Chapter 3


3.1 Symbolic Variables

Algebra is a lot like arithmetic but deals with symbolic variables in addition to numbers. Very often these include image, and/or z, especially for “unknown” quantities which is often your job to solve for. Earlier letters of the alphabet such as image are often used for “constants,” quantities whose values are determined by assumed conditions before you solve a particular problem. Most English letters find use somewhere as either variables or constants. Remember that variables are case sensitive, so X designates a different quantity than x. As the number of mathematical symbols in a technical subject proliferates, the English (really Latin) alphabet becomes inadequate to name all the needed variables. So Greek letters have to be used in addition. Here are the 24 letters of the Greek alphabet:


If you ever pledged a fraternity or sorority, they probably made you memorize these Greek letters (but your advantage was probably nullified by too much partying). Several upper-case Greek letters are identical to English ones, and so provide nothing new. The most famous Greek symbol is image, which stands for the universal ratio between the circumference and diameter of a circle: image

The fundamental entity in algebra is an equation, consisting of two quantities connected by an equal sign. An equation can be thought of as a two pans of a scale, like the one held up by blindfolded Ms. Justice (Figure 3.1). When the weights of the two pans are equal, the scale is balanced. Weights can be added, subtracted, multiplied, or interchanged in such a way that the balance is maintained. Each such move has its analog as a legitimate algebraic operation which maintains the equality. The purpose of such operations is to get the equation into some desired form or to “solve” for one of its variables, which means to isolate it, usually on the left-hand side of the equation. Einstein commented on algebra, “It’s a merry science. When the animal we are hunting cannot be caught, we call it x temporarily and continue to hunt until it is bagged.”


Figure 3.1 The two sides of an equation must remain balanced just like the Scales of Justice. We will remove her blindfold to help her keep the scales balanced.

3.2 Legal and Illegal Algebraic Manipulations

Let’s start with a simple equation

image (3.1)

Following are the results of some legal things you can do

image (3.2)

image (3.3)

image (3.4)

image (3.5)

image (3.6)

image (3.7)

Here is a very tempting but ILLEGAL manipulation:

image (3.8)

A very useful reduction for ratios makes use of crossmultiplication:

image (3.9)

Note that you can validly go in either direction.

For the addition and multiplication of fractions, the two key relationships are:

image (3.10)

The distributive law for multiplication states that

image (3.11)

This implies

image (3.12)

In particular

image (3.13)

Another useful relationship comes from

image (3.14)

These last two formulas are worth having in your readily available memory.

Complicated algebraic expressions are best handled nowadays using symbolic math programs such as MathematicaTM.

Cancellation is a wonderful way to simplify formulas. Consider

image (3.15)

where the symbols in gray boxes are to be crossed out. But don’t spoil everything by trying to cancel the bs or cs as well. The analogous cancellation can be done on the two sides of an equation:

image (3.16)

For your amusement, here is a “proof” that image. The following sequence of algebraic operations is entirely legitimate, except for one little item of trickery snuck in. Suppose we are given that

image (3.17)


image (3.18)


image (3.19)

Factoring both sides of the equation,

image (3.20)

We can then simplify by cancellation of image to get

image (3.21)

But since image this means that 2 = 1! Where did we go wrong?

Once you recover from shock, note that image. And division by 0 is not legitimate. It is, in fact, true that image, but we can’t cancel out the zeros!

3.3 Factor-Label Method

A very useful technique for converting physical quantities to alternative sets of units is the factor-label method. The units themselves are regarded as algebraic quantities subject to the rules of arithmetic, particularly to cancellation. To illustrate, let us calculate the speed of light in miles/s, given the metric value image. First write this as an equation

image (3.22)

Now image, which we can express in the form of a simple equation

image (3.23)

Multiplying Eq. (3.22) by 1 in the form of the last expression, and cancelling the units m from numerator and denominator, we find

image (3.24)

We continue by multiplying the result by successive factors of 1, expressed in appropriate forms, namely

image (3.25)

Thus we can continue our multiplication and cancellation sequence beginning with (3.24):

image (3.26)

a number well known to readers of science fiction. Note that singular and plural forms, e.g. “foot” and “feet,” are regarded as equivalent for purposes of cancellation.

As another example, let us calculate the number of seconds in a year. Proceeding as before:

image (3.27)

To within about 0.5%, we can approximate

image (3.28)

3.4 Powers and Roots

You remember of course that image and image, so image. It is also easy to see that image. The general formulas are:

image (3.29)


image (3.30)

For the case image with image, the last result implies the frequently encountered identity

image (3.31)

From (3.29) with image

image (3.32)

(not image). More generally,

image (3.33)

Note that image, an instance of the general result

image (3.34)

You should be familiar with the limits as image

image (3.35)


image (3.36)

Remember that image while image. Dividing by zero sends some calculators and computers into a tizzy.

Consider a more complicated expression, say a ratio of polynomials


In the limit as image, image becomes negligible compared to image, as does any constant term. Therefore

image (3.37)

In the limit as image, on the other hand, all positive powers of x eventually become negligible compared to a constant. And so

image (3.38)

Using (3.32) with image we find

image (3.39)

Therefore image must mean the square root of x:

image (3.40)

More generally

image (3.41)

and evidently

image (3.42)

This also implies the equivalence

image (3.43)

Finally, consider the product image. The general rule is

image (3.44)

3.5 Logarithms

Inverse operationsare pairs of mathematical manipulations in which one operation undoes the action of the other—for example, addition and subtraction, multiplication and division. The inverse of a number usually means its reciprocal, i.e. image. The product of a number and its inverse (reciprocal) equals 1. Raising to a power and extraction of a root are evidently another pair of inverse operations. An alternative inverse operation for raising to a power is taking the logarithm. The following relations are equivalent:

image (3.45)

in which a is called the base of the logarithm.

All the formulas for manipulating logarithms can be obtained from corresponding relations involving raising to powers. If image then image. The last relation is equivalent to image, therefore

image (3.46)

where base a is understood. If image and image, then image and image. Therefore

image (3.47)

More generally,

image (3.48)

There is no simple reduction for image—don’t fall into the trap mentioned in the Preface! Since image,

image (3.49)

The identity image implies that, for any base,

image (3.50)

The log of a number less than 1 has a negative value. For any base image, image, so that

image (3.51)

To find the relationship between logarithms of different bases, suppose image, so image. Now, taking logs to the base a,

image (3.52)

In a more symmetrical form

image (3.53)

The slide rule, shown in Figure 3.2, is based on the principle that multiplication of two numbers is equivalent to adding their logarithms. Slide rules, once a distinguishing feature of science and engineering students, have been completely supplanted by hand-held calculators.


Figure 3.2 Principle of the slide rule. Top: Hypothetical slide rule for addition. To add 2 + 3, slide the 0 on the upper scale opposite 2 on the lower scale and look for 3 on the upper scale. The sum 5 appears below it. Bottom: Real slide rule based on logarithmic scale. To multiply image, slide the 1 on the upper scale opposite 2 on the lower scale and look for 3 on the upper scale. The product 6 appears below it. Note that image and that adding image gives image.

Logarithms to the base 10 are called Briggsian or common logarithms. Before the advent of scientific calculators, these were an invaluable aid to numerical computation. Section 1.3 on “Powers of 10” was actually a tour on the image scale. Logarithmic scales give more convenient numerical values in many scientific applications. For example, in chemistry, the hydrogen ion concentration (technically, the activity) of a solution is represented by

image (3.54)

A neutral solution has a pH of 7, corresponding to image. An acidic solution has image while a basic solution has image. Another well-known logarithmic measure is the Richter scale for earthquake magnitudes. image is a minor tremor of some standard intensity as measured by a seismometer. The magnitude increases by 1 for every 10-fold increase in intensity. image is considered a “major” earthquake, capable of causing extensive destruction and loss of life. The largest magnitude in recorded history was image, for the great 1960 earthquake in Chile.

Of more fundamental mathematical significance are logarithms to the base image, known as natural logarithms. We will explain the significance of e a little later. In most scientific usage the natural logarithm is written as “ln”

image (3.55)

But be forewarned that most literature in pure mathematics uses “image” to mean natural logarithm. Using (3.52) with image and image

image (3.56)

Logarithms to the base 2 can be associated with the binary number system. The value of image (also written lg 2) is equal to the number of bits contained in the magnitude x. For example, image.

3.6 The Quadratic Formula

The two roots of the quadratic equation

image (3.57)

are given by one of the most useful formulas in elementary algebra. We don’t generally spend much time deriving formulas, but in this one instance, the derivation is very instructive. Consider the following very simple polynomial which is very easily factored:

image (3.58)

Suppose we were given instead image. We can’t factor this as readily but here is an alternative trick. Knowing how the polynomial (3.58) factors we can write

image (3.59)

This makes use of a strategem called “completing the square.” In the more general case, we can write

image (3.60)

This suggests how to solve the quadratic equation (3.57). First complete the square involving the first two terms:

image (3.61)

so that

image (3.62)

Taking the square root:

image (3.63)

then leads to the famous quadratic formula

image (3.64)

The quantity

image (3.65)

is known as the discriminant of the quadratic equation. If image the equation has two distinct real roots. For example, image, with image, has the two roots image and image. If image, the equation has two equal real roots. For example, image, with image has the double root image. If the discriminant image, the quadratic formula contains the square root of a negative number. This leads us to imaginary and complex numbers. Before about 1800, most mathematicians would have told you that the quadratic equation with negative discriminant has no solutions. Associated with this point of view, the square root of a negative number has acquired the designation “imaginary.” The sum of a real number with an imaginary is called a complex number. If we boldly accept imaginary and complex numbers, we are led to the elegant result that every quadratic equation has exactly two roots, whatever the sign of its discriminant. More generally, every nth-degree polynomial equation

image (3.66)

has exactly n roots.

The simplest quadratic equation with imaginary roots is

image (3.67)

Applying the quadratic formula (3.64), we obtain the two roots

image (3.68)

As another example,

image (3.69)

has the roots

image (3.70)

Observe that whenever image, the roots occur as conjugate pairs, one root containing image and the other, image.

The three quadratic equations considered above can be solved graphically, as shown in Figure 3.3. The two points where the parabola representing the equation crosses the x-axis correspond to the real roots. For a double root, the curve is tangent to the x-axis. If there are no real roots, as in the case of image, the curve does not intersect the x-axis.

Problem 3.6.1

Solve the quadratic equation image.


Figure 3.3 Graphical solution of quadratic equations.

3.7 Imagining i

If we an nth-degree polynomial does indeed have a total of n roots, then we must accept roots containing square roots of negative numbers—imaginary and complex numbers. The designation “imaginary” is an unfortunate accident of history since we will show that image is, in fact, no more fictitious than 1 or 0—it’s just a different kind of number, with as much fundamental significance as those we respectfully call real numbers.

The square root of a negative number is a multiple of image. For example, image. The imaginary unit is defined by

image (3.71)


image (3.72)

Clearly, there is no place on the axis of real numbers running from image to image to accommodate imaginary numbers. We are therefore forced to move into a higher dimension, representing all possible polynomial roots on a two-dimensional plane. This is known as a complex plane or Argand diagram, shown in Figure 3.4. The abscissa (“x-axis”) is called the real axis while the ordinate (“y-axis”) is called the imaginary axis. A quantity having both a real and an imaginary part is called a complex number. Every complex number is thus represented by a point on the Argand diagram. Recognize the fact that your name can be considered as a single entity, not requiring you to always spell out its individual letters. Analogously, a complex number can be considered as a single entity, commonly denoted by z, where

image (3.73)

The real part of a complex number is denoted by image (or image) and the imaginary part by image (or image). The complex conjugateimage (written image in some books) is the number obtained by changing i to image:

image (3.74)

As we have seen, if z is a root of a polynomial equation, then image is also a root. Recall that for real numbers, absolute value refers to the magnitude of a number, independent of its sign. Thus image. We can also write image. The absolute value of a complex number z, also called its magnitude or modulus, is likewise written image. It is defined by

image (3.75)


image (3.76)

which by the Pythagorean theorem is just the distance on the Argand diagram from the origin to the point representing the complex number.


Figure 3.4 Complex plane, spanned by real and imaginary axes. The point representing image is shown along with the complex conjugate image. Also shown is the modulus image.

The significance of imaginary and complex numbers can also be understood from a geometric perspective. Consider a number on the positive real axis, say image. This can be transformed by a image rotation into the corresponding quantity image on the negative half of the real axis. The rotation is accomplished by multiplying x by −1. More generally, any complex number z can be rotated by image on the complex plane by multiplying it by image, designating the transformation image. Consider now a rotation by just image, say counterclockwise. Let us denote this counterclockwise image rotation by image (repress for the moment what i stands for). A second counterclockwise image rotation then produces the same result as a single image rotation. We can write this algebraically as

image (3.77)

which agrees with the previous definition of i in Eqs. (3.71) and (3.72). A image rotation following a counterclockwise image rotation results in the net transformation image. Since this is equivalent to a single clockwiseimage rotation, we can interpret multiplication by image as this operation. Note that a second clockwise image rotation again produces the same result as a single image rotation. Thus image as well. Evidently image. It is conventional, however, to define i as the positive square root of image. Counterclockwise image rotation of the complex quantity image is expressed algebraically by

image (3.78)

and can be represented graphically as shown in Figure 3.5.


Figure 3.5 Geometric representation of multiplication by image. The point image is obtained by 90° counterclockwise rotation of image in the complex plane.

Very often we need to transfer a factor i from a denominator to a numerator. The key result is

image (3.79)

Several algebraic manipulations with complex numbers are summarized in the following equations:

image (3.80)

image (3.81)

image (3.82)

Note the strategy for expressing a fraction as a sum or real and imaginary parts: multiply by the complex conjugate of the denominator then recognize the square of an absolute value in the form of (3.75).

Problem 3.7.1

Find the real and imaginary parts of image, where image, x and y real.

Problem 3.7.2

If you don’t mind doing the algebra, analogously find the real and imaginary parts of image.

3.8 Factorials, Permutations and Combinations

Imagine that we have a dozen differently colored eggs which we need to arrange in an egg carton. The first egg can go into any one of 12 cups. The second egg can then go into any of the remaining 11 cups. So far, there are image possible arrangements for these two eggs in the carton. The third egg can go into 10 remaining cups. Continuing the placement of eggs, we will wind up with one of a possible total of image distinguishable arrangements. (This multiplies out to 479,001,600.) The product of a positive integer n with all the preceding integers down to 1 is called n factorial, designated image:

image (3.83)

The first few factorials are image. As you can see, the factorial function increases rather steeply. Our original example involved 12! = 479,001,600. The symbol for factorial is the same as an explanation point. (Thus be careful when you write something like, “To our amazement, the membership grew by 100!”)

Can factorials also be defined for nonintegers? Later we will introduce the gamma function, which is a generalization of the factorial. Until then you can savor the amazing result that

image (3.84)

Our first example established a fundamental result in combinational algebra: the number of ways of arranging n distinguishable objects (say, with different colors) in n different boxes equals image. Stated another way, the number of possible permutations of n distinguishable objects equals image.

Suppose we had started the preceding exercise with just a half-dozen eggs. The number of distinguishable arrangements in our egg carton would then be “only” image. This is equivalent to image. The general result for the number of ways of permuting r distinguishable objects in n different boxes (with image) is given by

image (3.85)

(You might also encounter the alternative notation image, or image.) This formula also subsumes our earlier result, which can be written image. To be consistent with (3.85), we must interpret image.

Consider now an alternative scenario in which the eggs are not colored, so that they remain indistinguishable. The different results of our manipulations are now known as combinations. Carrying out an analogous procedure, our first egg can again go into 12 possible cups and the second into one of the remaining 11. But a big difference now is we can no longer tell which is the first egg and which is the second—remember they are indistinguishable. So the number of possibilities is reduced to image. After placing 3 eggs in image available cups, the identical appearance of the eggs reduces the number of distinguishable arrangements by a factor of image. We should now be able to generalize for the number of combinations with m indistinguishable objects in n boxes. The result is

image (3.86)

Another way of deducing this result. The total number of permutations of n objects is, as we have seen, equal to image. Now, permutations can be of two types: indistinguishable and distinguishable. The eggs have image indistinguishable permutations among themselves, while the empty cups have image indistinguishable ways of hypothetically numbering them. Every other rearrangement is distinguishable. If image represents the total number of distinguishable configurations then

image (3.87)

which is equivalent to (3.86).

A simple generalization to distinguish permutations from combinations is that permutations are for lists, in which the order matters, while combinations are for groups in which the order doesn’t matter.

Suppose you have n good friends seated around a dinner table who wish to toast one another by clinking wineglasses. How many “clinks” will you hear? The answer is the number of possible combinations of objects taken 2 at a time from a total of n, given by

image (3.88)

You can also deduce this more directly by the following argument. Each of n diners clinks wineglasses with his or her image companions. You might first think there must be image clinks. But, if you listen carefully, you will realize that this counts each clink twice, one for each clinkee. Thus dividing by 2 gives the correct result image.

Problem 3.8.1

A basketball team has 12 players. How many different ways can the coach choose the starting 5?

3.9 The Binomial Theorem

Let us begin with an exercise in experimental algebra:

image (3.89)

The array of numerical coefficients in (3.89)

image (3.90)

is called Pascal’s triangle. Note that every entry can be obtained by taking the sum of the two numbers diagonally above it, e.g. 15 = 5 + 10. These numbers are called binomial coefficients. You can convince yourself that they are given by the same combinatorial formula as image in Eq. (3.86). The binomial coefficients are usually written image. Thus

image (3.91)

where each value of n, beginning with 0, determines a row in the Pascal triangle.

Setting image, the binomial formula can be expressed

image (3.92)

This was first derived by Isaac Newton in 1666. Remarkably, the binomial formula is also valid for negative, fractional, and even complex values of n, which was proved by Niels Henrik Abel in 1826. (It is joked that Newton didn’t prove the binomial theorem for noninteger n because he wasn’t Abel.) Here are a few interesting binomial expansions which you can work out for yourself:

image (3.93)

image (3.94)

image (3.95)

Each of the above series is convergent only for image.

Problem 3.9.1

Determine the first few terms in the expansion of image.

3.10 e is for Euler

Imagine there is a bank in your town that offers you 100% annual interest on your deposit (we will let pass the possibility that the bank might be engaged in questionable loan-sharking activities). This means that if you deposit $1 on January 1, you will get back $2 one year later. Another bank across town wants to get in on the action and offers 100% annual interest compounded semiannually. This means that you get 50% interest credited after half a year, so that your account is worth $1.50 on July 1. But this total amount then grows by another 50% in the second half of the year. This gets you, after 1 year,

image (3.96)

A third bank picks up on the idea and offers to compound your money quarterly. Your $1 there would grow after a year to

image (3.97)

Competition continues to drive banks to offer better and better compounding options, until the Eulergenossenschaftsbank apparently blows away all the competition by offering to compound your interest continuously—every second of every day! Let’s calculate what your dollar would be worth there after 1 year. Generalization from Eq. (3.97) suggests that compounding n times a year produces image. Here are some numerical values for increasing n:

image (3.98)

The ultimate result is

image (3.99)

This number was designated e by the great Swiss mathematician Leonhard Euler (possibly after himself). Euler (pronounced approximately like “oiler”) also first introduced the symbols i, image, and image. After image itself, e is probably the most famous transcendental number, also with a never-ending decimal expansion. The tantalizing repetition of “1828” is just coincidental.

The binomial expansion applied to the expression for e gives

image (3.100)

As image, the factors image all become insignificantly different from n. This suggests the infinite-series representation for e

image (3.101)

Remember that image and image. This summation converges much more rapidly than the procedure of Eq. (3.99). After just six terms, we obtain the approximate value image.

Let’s return to consideration of interest-bearing savings accounts, this time in more reputable banks. Suppose a bank offers X% annual interest. If image, your money would grows by a factor of image every year, without compounding. If we were able to get interest compounding n times a year, the net annual return would increase by a factor

image (3.102)

Note that, after defining image,

image (3.103)

Therefore, in the limit image, Eq. (3.102) implies the series

image (3.104)

This defines the exponential function, which plays a major role in applied mathematics. The very steep growth of the factorials guarantees that the expansion will converge to a finite quantity for any finite value of x, real, imaginary, or complex. The inverse of the exponential function is the natural logarithm, defined in Eq. (3.55):

image (3.105)

Two handy relations are

image (3.106)

When the exponent of the exponential is a complicated function, it is easier to write

image (3.107)

Exponential growth and exponential decay, sketched in Figure 3.6, are observed in a multitude of natural processes. For example, the population of a colony of bacteria, given unlimited nutrition, will grow exponentially in time:

image (3.108)

where image is the population at time image and k is a measure of the rate of growth. Conversely, a sample of a radioactive element will decay exponentially:

image (3.109)

A measure of the rate of radioactive decay is the half-lifeimage, the time it takes for half of its atoms to disintegrate. The half-life can be related to the decay constant k by noting that after time image, image is reduced to image. Therefore

image (3.110)

and, after taking natural logarithms,

image (3.111)


Figure 3.6 Exponential growth and decay.

No doubt, many of you will become fabulously rich in the future, due, in no small part, to the mathematical knowledge we are helping you acquire. You will probably want to keep a small fraction of your fortune in some CDs at your local bank. In better economic times there was a well-known “Rule of 72” which stated that to find the number of years required to double your principal at a given interest rate, just divide 72 by the interest rate. For example, at 8% interest, it would take about image. To derive this rule, assume that the principal image will increase at an interest rate of image to image in Y years, compounded annually. Thus

image (3.112)

Taking natural logarithms we can solve for

image (3.113)

To get this into a neat approximate form image, let

image (3.114)

You can then show for interest rates in the neighborhood of 8% (image), the approximation will then work with the constant approximated by 72.

Problem 3.10.1

Unfortunately, bank interest rates have lately fallen to the neighborhood of 1%. How many years would it take for you to double your money at this puny rate?

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