Variables and Motions

Algebraic letters are pure symbols; we see numerical relationships not in them, but through them; they have the highest “transparency” that language can attain.

—Susanne Katherina Langer (1895–1985), Philosophy in a New Key: A Study in the Symbolism of Reason, Rite, and Art

1

Numerical Symphonies

Music is the electrical soil in which the mind thrives, thinks and invents.

—Ludwig van Beethoven (1770–1827), letter from Bettine von Arnim to Johann Wolfgang von Goethe

Algebra is a vast and beautiful continent—at times serene and familiar, at other times mysterious and wild.

Despite the fact that it has been powerfully used for centuries, underwriting some of humanity’s most important innovations, serious questions and riddles remain: especially in regard to its essential nature, its place in education, and why so many intelligent people struggle to understand it.

Consider this an invitation to experience some of the vastness, aura, and beauty of this terrain.

But algebra does not reveal its scenery for free. It requires an intense cocktail of conceptual techniques to bring this beauty into sharp relief. Consequently, I will heavily employ some of the most powerful weapons of exposition available, including metaphor, analogy, history, and narrative.

Of metaphor, mathematics education researcher Anna Sfard states:

Metaphors are the most primitive, most elusive, and yet amazingly informative objects of analysis. Their special power stems from the fact that they often cross the borders between the spontaneous and the scientific, between the intuitive and the formal. Conveyed through language from one domain to another, they enable conceptual osmosis between every day and scientific discourses, letting our primary intuition shape scientific ideas and the formal conceptions feed back into the intuition.¹

Of history, mathematician J. W. L. Glaisher said, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”²

And of narrative, cognitive psychologist Steven Pinker says, “Cognitive psychology has shown that the mind best understands facts when they are woven into a conceptual fabric, such as a narrative, mental map, or intuitive theory. Disconnected facts in the mind are like unlinked pages on the Web: They might as well not exist.”³

Many experts share these sentiments.

In this book, we explore how far we can go with injecting these techniques (with a vengeance) throughout the discussion. My hope is that it will transform your conceptual and emotional understanding of this oft-maligned subject. In this chapter, we begin with music.

MUSIC

Music is one of the most remarkable of all the activities of humankind. Millions willingly subject themselves to its mood-altering effects day after day. Just a simple thirty-second ditty or jingle can launch back to life memories from decades past.

Mute the sound to a video of people vigorously dancing and their energized behavior looks fascinating at best, bizarre at worst. Go to the mall, ride an elevator, or watch a movie, and you will find it there lurking in the background. It is everywhere.

But what exactly is it? Why does it impact people in the ways that it does? How can it launch some into a state of almost pure euphoria while reawakening painful emotions, long thought extinct, in others?

It too is a vast expanse of familiarity, serenity, and mystery.

Of all its forms and manifestations, one of the most grand, vivid, and complex arises in the guise of the symphony: “a lengthy form of musical composition for orchestra, normally consisting of several large sections or movements…”⁴

The trajectory of sounds in a symphony can be extensive, wide-ranging, and dramatic. Reaching a profound and notably intense form in the work of Ludwig van Beethoven, it is said to be the medium in which many composers still choose to demonstrate their technical prowess and most expressive ambitions.⁵

Our interest with it here lies primarily in the great variety and scale that can arise around a central well-developed theme (or core)—we will see something similar happen repeatedly in a mathematical context.

One of the most fascinating things about music is that it is possible to capture the dynamic range of a symphony on flat sheets of paper. It is almost as if musicians can freeze the essence of an hour’s worth of lively music and hold it in suspended animation to be viewed later and analyzed at their leisure. This is no small thing and offers great benefits to those who choose to use it.

Written musical notation gave Beethoven (who could barely hear at all by his forties) the inspirational capacity to compose and share wonderful music right on up to the last years of his life, with the release of his highly acclaimed Ninth Symphony occurring in 1824 at age 53.⁶ It is hard to imagine him doing this without the aid of visual notation. To this very day, orchestras are still able to perform these masterpieces thanks, in large part, to their preservation in written form.

But the sounds of music in a complex performance are not the only phenomena that can vary in our world.

Variations in temperature, moisture, and so on connect up to collectively form the climate of a region.

Variations in events, political leaders, ideas, cultural norms, and so on combine to form the history of a place.

Artificial satellites soar through space constantly changing their individual locations, which, when taken together, collectively form an orbit; whereas weekly variations in time on the job join up to give the yearly earnings of an hourly employee.

Variations in nature differ in kind, too—with some variations being extremely simple (capable of complete description), some being more difficult to tame but still forecastable, and others seeming to totally defy prediction.

A central goal of this book is to learn more about variations of the numerical persuasion and to showcase their accompanying descriptions in symbols. We will find these variations to be relevant, often surprising, and more around us than we might think (often unrecognized). Moreover, we will find that their systematic description opens wide to us an entirely new and vast-reaching branch of mathematics: one that is distinct and separate from elementary arithmetic on the one hand yet critically fused at the hip with it on the other. Together these two branches will team up to form one of the most potent one-two punches in the history of human thought—creating, in the process, a quantitative version of Beethoven’s “electrical soil” in which the sibling spirits of mathematics and science can often materialize in, thrive, and discover masterful expression.

MAGICAL THREE-DIGIT NUMBERS

We now take a look at an artificially created numerical variation and observe how the values it produces can store wide-ranging information. To get a good feel for it requires your participation.

Pick the number of days you like to eat out in a week (choose from 1, 2, 3, 4, 5, 6, 7). Multiply this number by 4. Then add 17. Multiply that result by 25. Next add the number of calendar years it is past 2013 (e.g., if the year is 2016, then add 3). Now if you haven’t had a birthday this year, then add 1587, but if you have had a birthday this year, then add 1588. Finally, subtract the year that you were born from this.

After all is said and done, you should have a very personal three-digit number. Reading it from left to right, the first digit is the number of times you like to eat out in a week and the last two digits are your age. For those younger than 10 years of age, it also works if their age is simply interpreted as a two-digit number with a zero in front (i.e., interpreting 09 as 9 and so on). Try it again, using a different number of days and/or a different date and year. Save your efforts for we shall return to them in later chapters.

Here are a couple of examples of this in action:

1. Let’s say that the current date is November 5, 2030, that Abu Kamil’s birthdate is February 6, 1950, and that he likes to eat out five times a week. This scenario reads for him as follows:

a. Pick the number of days you like to eat out in a week: His number is 5.

b. Multiply this by 4: His number is now 4 × 5 = 20.

c. Add 17: He now has 17 + 20 = 37.

d. Multiply that result by 25: This gives him 25 × 37 = 925.

e. Add the number of calendar years it is past 2013: In 2030 this would be 17, which would give him 925 + 17 = 942.

f. If you haven’t had a birthday this year, then add 1587. If you have had a birthday this year, then add 1588: He had a birthday, so he adds 1588, which gives him 1588 + 942 = 2530.

g. Subtract the year that you were born from this: Since he was born in 1950, this gives him 2530 – 1950 = 580.

h. Reading from left to right, the first digit is 5 (the number of times he likes to eat out a week) and the last two digits are 80 (his age at the current date).

2. Let’s say the current date is May 16, 2018, that Pandrosion’s birthdate is December 26, 1980, and that she likes to eat out twice a week. Then this scenario reads for her as follows:

a. Pick the number of days you like to eat out in a week: Her number is 2.

b. Multiply this by 4: Her number is now 4 × 2 = 8.

c. Add 17: She now has 17 + 8 = 25.

d. Multiply that result by 25: This gives her 25 × 25 = 625.

e. Add the number of calendar years it is past 2013: In 2018 this would be 5, which would give her 625 + 5 = 630.

f. If you haven’t had a birthday this year, then add 1587. If you have had a birthday this year, then add 1588: She has not had a birthday, so she adds 1587, which gives her 1587 + 630 = 2217.

g. Subtract the year that you were born from this: Since she was born in 1980, this gives her 2217 – 1980 = 237.

h. Reading from left to right, the first digit is 2 (the number of times she likes to eat out a week) and the last two digits are 37 (her age at the current date).

There are hundreds of different values that can be generated by this process by varying the number of days, the current date, and the ages of the participating readers. Some such numbers include 123, 457, 720, 485, 323, 389, 717, 649, and 234, as well as possible different ones obtained by you and other readers—it is a veritable “symphony” of numbers!

However, the final scenario involving this three-digit number (and our interpretation of it) will fail for someone who is 100 years of age or older.

What in the world is happening? Why do these varying three-digit numbers simultaneously contain information that is personal to each of you yet different from other readers? Is it possible to describe this and to also show why it fails for centenarians? If so, how do we do it?

ARITHMETIC IS NOT ENOUGH

On its own, arithmetic will encounter great difficulty in conveniently describing what is happening in this number of days and age problem. There is simply too much going on—too much variety.

In generating each number, it is almost as if we are doing the same type of arithmetic process but on a different channel (identified by the number of days we like to eat out, the current year, our year of birth, and whether or not we have had a birthday). In the two examples we worked through above, one channel generates the number 237 while the other channel gives 580.

There are many other channels that generate all of the other values that can be produced.

Four channels of the hundreds that can be run and the resulting numbers generated

This then is a first example of the great variety (of numbers in this case) that can arise around a central theme or process. We don’t have enough tools yet to tackle this problem directly, so we will scale back to simpler scenarios and cut our teeth there first.

ORDINARY LANGUAGE IS NOT ENOUGH

Let’s begin by considering a different way to express a well-known and basic property of addition: the value we obtain when adding any two numbers doesn’t change if we reverse the order in which we add the two quantities. For example, the value we obtain when we add 3 + 5 is 8, and we obtain the same value if we reverse the order of the two numbers and add 5 + 3. This property is more formally known as the commutative property of addition.

You might ask, isn’t it enough to simply describe the property as we have above; what’s the point of searching for another way? The point being that, while describing mathematical concepts in plain English can be useful for representing and communicating ideas, it is not very useful for systematically rearranging them. Sometimes the key to grasping an idea or concept critically involves the ability to conveniently maneuver it into a simpler or more transparent form. Language simply is not always up to the task of doing this.

As an illustration of this point, let’s look at the simple problem of adding the three numbers one hundred sixty-seven, two hundred seventeen, and six hundred eighty-nine:

Written out this reads as: one hundred sixty-seven plus two hundred seventeen plus six hundred eighty-nine.

Using mathematical symbols this reads as: 167 + 217 + 689.

The mathematical form yields to easy manipulations (once we know the rules):

Conversely, the addition in English words alone does not yield to simple manipulations. That is, there is no realistic method to work our way to the answer using only what we are initially given—namely, the letters of the alphabet:

In practice, whenever we are given larger numbers in words to add, most of us resort to using numerals (whether on paper, mentally, or using a calculator) to complete the computation. We don’t think to compute by aligning the words and adding individual letters. That is, we don’t ask ourselves what adding the last letters of each number word (e + n + n) will equal and so forth—letters when used as language components simply don’t work that way.

In a similar fashion, looking at the number of days and age problem as it currently reads doesn’t give a clear idea of what is going on. We can certainly run the numbers as the procedure asks, but why they end up the way they do seems almost like magic.

We need a different way to express the problem. We need a method that transforms the problem the way the symbols “167 + 217 + 689” transform the English statement “one hundred sixty-seven plus two hundred seventeen plus six hundred eighty-nine.” In short, we need to take the entire problem itself, as stated in English, and recast its essence in a new form—into something that can be operated on and meaningfully rearranged.

In the rest of this chapter, we will focus on how to recast quantitative ideas and procedures that can vary or change value into a more malleable form, then in the next chapter we will turn our attention to how to successfully maneuver them after they have been converted to this new form.

STORING IDEAS

Let’s return to the commutative property of addition. We give four more examples of this property in action:

The total possible occurrences of this property are infinite. And once more we are faced with a “symphony of numbers”: this time involving sets of numerical expressions (such as I, II, III, and IV) as opposed to single numbers.

Lots of variety indeed, yet all of it tied together by a simple core theme—that of commutativity (i.e., the order in which we add two numbers gives us the same answer). Like the number of days and age problem, it is as if all of these different expressions are simply different channels of the same idea.

From here on out, when we refer to a “symphony” or “ensemble” it will mean a collection of numbers, expressions, or objects that are tied together around a specific procedure, rule, or theme.

Let’s now give chase to recasting this idea of commutativity into a more malleable shape by capturing its operational essence, which is that we have two slots, on the left-hand side of the equals sign, into which two numbers can be inserted and added, and then we reverse their positions on the right-hand side of the equals sign. We can describe this as

first number + second number = second number + first number.

An advantage of expressing the idea of commutativity this way (as opposed to writing out “that the order in which we add two numbers doesn’t matter”) is that the arrangement now has the same form as the property does when we write it out with numbers (e.g., 1 + 2 = 2 + 1). That is, the expression now is not very far removed operationally from the thing it is describing (this is not true of the written statement).

In the four numerical examples I, II, III, and IV, the slot described by “first number” takes on the values 1, 8, 452, and 11200, and the slot described by “second number” takes on the values 2, 12, 987, and 876543, respectively. Thus, this expression operationalizes, in a sense, the general idea of commutativity.

If we were text-messaging this idea to someone, we might choose to abbreviate it in either of the following ways:

first number + second number = second number + first number

becomes

fn + sn = sn + fn,

or taking this even further to

f + s = s + f,

with no loss of essential information. If we choose the latter, all of the variety that can be expressed with the different numerical instances of commutativity can be reproduced from this stripped-down alphabetic rendering in the following way:

Set f to:	Set s to:	f + s = s + f
1	2	1 + 2 = 2 + 1
8	12	8 + 12 = 12 + 8
452	987	452 + 987 = 987 + 452
11200	876543	11200 + 876543 = 876543 + 11200

If we let f = 300 and s = 987, then f + s = s + f becomes 300 + 987 = 987 + 300 and so on. The innumerable demonstrations (variations) of the commutative property in action can now all be obtained by simply setting “f” and “s” to the required numerical values in the expression f + s = s + f.

All of the infinite variation (hence the idea of commutativity itself) is now, in effect, captured and becomes stored or seeded in a single easy-to-read expression. This is a major conceptual shift as we are now looking at letters as platforms for storing changing numerical values as opposed to their traditional use as carriers of information for the spoken or written word.

There is great value in this. We store ideas in language, too.

In English, the word tree applies to trillions of distinct and varied plants on Earth.⁷ Each individual tree is a tangible example of the specific combination of qualities that we give to the word t-r-e-e: meaning that the word serves as a symbolic storage device for every one of these plants. They have properties in common that allow us to quickly refer to the majority of them as trees.

We gain tremendous advantages by being able to refer to lots of different and distinct things by the same expression or name. In this particular case, we can in one statement (“Trees help to remove carbon dioxide from Earth’s atmosphere”) communicate something that applies equally well to processes involving every single living tree on the planet. A single innocent sentence, simple enough to be taken in with a single sweep, is still broad enough to say, all at once, something that is true about trillions of different plants.

Languages, in general, give us this wide-ranging ability to describe lots of objects and ideas with a relatively small glossary of words. Taking these words, then, in combination to form sentences—language expressions—gives us the breathtaking ability to describe nearly everything that we experience in life or are able to think about in the world around us. We seek the same in the world of numerical variations.

OTHER ENSEMBLES OF NUMBERS

There are other numerical ensembles out there awaiting description. Let’s look at a few.

Consider a plane flying at a height of 35,000 feet that is traveling west for four hours at the constant speed of 450 miles per hour. During this time, it travels a total distance of 1800 miles, meaning that, in theory, every distance from 0 to 1800 miles is covered at some point over the four-hour journey.

Now this might be starting to sound like the word problems you remember, and maybe dreaded, from algebra class. However, based on the principles we’ve just established, this is nothing more than another kind of numerical symphony. Let me show you what I mean by including a few members of this ensemble: 450 miles (for the distance traveled in 1 hour), 900 miles (for the distance traveled in 2 hours), 1080 miles (for the distance traveled in 2.4 hours), 1350 miles (for the distance traveled in 3 hours), and 1800 miles (for the distance traveled in 4 hours).

We can visually represent this:

A nice way to store and operationalize all of this variety is by again identifying its core, which in each case here involves multiplying 450 by the time of travel in hours. Here, we will abbreviate “the time of travel” with the letter “t” (for time), and then we can reproduce all of the numbers in the previous diagram by simply writing

450t

(which means 450 multiplied by t) and setting t to the following values:

t = 1; t = 2; t = 2.4; t = 3; t = 4.

The expression 450t stores in a leaner form the variation contained in this situation. We have shown only five distances produced by five different values for t, but there are a host of others. For instance, if t = 3.7 hours, 450t would become 450 multiplied by 3.7, which equals 1665 miles. Or, put another way, the plane travels 1665 miles in 3.7 hours of flight. We could do this, in theory, for any of the values of t between 0 and 4. It is as if the expression acts like a seed or computer folder containing all of the information regarding all possible distances of travel from 0 to 4 hours in this scenario.

A second symphony: suppose we want to calculate the amount of money that we would earn when paid an hourly wage of $16 an hour. This generates the following numerical ensemble:

As before, there are a host of other values for wages earned based on the possible hours worked, which could eventually amount to tens of thousands of hours for a given individual. We can store all of these in the following expression, where “h” stands for hours worked:

16h,

or 16 multiplied by h. For example, all of the earnings in the previous diagram can be easily reproduced by simply setting h equal to each of the following values:

h = 10; h = 25; h = 45; h = 68; h = 237.

Other everyday situations that yield groups of numbers orbiting a common theme include the following:

• The amount owed on a 30-year house loan of $900,000 after making payments of $5000 for a given number of months. To generate the ensemble, let the number of months vary—then compute how much is still owed after each month.

• The amount of sales tax (at 7%) paid by each individual in a certain town for a given year. To get the ensemble, multiply the total retail spending of each person for the year by 0.07.

• The numerical position of Earth as it travels in its orbit around the sun. To get the ensemble, choose different dates and times throughout the year, then locate the position of Earth.

• The batting average of a baseball player for a given season based on the number of hits obtained in so many at-bats. To get the numerical ensemble, choose the number of hits and number of at-bats for a given player—then calculate .

The individual batting average of each of the thousands of MLB baseball players puts a face on many of the values in this numerical symphony (e.g., 0.344 for Miguel Cabrera in 2011, 0.406 for Ted Williams in 1941, and so on).⁸

All of the numerical variation possible in these four scenarios can also be captured and stored with expressions that use letter abbreviations rather than words. The expressions in some cases will be much harder to obtain and more complicated than before, but they still accomplish the same goal.

A NEW WAY OF THINKING

Though numbers have been ever-present in these examples, we are no longer dealing with simple arithmetic. Something more is going on now.

Imagine if you will a child engaged in word play. Such a child might play with words in various combinations and, upon stumbling onto the word ram and liking the sound of it, decide to explore further to find other words that sound the same—eventually discovering words like bam, clam, dam, gram, ham, jam, spam, and yam. After this first successful exploration, they may substitute some letters to create new rhymes and eventually learn dozens more new words through this study.

So, the accidental discovery of an unknown word and its pleasant sound has suddenly opened wide, for the child, a whole new way of thinking about words. They can now systematically search for new words that sound alike as well as search for words that have the same meaning.

We are presently at a similar place. But what is it that we have discovered? Is it that we can abbreviate words? Surely there must be more, as abbreviation is not a new technique. In fact, numerals (which go back thousands of years) are themselves shorthand symbols for the quantities they express. And though abbreviation has a significant presence in algebra, it is not what fundamentally differentiates the subject from arithmetic.

What we have inaugurated here that is truly different from arithmetic is a new and deeper way of thinking about certain types of mathematical problems.

For instance, in arithmetic you might be interested in computing your earnings from an hourly wage of $16 an hour. If you worked 40 hours, you would simply multiply 16 by 40 to conclude that you will earn $640. If you work 56 hours another week, then you would calculate 16 × 56 and move on. Question answered.

What we are doing now is not just looking at a single situation and making a computation, but establishing a rule that holds for the wider variety of situations possible (earnings in this case) and how they relate to one another—think weather on a given date versus climate over a decade. If we can do that, then we can generate any value that we care to know about (e.g., 16h).

The possibilities are immense. Now that we know that some variable phenomena—like money earned from an hourly wage, distance traveled by a plane, and the commutative property of addition—can be readily described by written, abbreviated expressions in this new way of thinking, could it be possible that if we reverse the process and first create abbreviated expressions of our own choosing, then we might eventually be able to describe novel things—presently unknown to us?

For example, we have established that “16h” can be used to help an employee who works h hours at $16 an hour calculate their total wages over a specific period of time. What if we now, from simply looking at this object, decide to find other objects that “rhyme with it” by raising the h to the second, third, or fourth power, obtaining 16h², 16h³, or 16h⁴, respectively? Could these new expressions possibly describe some kind of variable behavior that we don’t know about yet?

If so, it could be very worthwhile to study these expressions in their own right. This is an exciting prospect as it suggests that we can learn more about sophisticated real-world phenomena by simply studying abbreviated expressions, through the prism of this enlarged outlook, on paper.

However, if we decide to do this, then it leads to an interesting situation. If a car is traveling at the steady speed of 16 miles per hour, we can describe the distance it travels after t hours by 16t. We already know that 16h can be used to describe the earnings of a person who has worked h hours at a rate of $16 per hour. This gives us two separate expressions (16t and 16h) that look different, but we must ask, are they really different?

It turns out that despite their different contexts, they produce the same numerical values when we evaluate them for t = 2, 5, 10 or h = 2, 5, 10. In both cases, we find that we obtain the same numerical values (32, 80, 160) with albeit different interpretations: miles in the first case and dollars in the second.

In fact, this occurs here whenever t and h are set to the same number. So these two expressions in effect generate the same numerical ensemble (in the same way)—meaning that if we divorce them of their interpretations (looking only at the values that they generate from their numerical inputs), they are essentially identical. This scenario will repeat itself with other expressions as well (e.g., “16h² + 70h” and “16t² + 70t”).

Given that different-looking expressions can produce equivalent values from the same input numbers, you can see how it could be useful and perhaps less confusing, at first, to standardize the letters we use for this purpose. Think of it as putting the letters we use in the same font and size. This will allow us to focus most of our initial attention on how the expressions behave as opposed to being distracted by their appearance.

We do this with language, too. Sometimes we want to be specific and talk about five apples or five cars or five phones, but sometimes we want to be more general (divorcing the objects from any specific interpretation) and simply say we have five things. In business or economics, the term widget is sometimes used to represent a generic product.

In mathematics, various names have been given to the unspecified object over the centuries. The medieval Muslims sometimes used the word shay to represent unspecified information. Some in India used the abbreviated term yā, whereas the Italians of the Renaissance often used the term cosa.

Once the idea of systematically abbreviating terms took firm hold in the late 1500s and early 1600s, the unspecified entity took several shapes. One of the earlier suggestions, known as the Viète/Harriot protocol, was that vowels in the Latin alphabet (e.g., A, E, and I) be used; but this idea didn’t stick for long.

A later idea employed in the mid-1600s by the French philosopher and mathematician René Descartes was to represent the primary variations in a problem by using letters late in the Latin alphabet (x, y, and z as needed). This is the idea that stuck and is still most often employed in most elementary algebra texts today.

Using this standard means that the expressions we used earlier could translate to the following:

Commutative property of addition	f + s = s + f becomes x + y = y + x, where x represents the first number and y represents the second number
Distance traveled by plane	450t becomes 450x, where x represents the time of travel in hours
Amount earned	16h becomes 16x, where x represents the number of hours

Notice that it is less distracting mathematically to compare 450x to 16x than it is to compare 450t to 16h.

So we have in a sense two ways to express and operationalize situations involving quantities that can vary: the generic sense, in which case we generally employ letters such as x, y, and z; and the interpretive sense, where we abbreviate the variable quantities that we want to describe, using whatever letters work naturally. The generic sense is more commonly used when we are doing a general study of how to represent and manipulate variation. By contrast, the interpretive sense is used more commonly when we employ abbreviations to describe a specific scenario.

Let’s look at an example of this principle in action. If we want to understand the relationship between the area of any rectangle and its length and width, the standard way to represent the numerical ensemble generated by the interaction of these values would be to use abbreviations for each of the words. We can see this in the formula “area equals length times width,” which we shorten to A = lw. Though we could write this as A = xy, where x and y stand in for length and width, respectively, we rarely do so unless we want to operate on the expression as part of a larger problem where there is some benefit to being more generic.

In most applications (such as physics, engineering, and statistics), x, y, and z are usually avoided and single-letter abbreviations are preferred so that the quantities being related to each other are easier to remember. For instance, in Einstein’s famous equation E = mc², E stands for energy and m for mass, while c follows the universal convention for representing the velocity or speed of light.⁹ We will come back to this distinction between standard and interpretive notation in later chapters.

CONCLUSION

We have shown that it is possible to capture, in writing, the essence of many phenomena that vary in value. We can then use abbreviations to further simplify what we have captured with no critical loss of information. This can be thought of as creating a written notation, if you will, for describing on paper numerical phenomena that can change value—just as we already have a written notation for music that allows us to describe on paper something as complicated and varied as the sounds from an hour-long Beethoven symphony.

However, this just barely scratches the surface, for we will soon discover that these written expressions truly distinguish themselves through their dazzling capacity for interacting with each other (and numbers) in ways that allow them to systematically discover unknown facts about the world—like almost nothing else. Taking advantage of this ability for interaction will give us the precise tools we need to completely understand and easily dominate the number of days and age problem.

We shall also find that their capacities for representation, combination, rearrangement, and generalization were ultimately the engines that gave rise to such expressions having an immense expanse all their own—one that has been pivotal in the mathematical, scientific, and technical applications of the last half of the second millennium and on into the third. Called Hisab al-jabr w’al-muqabala [calculation by restoration (al-jabr) and reduction (al-muqabala)] by its ninth-century Persian/Arab father, Al-Khwarizmi, and The Analytic Art by its Renaissance European father, François Viète, it is the vast conceptual continent we know today as algebra.