4
The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization.… The separation of relevant from irrelevant factors is the beginning of knowledge.
—Hans Reichenbach (1891–1953), The Rise of Scientific Philosophy
In the background of any area of mathematics lurks a most remarkable and pervasive presence. Mathematician Morris Kline hints at it when he writes:
… A study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning… were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explication of this marvelous power is called for.^{1}
Whatever the full extent and nature of this “marvelous power” of mathematics, traces of its presence should be visible even in the most fundamental areas of the subject. As these are the areas where we reside in this book, part of our task in exploring algebra should be to give a glimpse of this potency in its pages. A few observations are the order of the day.
Imagine you knew nothing about mathematics and were playing around with pebbles on the ground. It would still be possible to learn basic facts, such as two pebbles added to four pebbles yields six pebbles, or taking seven groups of three pebbles together yields a total of twenty-one pebbles. In fact, it would be possible to learn a great deal about arithmetic from tinkering around with pebbles only.
But the wonderful thing then is that what is true of pebbles is true of so much more, for if you replace the pebbles with people, dollars, or cars instead, you still obtain true results. That is, it is not just true that 2 pebbles + 4 pebbles = 6 pebbles or that 7 × 3 pebbles = 21 pebbles, but it is also true that 2 people + 4 people = 6 people and that 7 × 3 people = 21 people—and the same with dollars, cars, and so many more things.
In short, by playing with pebbles, one can not only discover arithmetic facts that are true of pebbles, but one can in effect discover arithmetic facts that are true of infinitely many things! It is almost as if the universal laws of addition and multiplication in arithmetic momentarily materialize in the guise of the pebbles in your possession, uncloaking some of their general properties to any and all who would take notice.
Moreover, this remarkable ability to generalize from a specific experience is not unique to mathematics. We may learn the rules for how to safely cross a street on one near where we grew up, yet those same basic principles and behaviors can be used to successfully cross a similar street on the other side of town, in Waterbury (Connecticut), in Trondheim (Norway), or on practically any similar street on the face of the Earth!
From learning how to play an instrument, a board game, a video game, or a sport to learning how to type, drive a car, or cook, this ability to project our knowledge and skills from the context in which we first acquired them, and extend them to millions of other similar circumstances, is everywhere. Amazingly, we all have been granted the ability to connect our individual experiences to something much greater than ourselves—touching, in a sense, eternity itself from the confines of home.
Perhaps the most compelling manifestations of this phenomenon occur in the traditional scientists’ and inventors’ laboratories, where breakthroughs such as the telephone, the light bulb, X-rays, radioactivity, and the transistor first were discovered or developed in the small, then have ultimately come to affect millions and in some cases billions of people in the large. And this ability to flirt with the eternal from home is also present right here in elementary algebra, where the time has come to spend more of the algebraic capital that we acquired in the first three chapters. Here, we begin with a simple situation from business in the small. Let’s see where it takes us.
MONEY STREAMS
Let’s consider a situation involving a newly formed small business venture where hamburger meals will be sold. After doing a bit of research, we have determined that overhead (rent, maintenance, utilities, etc.) every two months will cost around $3100. We have also learned that it will cost about $4.75 for the time and ingredients to make each hamburger meal. Based on this information, we have settled on a selling price of $6.00 for each meal. This guarantees that we will make more money from each sale than we spend.
What we would like to know is how many of these meals we have to sell every two months so that the amount of money we bring in (revenue) matches the amount of money we spend (total costs). For the purposes of simplifying the problem, we will make the assumptions that our cost estimates are exact and that we sell all of the food we buy.
To get a feel for the problem, we will start by playing with a few hypothetical situations. What happens, for instance, if we were to make and sell 100 meals? We have two cases to consider: the amount of money brought in and the total amount spent. Based on the conditions just listed, these are both straightforward to obtain:
Revenue (Money Made) |
Total Costs (Money Spent) |
(Selling price per meal) times (number of meals made and sold) |
(Cost to make each meal) times (number of meals made and sold) + overhead |
$6 per meal × 100 meals = |
($4.75 per meal × 100 meals) + $3100 = $475 + $3100 = |
$600 |
$3575 |
Situation 1: Make and sell 100 hamburger meals
So, we see in this situation that the revenue is $2975 less than the total costs ($3575 – $600), which is how far we still are in the hole. But this is already an improvement over the $3100 overhead debt we started with before making and selling any meals.
What if we make and sell 400 meals?
Revenue |
Total Costs |
$6(400) = |
$4.75(400) + $3100 = $1900 + $3100 = |
$2400 |
$5000 |
Situation 2: Make and sell 400 hamburger meals
Here, we see that both revenue and costs have grown, but the gap is closing as the difference between them now is only $2600 versus $2975 from the first situation.
We continue our testing for the situation where we make and sell 1000 meals:
Revenue |
Total Costs |
$6(1000) = |
$4.75(1000) + $3100 = $4750 + $3100 = |
$6000 |
$7850 |
Situation 3: Make and sell 1000 hamburger meals
Now the gap between revenue and total costs has closed to only $1850. Thus, the two money streams seem to be converging together—with the revenue stream gaining progressively more ground.
We can try to speed things up by using these trials as a springboard to make even better guesses until the desired point is found (where dollars earned equal dollars spent). Many will undoubtedly succeed in this manner. But because we are in the algebraic frame of mind, let’s try to use this evidence in a different way to see if we can gain a more comprehensive overview of the forces at work in this problem.
What we have here are two quantities—revenue and total costs—that change depending on the quantity of meals made and sold. These are varying amounts (or numerical symphonies) based on systematic procedures. A few values for each ensemble are given here:
Revenues for number of meals sold
Total costs for number of meals produced (with overhead included)
Can we come up with algebraic expressions that generate each of these numerical ensembles? To do so requires that we separate out the variable components from the constant components. As with the number of days and age problem in Chapter 1, we will do this by tagging the variable terms with letters to allow them to hold on to their separate identities.
Looking at the previous tables and diagrams, we see that the quantities that change from example to example are the number of meals, revenue, and total costs. However, because the variations in the revenue and total costs depend on the number of meals, we will simply tag the number of meals and see what happens. Representing the number of meals made and sold by x gives the following:
Revenue |
Total Costs |
$6(x) = |
$4.75(x) + $3100 = |
$6x |
$4.75x + $3100 |
Variable situation: Make and sell x hamburger meals
Notice that we can generate the previous three situation tables by respectively setting x to 100, 400, and 1000 in our variable situation table. We can generate new situation tables as well (e.g., 897 meals sold or 1400 meals sold) by setting x to the number of meals sold. Thus, we have completely captured all of the business situations possible for this hamburger venture—meaning that whatever we decide to do to our variable table, we are doing, simultaneously, to all of the potential sales scenarios in one grand maneuver. This is the first drama.
Our task here, however, is not to store or generate all possible outcomes, but to identify that special one in which the revenue is equal to the total cost. This special point for a business is called the break-even point—after which point additional sales in the two-month period produce a net profit. The good news is that the tools to handle this situation have already been developed. We need only elevate this task of ours to the platform of equations and then unleash the techniques from Chapter 3 on it. This is the second drama. Thus,
revenue equal to total costs
becomes in algebraic equation language
6x = 4.75x + 3100.
Solving this using reduction diagrams yields this:
Because the solution to this equation is 2480, we have to sell 2480 hamburger meals in order for revenue to equal total costs over a two-month period. Let’s check our work.
Revenue |
Total Costs |
$6(2480) = |
$4.75(2480) + $3100 = $11780 + $3100 = |
$14,880 |
$14,880 |
Solution situation: Make and sell 2480 hamburger meals
So, we see that once the hamburger meal problem is stripped down to its bare essentials, it lends itself to a very straightforward algebraic treatment. This treatment allows us to almost automatically home in on the answer through the routine solving of a basic equation—swiftly and completely eliminating the need for clever guesswork!
The procedure is so routine, in fact, that it is easy to gloss over what is going on behind the scenes—which is that algebra has allowed us to capture the essence of a task, then gifted us with the ability to efficiently maneuver it in ways, not only saving us from a lot of potential work using trial and error, but also allowing us to build a stage upon which to showcase a strategy for dealing with the entire category of break-even business problems.
A DIFFERENT KIND OF VARIABLE
Just as with the arithmetic rules learned using pebbles, so too does our solution to this break-even problem (learned using hamburger meals) generalize to other scenarios. For as long as the conditions remain the same ($6 selling price, $4.75 cost to make each item, and an overhead of $3100 every two months), it is clearly immaterial if our product changes to chicken meals, packets of writing pens, or any other item. The break-even point will remain 2480 chicken meals, 2480 packets of pens, or 2480 of whatever item we are selling.
This is separating the mathematically relevant components of a problem (selling price and costs) from the mathematically irrelevant details (name and type of product). But this is just the beginning. The algebraic mode of thinking about this problem now comfortably lays before us the potential to bag much larger game. To do so, however, will require the identification of a new type of “variable.”
Let’s start by going back to the top of our problem to see how revenue and total costs appear in words:
• Revenue = (selling price per meal) times (number of meals made and sold).
• Total costs = (cost to make each meal) times (number of meals made and sold) + overhead.
Since we are now interested in working with the mathematically relevant components, we know (at least as far as the algebra is concerned) that it doesn’t really matter what the name is of the product being made and sold. As long as the relationships stay the same, so too does the mathematics. To capture this understanding, we will replace the word “meal” by the more general word “thing.” This gives the following:
• Revenue = (selling price per thing) times (number of things made and sold).
• Total costs = (cost to make each thing) times (number of things made and sold) + overhead.
Tagging the number of things made and sold with x gives this:
• Revenue = (selling price per thing) times (x).
• Total costs = (cost to make each thing) times (x) + overhead.
Now for a given business scenario, the number of things made and sold can vary while the selling price, cost per thing, and overhead remain the same (at least over a short period). We saw this happen in our hamburger example—the number of meals varied, while the selling price at $6, cost per meal at $4.75, and overhead at $3100 remained the same. The bigger game we want to bag now is the more general case, where the business scenarios themselves can change (i.e., where the selling price, cost per thing, and overhead change value). We list two such scenarios:
• Selling price per set of knives = $12, cost to make each knife set = $9, overhead = $4000 per month, and x = number of knife sets made and sold.
○ Revenue = 12x.
○ Total costs = 9x + 4000.
○ Break-even equation (revenue = total costs): 12x = 9x + 4000.
• Selling price per purse = $52.25, cost to make each purse = $38.15, overhead = $11,985 per month, x = number of purses made and sold.
○ Revenue = 52.25x.
○ Total costs = 38.15x + 11985.
○ Break-even equation (revenue = total costs): 52.25x = 38.15x + 11985.
Placing the various scenarios in a symphony diagram yields the following:
Let’s zero in on the “General Business Scenario” at the bottom of the diagram. Notice that, in addition to the x, it has three other quantities that can change: the selling price, the cost to make each thing, and the overhead. These quantities, however, vary in a different way than does the quantity represented by x (the number of things made and sold). These three general quantities stay the same for a specific business scenario such as in the hamburger case, but they can vary from scenario to scenario as in the diagram. Conversely, the x has no such restrictions and can vary within a given scenario, as we saw in the situation tables.
So, the selling price per thing, cost to make each thing, and overhead cost possess both a variable aspect (from scenario to scenario) and a constant aspect (within a given scenario). This makes these three a different sort of variable—a variable that we tune or fix to a certain value for a given scenario, but when the scenario changes, we tune it to other values.
Scenario |
Selling Price |
Cost Per Thing |
Overhead |
Break-Even Equation (Selling price)x = (cost per thing)x + overhead |
Hamburger |
$6 |
$4.75 |
$3100 |
6x = 4.75x + 3100 |
Knives |
$12 |
$9 |
$4000 |
12x = 9x + 4000 |
Purses |
$52.25 |
$38.15 |
$11,985 |
52.25x = 38.15x + 11985 |
Mathematicians have special names for tuning variables such as these: one of the terms they commonly use is parameters. Parameters represent quantitative concepts as well. So, just as we label the regular, less restricted variable (number of meals made and sold) with the letter x, the convention is to represent these parameters by letters, too (often generic ones earlier in the alphabet, but sometimes just a direct, one-letter abbreviation of the quantity name; much more on this later in the book).
For this particular circumstance, we will employ the following abbreviations: selling price = P, cost per thing = C, and overhead = F. In the case of overhead, using an O as an abbreviation could potentially cause confusion with zero, and because overhead costs can also be described as fixed costs like rent and utilities, we have chosen instead to abbreviate the “fixed” with an F for the sake of clarity. If we employ these, our general break-even equation becomes Px = Cx + F. And we can rewrite the symphony diagram from earlier with parameters (P, C, and F):
Algebra has lifted us to a high place here. Now when we encounter a break-even problem of this type, instead of using trial and error, we simply ask the following of the problem: What is the selling price per product, what is the cost to make each product, and what is the overhead (or fixed costs)? Then, all we need to do is plug these values into the equation Px = Cx + F, solve it, and we are done.
As an example, let’s find the break-even point for the purses scenario (P = 52.25, C = 38.15, and F = 11,985):
We see for this business model that the point where revenue matches the total costs occurs once 850 purses have been sold. That’s it, no situation tables and no guessing. We are done. Period!
And so, algebra allows us to organize and completely dominate this kind of break-even problem. Of course, the details of how to pin down the values for the parameters can become quite involved—especially when inventory, warehousing, and marketing issues are also included—requiring the care and attention of a skilled businessman. But the algebraic way of conducting affairs has allowed us to at least glimpse the soul of the problem and maneuver it to great advantage.
OTHER EXAMPLES OF PARAMETERS
Parameters are one of the most useful devices in the toolkit of anyone trying to capture an entire category or classification of varying behavior. Let’s take a look at a few more examples.
Consider three towns with different sales tax rates of 5%, 7%, and 8.2%, respectively. The following diagram shows expressions for calculating the sales tax in a given town, for a given amount of money spent (represented by x):
For example, if you buy $3000 worth of goods in Town Two, then x = 3000 and the sales tax you pay is (0.07)(3000) = $210.
Here, we see that the sales tax rate acts as a parameter. It is constant in a given town (over a period of time, at least) but varies from town to town (scenario to scenario). The amount of money we spend can, of course, vary within a given town, and so we think of this as the regular variable and represent it by x.
Though we already have a general formula for calculating the sales tax for any town, we can make it more user-friendly by representing the tax rate by a letter. We make the obvious abbreviation: Tax rate as a decimal = r. This makes our general expression become sales tax = rx. Now that we have conveniently captured the sales tax due on any amount of goods bought in any town, all we need do is substitute the value of the parameter r for a given town and we are ready. In these examples, r would equal 0.05, 0.07, and 0.082, respectively, for the three towns.
We encounter another parameter in physics when we want to calculate the speed of an object dropped from a certain height. After a great deal of experimentation, Galileo theorized that freely falling objects sped up at the same regular rate. In modern parlance, their speed (in feet per second) can be approximated well by the expression 32t (32 multiplied by the time in seconds after being dropped). Neglecting the effects of air resistance (also called drag or friction), this means that after 4 seconds of falling, a freely falling object will be traveling at approximately 32(4) = 128 feet per second, and after 7 seconds will be traveling at approximately 32(7) = 224 feet per second, and so on.
Galileo’s experiments, though more numerous and far more involved, are in some ways similar to the hypothetical situations that we tested in our hamburger meals break-even scenario. And just as we did there, it is possible to explain the results of his experiments with algebraic expressions.
Moreover, as science progressed into the 1600s, visionaries such as Isaac Newton realized that the algebraic expression for approximating the speed of a falling object on Earth was a special case of a more general situation. Objects on other planets should fall in a similar way as they do on Earth, just at different speeds. Thus, similar but distinct formulas should exist to describe these speeds, too. The following table lists formulas for a few bodies, ignoring the effects of air resistance where necessary:
Celestial Body |
Formula for Approximate Speed (of a Dropped Object) |
t = 7 sec. |
Approximate Speed When t = 7 Seconds |
Earth |
32.2t |
32.2(7) |
225 feet per second (~153 mph) |
Venus |
29.1t |
29.1(7) |
204 feet per second (~139 mph) |
Mars |
12.2t |
12.2(7) |
85 feet per second (~58 mph) |
Moon |
5.3t |
5.3(7) |
37 feet per second (~25 mph) |
Jupiter |
85.1t |
85.1(7) |
596 feet per second (~406 mph) |
Speed of falling objects on different celestial bodies (these expressions are useful for heights above the surface of body that are small relative to the size of the body; modern-day values rounded to one decimal place have been used; Jupiter, a gaseous planet, doesn’t have a solid surface as do the others)
From the table, we see that the generalized expression for approximating the speed after t seconds takes the form
speed of fall = (a number characteristic of each celestial body) times t.
This numerical characteristic of each celestial body is a parameter that physicists call the acceleration due to the gravity of that celestial body. This parameter is often abbreviated as g. Using this abbreviation, it follows that the speed of an object’s fall is close to gt. Here, the notion of parameters changing values as we go from scenario to scenario now translates to the parameter g changing values as we go from planet to planet (or world to world).
CONCLUSION
The great abilities of algebra to decisively organize, maneuver, and generalize information are on full display here. If you recall, we started out with the relatively pedestrian task of finding out how many hamburger meals we needed to sell so that the money brought in equaled the total amount spent. This we have done with the crucial assistance of basic algebra, and now, because of it, we find ourselves in a commanding position.
From relatively humble beginnings, we now have a general equation (Px = Cx + F) that allows us to quickly arrange and solve break-even problems of this type from all over the business landscape—totally eliminating (at this algebraic stage) the need for trial and error. Moreover, along the way we have discovered and made explicit use of an entirely new species of objects called parameters. The systematic introduction and exploration of this class of variables is certainly one of the watershed events in a watershed century (the sixteenth) for algebra.
In the minds of many modern historians of algebra, it was François Viète’s understanding and explicit use of parameters that was one of the final decisive steps in propelling algebra into the systematic and broad discipline that it is today.^{2} Their introduction helped to literally break the subject wide open and expand the scale of its usefulness—turning small algebra into big algebra.
This is why Viète, more than any of the other great sixteenth-century algebraists before or contemporary with him—including Scipione del Ferro, Niccolò Fontana (Tartaglia), Lodovico Ferrari, Christoff Rudolff, Michael Stifel, Rafael Bombelli, Thomas Harriot, Simon Stevin, or even the talented Girolamo Cardano—is sometimes considered to be the father of symbolic algebra.^{3}
For another example of the value of parameters as a tool for studying entire categories of equations and their general properties, see Appendix 1, which includes a brief discussion on the quadratic equation.
To a large extent, before the late 1500s, algebra was seen primarily as a tool for determining unknown values starting from known numerical quantities. This is how the great medieval Islamic mathematician Al-Karaji essentially described the subject in the early eleventh century.^{4} Problems were more often than not either phrased in ways that were directly about knowns and unknowns from the start, or in ways that could be translated as such. Once so translated, the techniques of rhetorical algebra (and the slowly emerging symbolic algebra) were then employed to solve them.
Yet Viète saw that algebra still had far more to say about the world. What he saw more clearly than his contemporaries or near predecessors (at least as expressed in print) was that there were all sorts of other situations out there that didn’t on the surface look anything at all like the typical problems, yet which were in fact amenable to the same mathematical treatment. In other words, he saw that they were really problems about algebra cloaked in camouflage.
This gave him an expansive view of what algebra really was capable of doing—making him realize, armed with his parameters, that it was an extremely general tool that could aid in the understanding of all kinds of investigations (especially geometrical ones). He gave this enlarged view of the subject the name The Analytic Art, and modern mathematics was well on its way to impacting the world as never before.^{5}
Compare this to the rise of computers in the twentieth century. In the first half of the century, the primary function of a computer was to perform highly technical numerical calculations for computationally intense endeavors, such as those encountered in making calculations for artillery firing tables or in designing the hydrogen bomb. But as time passed, these machines showed themselves to be capable of much, much more. Like mechanical actors, they became capable of imitating all sorts of machines.
Soon computers were not just lightning-fast calculators, they were becoming typewriters with memory (word processors), spreadsheets, databases, flight simulators, road maps, and social media platforms. Eventually, these machines were able to be miniaturized, and their storage capacities enlarged to such an extent that even the average consumer could afford to purchase one and bring this incredible power and worldwide access into their home.
Computers have changed the way we communicate with one another, creating whole new industries and ways of interacting with the world. All of this has made their impact on human affairs orders of magnitude greater than many of their original practitioners ever imagined.
Algebra has been transformed in a similar fashion and scale, from the early days of finding unknowns in basic numerical and recreational problems to the subject forming critical fuel for calculus (and much of modern mathematics) with the subsequent spectacular applications in physics, chemistry, the sciences in general, engineering, economics, finance, and ultimately to the very computer itself. The uses of algebra continue to expand to this very day.