APPENDIX II:

MISTER ANTHONY'S LECTURE

ON

EUCLIDEAN ALGEBRA

In mathematics when we take the next step beyond basic arithmetic we come into the realm of algebra. If we hadn't chosen to use the Arabic name for the topic, we might have more accurately called it abstract arithmetic. It does, indeed, give us arithmetic without numbers, but in taking out the numbers mathematicians have created an abstract arithmetic that enables us to do things that we could never have hoped to do with our everyday grammar-school arithmetic.

To understand
algebra you need to understand only two basic concepts - those of formulae and
of equations, in both of which we represent the absent numbers with letters of
the alphabet. We may rightly call this basic algebra Euclidean, because the
fundamental operations by which we solve problems conform to the five common
notions that Euclid included in his "Elements", the basic text on plane
geometry.

Formulae

In an algebraic formula we simply have a recipe for applying the operations of arithmetic to unknown numbers. To see what that means consider the following example:

Take a velocity and divide it by the speed of light; square the resulting quotient and subtract the square from one; extract the square root of the remainder; multiply the result by the length of an object.

There you have the recipe for the Lorentz-Fitzgerald contraction, the phenomenon of Relativity in which an object appears shorter when moves than it does when it does not move. That recipe gives valid results for all possible velocities between zero and the speed of light and for all possible lengths of any objects you care to imagine. Because nothing obliges us to put actual specific numbers into them right away, we can see such recipes as generalizations that reveal something of the essence of the phenomenon they describe. That's why physicists express the laws of physics as algebraic recipes.

But we usually
don't write algebraic formulae out in words , as above. We represent them,
instead, by strange-looking, highly simplistic pictures. We draw the formula
described above, for example, in arithmetic cartoon form as

In order to carry out the calculation that we have represented by that formula, we need only apply four simple rules:

1) Replace each letter by the number that it represents. Think of each letter as labeling a blank space that you have reserved in the recipe for a specific number: it simply serves us as a reminder of which number goes into which blank. In our example vee labels the blank into which we must insert the velocity between two inertial frames (when we know what it will be) and the upper-case ell represents the length that the object under consideration has when it is at rest.

But if letters represent numbers that we don't know yet, then why do we use lower-case cee to represent the speed of light? We already know the number that tells us how fast light moves and we know that the speed of light never changes in a vacuum, so why not write it into the formula directly? Our motive is the desire for convenience (also known as sheer laziness). The speed of light appears in many of the formulae of physics and physicists often write those formulae over and over again in their work. Those circumstances make it much more convenient to write the speed of light as cee rather than as "299,792.458 kilometers per second" (or equivalently '186,234.709 miles per second").

2) When we see two letters written together, we multiply the numbers that they represent, one by the other. Because we have the ell written right up against the square root sign in our example, we must multiply the at-rest length of our chosen object by whatever number we obtain when we extract the square root. We have, as you might suspect, many exceptions to this rule, but mostly they don't appear until you get into trigonometry and the calculus, disciplines that come from taking the next steps beyond simple algebra. As one common example I offer logA, which means find the logarithm of the number represented by A, not multiply together four numbers represented by ell, oh, gee, and upper-case ay. (If we did have such a multiplication of numbers represented by those four letters, we would write them in a different order in the formula in order to avoid making the product look like some other function.)

3) When we write two letters one over the other, in the manner of a fraction, we mean to divide the number that the upper letter represents by the number that the lower letter represents. Our example instructs us, as part of the calculation, to divide the velocity between our two inertial frames by the speed of light.

4) When we enclose part of a formula in parentheses (or in a root sign), we must carry out the operations indicated within the parentheses before carrying out the other operations indicated by the formula. Our example instructs us to divide the velocity between our inertial frames by the speed of light before performing the squaring operation and then to subtract the resulting square from one before extracting the square root.

We use parentheses in algebraic formulae to avoid confusion where a formula might otherwise be ambiguous. Consider the statement "A plus B times C": does that statement instruct us 1) to add A to B and then to multiply the sum by C or 2) to add A to the product of multiplying B by C? With parentheses we can eliminate the ambiguity: we write Option 1 as (A + B)C and Option 2 as A + (BC). Usually we would leave the parentheses off of Option 2, but we definitely need them for Option 1.

Those rules for
organizing the operations of arithmetic give you all the information that you
need to figure out how to calculate a number that an algebraic formula
represents.

Equations

An algebraic equation simply asserts an equality between two algebraic formulae. For the purpose of making that definition apply to all possible equations, we claim that a single number can be an algebraic formula. We can see the basic concept of the equation displayed clearly in the most famous equation of all,

E = mc^{2},

(Eq'n 1)

which equation translates into plain English as "the total amount of energy contained in a body equals the body's mass multiplied by the square of the speed of light". Please notice also the parenthetic label under the equation identifying it as Equation 1; that label allows us to refer to the equation in the text without actually bothering to redraw it.

Once we have obtained an equation, by whatever means, we want, more often than not, to solve it. That means that if we have an equation whose formulae include the letter eks (x) and we want to "solve the equation for x", then we must manipulate the equation's formulae in such a way that we end up with a new equation that has x all by itself on one side of the equality sign and all of the other letters and numbers, except x, in a formula on the other side of the equality sign. Solving an equation, then, gives us a game analogous to a word ladder, in which we must transform one word into another word by changing one letter at a time.

As we do in any other game, we solve an equation according to a set of rules and a playbook. It's the study of the basic playbook that takes up most of the high school course in algebra. The rule of the game (yes, we have only one rule) is simple: in order to maintain the equality that the equation expresses, whatever operation we perform on the formula on one side of the equality sign we must perform the same operation on the formula on the other side of the equality sign. Thus, if we add something to the formula on the left side of the equality sign, we must add the same thing to the formula on the right side of the equality sign; if we divide the formula on the left side by something, we must divide the formula on the right side by the same thing; and so on.

Here we can see a reflection of Euclid's common notions. In his treatise on plane geometry, the famous "Elements", Euclid listed the five common notions as:

1. Things equal to the same thing are also equal to one another.

2. If equals are added to equals the wholes are equal.

3. If equals are subtracted from equals the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

Because we base our algebra on those common notions, I have referred to the subject as Euclidean algebra.

As an example let me show you how to solve the infamous quadratic equation, the equation that stymies so many students in first-year high school algebra classes. I'll start by pointing out that we have in solving the general quadratic equation no trivial exercise in esoteric mathematics. The quadratic equation actually shows up in physics; specifically, it shows up in descriptions of oscillating systems, whether those systems display mechanical or electrical oscillations.

For our example we will contemplate an electrical pathway that includes an inductor (whose inductance we represent by I), a resistor (whose resistance we represent by R), and a capacitor (whose capacitance we represent by C). If an electric current, however temporary, flows along that pathway, then each of those objects contributes a voltage difference to the total voltage difference between the ends of the pathway. If we have no voltage difference between the ends of the pathway (that is, if we have removed the battery that set the current in motion from the circuit), then those contributions must add up to zero. In such an open circuit the current can only slosh back and forth within the pathway. When we describe the contributions to the voltage difference in terms of the frequency of that putative sloshing, we obtain the equation

(Eq'n 2)

in which x represents the frequency. Because we want to calculate the value of that frequency, we must solve Equation 2 for x.

In the first step toward the solution I subtract C from both sides of Equation 2. That gives us

(Eq'n 3)

In the next step I divide Equation 3 by I and obtain

(Eq'n 4)

Now we seem to be stuck. We have no obvious way to manipulate Equation 4 in a way that will leave the first power of x all by itself on the left side of the equality sign and put a formula involving I, R, and C, but not x, on the right side of the equality sign. Now we have a situation in which a knowledge of the playbook leads us readily to the solution, though, given enough time, we could actually figure out a way to proceed to the solution (after all, someone had to do it a first time in order to put the ploy into the mathematicians' playbook in the first place).

If the formula on
the left side of Equation 4 gave us a perfect square, we could merely extract
the square root of both sides of the equation and then proceed to the solution
in one obvious and easy step. Now, what does it mean for a formula to be a
perfect square? It means that we can write the square root of the formula
without introducing a square root sign. Thus, for example, x^{2} is a
perfect square because its square root is x, but x^{3} is not a perfect
square because its square root is
. The formula on
the left side of Equation 4 involves two numbers, x and R/I, so we can gain some
knowledge of what a perfect square involving two numbers looks like by
multiplying the sum of two numbers by itself. We get

(Eq'n 5)

If we compare that pattern on the right side of Equation 5 with the formula on the left side of Equation 4 and match up the pieces (properly called terms) that carry the same power of x, then we infer that we must make 2y = R/I. Now we know how to make the formula on the left side of Equation 4 into a perfect square: we divide that little equation in the previous sentence by 2, square the result, and add that square to both sides of Equation 4. Now we have

(Eq'n 6)

Before I extract the square root of that equation I want to clean up the formula on the right side by giving both terms a common denominator. If we multiply both the numerator and the denominator of a fraction by the same number, we don't change the value of the fraction; thus, we can multiply the numerator (C) and the denominator (I) of the second term on the right side of Equation 6 by 4I and obtain

(Eq'n 7)

We then have square root of that equation as

(Eq'n 8)

Before I finish up
the solution, I want to carry out one more simplification of the right side of
the equation. Conveniently enough, we can carry out the operations of division
and extracting a square root in either of the two possible orders; that is, we
could calculate the fraction indicated in the formula and then extract the
square root of the result or we could extract the square roots of the numerator
and of the denominator of the fraction and then divide the square root of the
numerator by the square root of the denominator. I want to represent that latter
option in my equation, because the square root of 4I^{2} is simply 2I.
Thus we can rewrite Equation 8 as

(Eq'n 9)

In making that change I have given the formula on the right side of the equation a common denominator with the second term on the left side of the equation.

Finally, I subtract R/2I from both sides of Equation 9 and thereby obtain the equation expressing the classical quadratic formula;

(Eq'n 10)

In that equation the plus sign drawn above a minus sign means "plus or minus" (or "give or take") and it acknowledges that extraction of the square root can give us either a positive number or a negative number. So now I can calculate the frequency at which the electric current in my simple electronic pathway will vibrate. If we connect that pathway to an antenna, I now know also the frequency of the radio waves that the circuit will radiate or receive.

Just that kind of connection between the equations and the things that they represent mitigates the aridity of abstract mathematics. The things that the equations represent may themselves be fairly abstract, but, nonetheless, having some anticipation of a use for the equations makes them more welcome as guests in our minds. Pure mathematics, with no reference to a possible use, might be likened to Zen, that Buddhist discipline that enables the disciple to empty their mind of all distractions and allow enlightenment to enter: think of pure mathematics as a kind of Rationalist Zen. It's definitely not for everyone. I always found that I learned more mathematics from my physics textbooks, precisely because it was presented there in a context of immediate application. But as I have grown older I have developed a deeper appreciation of the more abstract mathematics, developing through practice, I suppose, a greater tolerance for the drier, more aetherial stuff. But, then again, I may have merely built my tolerance of abstraction upon seeing, however faintly, a more abstract application for it.

This algebra, this
weird little game of devising formulae and of solving equations, stands before
us as the sine qua non of modern physics. The use of algebra (in both its basic
and advanced forms), above all else, has made physics successful in describing
Reality with astounding accuracy and precision and, thereby, in allowing us to
manipulate Reality technologically. The discovery that we can quantify many
features of Reality, either through direct counting or through comparison with
defined standard units, does not astonish us. After all, we can assign numbers
to things every bit as arbitrarily as we assign names to things. What should
definitely astound us is the fact that we can organize the numbers that we so
obtain into formulae that have consistent relations with other formulae devised
by other means. That we can deduce many formulae describing Reality from a few
axioms by way of purely imaginary experiments carries a strong implication that
algebra expresses some profound fundamentum of Reality, that through this silly
alphanumeric game we may discern faintly the thought processes (if we can
legitimately so describe them) of the Thing That Created The Universe.

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