathematics. In these subjects, e is an extremely important numerical constant.

  The Swiss mathematician Leonhard Euler (1707-1783) was one of the most prolific in all of history. Among a great many other major contributions, he was the one who assigned the symbol e to this famous number and proved the relationship

  This expression says that as the whole number n increases in magnitude, the value of the quantity in parentheses—call it S if you like—approaches the number e = 2.81828. For example, if n = 10, then S = 2.59374; if n = 100, S = 2.70481; if n = 1,000, S = 2.71692; and so on to n = ∞.

  Another great mathematician, England's Isaac Newton (1643-1727), showed that

  This expression is another example of an infinite series. From either equation (7.5) or equation (7.6), we can easily compute the value of e. As is true of π, the quantity e is an irrational and also transcendental number.

  The topic of logarithms was mentioned above. You probably remember logarithms from your course in elementary algebra though you may not recall that you probably dealt only with so-called “common” logarithms. This is the system in which the number 10 is used as the base of the logarithm—probably because humans have always had 10 fingers.

  Now any number can serve as the base of an arithmetic system including operations involving logarithms. In information theory and computer science, the number 2 (“binary”) is generally used although sometimes the numbers 8 (“octal”) and 16 (“hexadecimal”) are employed. In contrast to the use of the number 10 as the base for “common” logarithms, the number e is used as the base for so-called “natural” logarithms. Fine. But why is such a strange number used for this purpose?

  A complete answer to this question involves the limiting value property of e expressed by equation (7.5). To be brief, it is simply pointed out that if we use the numerical value e = 2.71828…, then the calculus operations called differentiation and integration are greatly simplified.

  For example, suppose we have the simple equation y = cx, where c is a positive constant. If we set c = e, then clearly y = ex. This is called the exponential function. Here comes the calculus. The derivative of this equation is dy/dx = ex. In other words, the derivative of the exponential function is equal to the function itself. By the same token, the integral of ex takes on the same form, that is, ∫ydx = ex. Thus, by using this particular numerical value for e, the function ex, its derivative, and its integral are all equal. For calculus operations, this represents an enormous simplification.

  Furthermore, with y = ex, then taking logarithms, we have x = logey, where the subscript e means that the base of the logarithm is e. This is a “natural” logarithm. Incidentally, logey is sometimes written In y to avoid confusion with log10y, the so-called common logarithm. Your calculator probably has keys for both types of logarithms.

  An Example: Earning Interest on Your Savings Account

  We take a quick look at a topic in which all of us are interested: how much money can you earn on your savings account? To answer the question, we utilize equation (7.5) but change it slightly to the form

  in which P0 = $1,000 is the amount of money you have in your savings account at the beginning of the year, that is, the original principal; r = 6% = 0.06 is the annual interest rate paid by your bank; n is the number of “compounding periods” during the year; and P is the amount of money in your account at the end of the year, i.e., 12 months later. The difference between P and Po is the amount of interest you earned during the year.

  Now if the bank compounds your interest earnings only once a year, then n = 1. Accordingly, from equation (7.7), at the end of the year you will have P = $1,000(1 + 0.06/1)1 = $1,060. This is called simple interest. Alternatively, suppose the bank compounds semiannually. In this case, n = 2 and equation (7.7) becomes P = $1,000(1 + 0.06/2)2 = $1,060.90. Next, assume that the bank compounds quarterly. Therefore, n = 4; substituting into equation (7.7) gives P = $1,000(1 + 0.06/4)4 = $1,061.36, and so on. The results of our calculations are listed in table 7.2.

  TABLE 7.2

  Observation number 1: Suppose that your bank advertises and applies “daily compounding” of interest on your savings account. Then, after 12 months, according to the table, your principal is $1,061.83 and you have earned P – Po = $61.83. Dividing this by the original principal and converting to a percentage gives r* = 6.183%. This is called the yield.

  Observation number 2: If there is instantaneous compounding, equation (7.7) becomes P = Poer, which gives P = $1,061.84. Clearly, as far as your interest earning is concerned, it makes essentially no difference whether your bank compounds your savings account daily, hourly, or instantaneously.

  Another Example: Geometrical Interpretation of the Number e

  Everyone knows that π has a very simple geometrical interpretation. It gives the circumference of a circle in terms of its diameter, C = πD. Alternatively, it expresses the area of a circle in terms of its radius, A = πR2. SO a logical question to ask is, what is the simplest geometrical interpretation you can devise involving that other very important number, e?

  FIG. 7.1

  A plot of the exponential function showing the geometrical interpretations of the ordinate, slope, and area under the curve.

  To get things started, here is one idea. As we know, the equation of the exponential function is y = ex; this expression is displayed graphically in figure 7.1. Now since y = ex, when x = 1, y = e. So, as shown in figure 7.1, e is the value of the ordinate, y, when the abscissa, x = 1. Further, as pointed out earlier, if y = ex, then the first derivative is dy/dx = ex. It turns out that the first derivative of a function represents the slope of the function. Accordingly, at x = 1, dy/dx = e is the slope of the curve, as shown in the figure. Well, here are two (not terribly exciting) geometrical interpretations of e: the ordinate and the slope of the exponential function at x = 1.

  Going further, if y = ex then an integration gives the result ∫ydx = ex if the lower limit of the integral is x = – ∞. The geometrical interpretation of this result is that the total area under the curve between x = – ∞ and x = x is A = ex. If we again select x = 1, then A = e is the area under the curve between x = –∞ and x = 1. This is another geometrical interpretation of e.

  You can do much better than this and here is your chance to prove it.

  ANNOUNCEMENT OF A GREAT CONTEST INVOLVING THE NUMBER e

  Part 1. Devise some kind of relationship that gives a simple geometrical interpretation of e.

  Part 2. Manufacture a mnemonic device for e along the lines given earlier for π. For your information, e = 2.71828 18284 59045.

  An interesting reference, somewhat analogous to the one by Beckmann (1971) concerning π, is the book by Maor (1994), which deals with our other famous number, e.

  Three Other Famous Numbers

  We have taken quick looks at the two most famous numbers in mathematics: π and e. There are numerous others; if you are interested in delving further into the matter, a good place to start is the little book by Wells (1986). For the present, we mention three other important numbers.

  Golden Ratio, ø = 1.61803

  This number defines t
he ratio of the length L and width H of a rectangle that allegedly gives the most esthetically attractive appearance. Its precise value is L/H = . We examine this number in detail in chapter 9, “Great Number Sequences: Prime, Fibonacci, and Hailstone.”

  Euler's Constant, γ = 0.57721

  This important number, devised by Leonhard Euler around 1750, is defined by the equation γ = (1 + 1/2 + 1/3 +…+ 1/n – loge n) as n becomes infinite. It makes its appearance in many problems in mathematics and statistics.

  Feigenbaum Number, δ = 4.66920

  This is the newest of the important numbers. It was discovered in 1978 by the American mathematician Mitchell Feigenbaum, in his early studies of chaos theory. This number, δ, is the ratio of the spacings of successive intervals of period doubling in a process leading to chaos. An easily understood description of the nature of δ is presented by Addison (1997).

  Some References for More Information

  In this brief look at some of the famous numbers of mathematics, we have run across the names of Archimedes, Euler, Gauss, and Newton. These are probably the four greatest mathematicians of all time. If you would like to learn more about them, you will find the book by Muir (1996) to be helpful. It presents interesting biographies of these four notable mathematicians and quite a few others. In addition, the comprehensive work of Boyer (1991) presents a great deal of information about the matter.

  Annual Celebration Days for π and Perhaps e

  In the preceding sections, we have examined two famous numbers, π and e, and looked briefly at three others, ø, γ, and δ. Noteworthy is the fact that of these five important numbers, only π can be expressed as a date if we require that three significant figures be utilized. That is, π = 3.14 which, of course, is March 14.

  It is fitting, therefore, that this most important number of all should be celebrated each year. Accordingly, shall we declare March 14 to be π-Day? If this meets with success, we could, later on, celebrate e-Day on February 7 and perhaps ø-Day on January 6. Suggested first steps: Contact the White House and the greeting card manufacturers.

  8

  Another Look at Some Famous Numbers

  In the preceding chapter we examined several of the most important numerical constants appearing in mathematics. The best known of these “famous numbers” are, of course, π (the ratio of the circumference and diameter of a circle) and e (the base of natural logarithms). We also looked briefly at three other important numbers: ø (the golden ratio), γ (Euler's constant), and δ (the Feigenbaum number).

  In this chapter, the subject of famous numbers is continued but aimed in a somewhat different direction. We begin with what are called real numbers, imaginary numbers, and complex numbers. First, some rules are given for procedures such as the addition of complex numbers—so-called vector addition. Then a very interesting equation, originally given by Leonhard Euler, is introduced to generate some remarkable relationships involving real and imaginary numbers, including the one presented in the opening paragraph of the previous chapter. Using the methods of complex number addition, and calling on some things we know about π and e, we construct a polygonal spiral which we then use wickedly to strangle a negative number.

  Finally, suppose you had to make a quick decision regarding a gift from your wealthy uncle. Which would you rather receive from him: 43 dollars or 34 dollars? Expressing this question in general terms and assuming that a is larger than b, which is the greater quantity: ab or ba?

  Numbers: Real, Imaginary, and Complex

  We start with a very simple algebraic equation: x2 – 1 = 0. What is its solution? Easy: We move the 1 to the other side of the equation to get x2 = + 1 and then take the square root of both sides. This yields . However, don't forget that it also yields x = – 1, because (– 1) × (– 1) = + 1. So the solution to this little equation is .

  Fair enough. Now suppose we start with another very simple equation: x2 + 1 = 0. Moving the 1 to the other side gives the expression x2 = –1, and taking the square root yields x = . What in the world does mean? Well, this is called an “imaginary” number; it may or may not be a good name but we're stuck with it. Very simply, an imaginary number is the square root of a negative number. For hundreds of years, mathematicians did not know how to handle imaginary numbers, though nowadays they present no particular difficulties. Indeed, in many areas of mathematics, imaginary numbers are extremely useful quantities. The letter i has been adopted as the symbol for imaginary numbers. That is, . Furthermore, an entire methodology has been developed to handle arithmetical and algebraic manipulations involving imaginary numbers.

  For example, consider the following equation: z1 = + 5 – 2i. In this expression, the + 5 is called a real number. On the other hand, the –2i is termed an imaginary number because it contains the quantity i. The indicated sum, z1 is called a complex number.

  Now consider another equation: z2 = – 3 + 4i. Suppose we want to add z1 and z2. To do this, we simply add the real numbers of the two equations and then add the imaginary numbers of the two equations. In our example, this operation yields the equation z1 + z2 = + 2 + 2i. All this is shown graphically in figure 8.1, in which the real numbers are plotted along the x-axis and the imaginary numbers are plotted along the y-axis. The z = x + iy plane is called the complex plane. We sometimes refer to z1 and z2 as vectors. We have just carried out a simple example of vector addition. We can also perform vector subtractions, multiplications, and divisions. The display shown in figure 8.1 is sometimes called an Argand diagram.

  FIG. 8.1

  An example of vector addition. z1 = 5 – 2i, z2 = – 3 + 4i, so z1 + z2 = 2 + 2i. This plot is an Argand diagram.

  For use in an example we shall look at in a moment, we summarize the immediately preceding analysis. In a rectangular (x, y) coordinate system we plot real numbers in the plus or minus x-direction depending on whether they are positive or negative quantities. In the same way, we plot imaginary numbers in the plus or minus y-direction depending on whether they are positive or negative quantities. What could be simpler? Incidentally, a suggested reference for an elementary presentation of various topics concerning imaginary and complex numbers is Gardner (1992).

  Some Amazing Mathematical Relationships

  In 1746, that incredibly prolific mathematical genius named Leonhard Euler presented the following identity:

  in which cos θ is the real part and sin θ is the imaginary part of the complex equation. As we shall now see, this equation yields some really amazing relationships.

  Amazing Relationship 1

  Let θ = π (i.e., 180°) in equation (8.1). Then since cos π = – 1 and sin π = 0, equation (8.1) gives

  This is an extremely remarkable result. Here is an equation that uniquely relates the five most important numbers in all of mathematics: e, i, π, 1, and 0.

  Amazing Relationship 2

  Let θ = π/2 (i.e., 90°) in equation (8.1). Then since cos(π/2) = 0 and sin(π/2) = 1, we get eiπ/2 = i. Multiplying the exponents of both sides of this equation by i yields e–π/2 = ii. Since e–π/2 = 0.2079, we obtain the very remarkable result that ii = 0.2079. This relationship says that i raised to the ith power is equal to a real number. Totally crazy! How can this possibly be? This is much too weird to even think about.

  How to Strangle a Negative Number

  We are now going to strangle the number – 1, and here is how we are going to do it. First, the following is a generalization of the infinite series given by equation (7.6):

  Next, from equation (8.2) we get the relationship eiπ = – 1. So if we substitute θ = iπ into equation (8.3) we obtain

  FIG. 8.2

  The polygonal spiral created by vector addition of the infinite-series form of eiπ. Note that the spiral converges on the point (– 1, 0).

  Now remember that . Therefore i2 = – 1, i3 = – i, i4 = + 1, i5 = + i, i6 = – 1, and so on. Substituting these values into equation (8.4) yields

  We now prepare a graphical display of this equation, following the rul
es we presented and utilized in constructing figure 8.1, in which we simply added z1 and z2. The results of this exercise in vector addition are shown in figure 8.2. The angles at the corners of the resulting polygon are all 90°. The lengths of the sides are 1, π, π2/2, π3/6, π4/24, and so on. The coordinates of the first two corners are indicated in the figure; the others are easily calculated.

  The graphical plot of figure 8.2 should be clear. As a result of the repeated vector addition, our “polygonal spiral” gets closer and closer to the point (– 1, 0) and eventually “strangles” the point. The total length of the spiral, without regard to the direction of the vectors, is simply L = eπ = 23.1407.