2
The mathematics of our day appears to me like a large weapon shop in peace time. The store window is filled with showpieces whose ingenious, artful, and pleasing design enchants the connoisseur. The real origin and purpose of these things, to attack and defeat the enemy, has retreated so far into the background of consciousness as to be forgotten.
—Felix Klein (1849–1925), Development of Mathematics in the 19th Century
In walking, it means going around the muddy puddle to get to the store dry and clean. In photography, it entails looking for the best vantage point to make a sunset sing. To sports goers, it can refer to leaving the game early to avoid traffic. To running back Barry Sanders, it meant getting around defenders in ways that were almost choreographic.
Maneuver is all around us in a wide variety of manifestations. In some domains, its presence is so pervasive and overwhelming that its very name simply cannot be hidden from view—taking on the mantle of entire doctrines even. Military thinking is one such arena, where the original meaning of the word was closely allied with the notion of moving forces on the ground into favorable positions that hastened the defeat of the enemy.^{1}
In its landmark philosophical document, Warfighting (1989), the US Marine Corps gives the following description: “maneuver warfare is a philosophy for generating the greatest decisive effect against the enemy at the least possible cost to ourselves.”^{2} Methods to achieve this effect now also include deception, surprise, shock, and speed not only on the ground but from the air and from the sea.
Two of the foremost military thinkers in history, Sun Tzu and Carl von Clausewitz, spend attention to the idea in their respective famous works, The Art of War and On War. It remains a hotly debated and energetic topic in strategic circles.
The word maneuver, however, is quite versatile in its other wideranging uses and definitions, among them “an action taken to gain a tactical end,” “a clever or skillful action or movement,” and “doing something in an effort to get an advantage or get out of a difficult situation.”^{3} Our use of the word in this book will contain aspects of each of these definitions. For our purposes, we summarize with the following general description:
Symbolic maneuver includes any introduction, combination, movement, and/or manipulation of symbols (including diagrams) to gain an advantage in knowledge, insight, organization, clarity, efficiency, etc.
Undoubtedly both too broad a description and simultaneously not comprehensive enough, this definition will serve the goodenough purpose here to characterize what is one of the central and foundational features in algebra. In this chapter, we place a magnifying glass on this cornerstone idea.
MANEUVER IN ARITHMETIC
Symbolic stratagems to gain an edge extend beyond mathematics. The use of symbols in language can also be looked upon as a form of sophisticated maneuver. When we tell someone about a vacation last summer photographing waterfalls in Wells Gray Provincial Park (British Columbia), we are actually using language symbols to get around limitations.
We cannot physically recreate the waterfalls, forests, rivers, mountains, wildlife, adventures, and interactions with people that we experienced on the trip, but we can symbolically share them with others using the words we give them in language—to tell stories of our experiences.
Similarly, many of the techniques and algorithms we employ in elementary arithmetic can be looked upon as symbolic maneuvers. Consider multiplication: How much in ticket sales might we expect to earn from an event that is scheduled to be attended by 965 people at a cost of $175 per person? We can obtain a reasonable estimate of what the revenue should be by simply finding the answer to 965 × 175. One way to get this would be to take nine hundred sixtyfive 175s (one for each attendee) and add them together:
Though we can obtain the answer this way, almost no one would do so; it is simply too slow and far too painful. What we generally do is take the information (965 × 175) and maneuver it into another form. A millennium ago, folks might have used some sort of device such as the abacus or counting board to find the answer, but today we have several options.
We could take the numerals and directly key them into a calculator and have the machine do the multiplication for us, using procedures coded in electricity. Or we could reformat the information in a way that allows us to perform swift moves in writing (with the aid of a multiplication table) like so:
This is a symbolic maneuver that keeps us from having to do anything close to the original 965 additions—conveniently showing us that the revenue should be $168,875.
Symbolic maneuvering also happens when we add fractions with unlike denominators. Consider the addition of onehalf to onethird expressed as . Now, we may want to simply add the top two numbers together and the bottom two numbers together to obtain , but this value is not accurate in the most common interpretation of fractions.
Think of how much pie you will have if you take half of a pie and add it to a third of another—it is certainly more than twofifths of the samesized pie.
A way to obtain the correct value is to transform the two fractions into an equivalent form where they both have the same denominators and to add the top numbers together while leaving the bottom number the same. Doing so in this case will yield denominators of sixths, giving
This is different from the multiplication example, because converting the problem into a symbolic form is not enough. To keep the ball rolling requires that we further convert the fractions into their equivalents in sixths.
This can be likened to trying to do laundry in a coinoperated machine that only accepts quarters when you have a tendollar bill. We have to change the form of the money to make the machine work, but not its value or worth, which in this case translates to 40 quarters.
Let’s finally suppose that we’re asked to add up all of the counting numbers from 1 to 1000 (1, 2, 3,…, 499, 500, 501,…, 999, 1000) to obtain their total. Straightforwardly, this looks like
1 + 2 + 3 + 4 +… + 499 + 500 + 501 +… + 997 + 998 + 999 + 1000.
This would be quite a bit of work even with the assistance of a calculator. However, with a little bit of maneuvering, we can rearrange and rewrite the problem as
1 
+ 
2 
+ 
3 
+ 
4 
+ 
… 
+ 
497 
+ 
498 
+ 
499 
+ 
500 

+ 
1000 
+ 
999 
+ 
998 
+ 
997 
+ 
… 
+ 
504 
+ 
503 
+ 
502 
+ 
501 
In this form a symmetry is made bare, which shows us that if we add vertically first, as opposed to horizontally, we obtain repeated copies of 1001 (e.g., 1 + 1000 = 1001, and so on):
1 
+ 
2 
+ 
3 
+ 
4 
+ 
… 
+ 
497 
+ 
498 
+ 
499 
+ 
500 

+ 
1000 
+ 
999 
+ 
998 
+ 
997 
+ 
… 
+ 
504 
+ 
503 
+ 
502 
+ 
501 
1001 
+ 
1001 
+ 
1001 
+ 
1001 
+ 
… 
+ 
1001 
+ 
1001 
+ 
1001 
+ 
1001 
There are a total of 500 copies of 1001 in the addition (you can see this by looking at the entries on the top row from 1 to 500), which means that we can swiftly obtain the answer now by simply multiplying 500 × 1001 to obtain 500,500.
Here, a little maneuvering has turned a problem, which would take longer than 25 minutes for most to directly do even with the help of a calculator, into one that can be done by hand in as quickly as a minute.^{4}
These examples point to the fact that maneuvering in mathematics is no small thing and can lead to spectacular savings in the time it takes to find answers to certain types of problems. No less significant is the fact that maneuvering symbols (and thus the ideas they represent) can also lead to sensational gains in insight, clarity, organization, identification, and generalization, too!
The types of maneuvers just discussed depend critically on the specific situation at hand. Imagine a scenario in which we could standardize a much larger class of maneuvers, maneuvers that grant us the ability to systematically solve all kinds of seemingly sophisticated and unrelated problems—enabling us to convert some of the elegance and magic of mathematical ingenuity into routine. In a sense, this is what algebra injects into the mathematical bloodstream: providing a method to reduce the brilliant and extraordinary into the ordinary and reproducible.
Many sixteenthcentury mathematicians were simply awestruck by this gift that had fallen into their hands. François Viète was so taken with the sweeping possibilities of algebra that he stated, “The analytic art… claims for itself the greatest problem of all, which is: To solve every problem.”^{5} Another founding father of modern algebra, Girolamo Cardano, called algebra a “truly celestial gift” and boldly proclaimed that “whoever applies himself to it will believe that there is nothing that he cannot understand.”^{6}
Famed twentiethcentury college basketball coach John Wooden (ten national championships at UCLA) spoke of the phenomenon, of turning the sensational into the routine, in the context of his sport. Not a fan of using emotion and other devices to rise to the occasion in a game, Wooden preferred his teams to achieve a consistently high level of excellence through selfdiscipline, intelligence, and hard practice. His philosophy was to “let others try to rise to a level we had already attained.”^{7}
Let’s now see how learning the art of maneuver in algebra makes it possible to attain a consistently higher level of excellence in our mathematical game.
APPLES AND ORANGES
If we add 3 apples to 4 oranges, what do we get: 7 appleoranges? Clearly not, but if we add 3 apples to 4 apples, we do get 7 apples. In each case, we can physically join the collections together to obtain seven distinct objects: apples and oranges in the former case and just all apples in the latter. So why can we simplify the latter description and not the former? What’s the difference?
One key difference is that “apple/apples” can refer to any number of apples, and this allows us to absorb two separate descriptions (3 apples and 4 apples), which differ only in number not type of fruit, into the single description of 7 apples. Conversely, apples and oranges are fundamentally different fruits and no such absorption (combining 3 apples and 4 oranges into a single numerical description) is possible if we want to retain their distinction.
So, the fundamental fact that apples and oranges are two different types of fruit is encoded symbolically by the fact that their numerical descriptions can’t be absorbed into a single one.
In adding or subtracting the lettered expressions that represent numerical variation, we will be confronted by similar circumstances (from now on we will call such expressions varying, variable, or algebraic expressions). This will turn out to be both a great strength of algebra, in giving the subject its wide scope for handling the mixing of different and same types of behavior, and a great weakness, in that these new rules of operation can be overwhelming in education.
Also, because our native powers of recognition won’t be as automatically kind to us as they are in distinguishing different kinds of fruit, it will be harder to initially sift out and work with the various types of objects we will encounter. Let’s begin our investigation.
Can we simplify either of the following expressions: (a) 3x^{2} + 4x or (b) 3x + 4x?
Before answering, let’s first discuss what we mean by the phrase “simplify the expression.” To return to apples, we were able to simplify “3 apples + 4 apples” to “7 apples”: meaning that two descriptions/terms (3 apples, 4 apples) combine to become a single term (7 apples). In working with different fruits, we were not able to symbolically combine the two terms (3 apples, 4 oranges) into a single term without losing essential information.
In the algebraic cases, by “simplify” we mean can we combine either of the sums involving x’s and x^{2’}s into a single term. This is a tall order, for remember that, unlike fruit, both of these expressions can take on a myriad of different values, so the simplification must be equal to the original expression for each and every value that x can represent.
On a first attempt, one might try to combine the expression 3x^{2} + 4x into 7x^{3}: a common student choice. However, consider the following table outlining the results of these expressions for the four values 0, 1, 2, and 3:
Set x to: 
3x^{2} + 4x 
7x^{3} 
Same Value? 
0 
3 · 0^{2} + 4 · 0 = 3 · 0 + 0 = 0 + 0 = 0 
7 · 0^{3} = 7 · 0 · 0 · 0 = 7 · 0 = 0 
Yes 
1 
3 · 1^{2} + 4 · 1 = 3 · 1 + 4 = 3 + 4 = 7 
7 · 1^{3} = 7 · 1 · 1 · 1 = 7 · 1 = 7 
Yes 
2 
3 · 2^{2} + 4 · 2 = 3 · 4 + 8 = 12 + 8 = 20 
7 · 2^{3} = 7 · 2 · 2 · 2 = 7 · 8 = 56 
No 
3 
3 · 3^{2} + 4 · 3 = 3 · 9 + 12 = 27 + 12 = 39 
7 · 3^{3} = 7 · 3 · 3 · 3 = 7 · 27 = 189 
No 
Note that the dot (·) is a streamlined symbol for multiplication that avoids the potential confusion of the cross symbol (×) with x. Also, the exponent x^{2} means x · x and x^{3} means x · x · x.
The table shows that the two expressions give the same values when x is 0 or 1, but different values when x is 2 or 3. This means that these two expressions are not equivalent, and thus we can’t faithfully preserve all of the information stored in 3x^{2} + 4x by simplifying it to 7x^{3}.
This signifies that, mathematically, 3x^{2} + 4x ≠ 7x^{3} for most values of x (where “≠” means “is not equal to”). It can be shown in a similar fashion that 3x^{2} + 4x can’t be simplified to other expressions such as 7x^{2}, either. The variables x^{2} and x are like our apples and oranges—they represent fundamentally different types of variation.
What about the expression 3x + 4x? Can it simplify to 7x? Let’s construct another table like the previous one and see what happens:
x 
3x + 4x 
7x 
Same Value? 
0 
3 · 0 + 4 · 0 = 0 + 0 = 0 
7 · 0 = 0 
Yes 
1 
3 · 1 + 4 · 1 = 3 + 4 = 7 
7 · 1 = 7 
Yes 
2 
3 · 2 + 4 · 2 = 6 + 8 = 14 
7 · 2 = 14 
Yes 
3 
3 · 3 + 4 · 3 = 9 + 12 = 21 
7 · 3 = 21 
Yes 
This time, we see that the two expressions are equal for all four given values of x. In fact, it turns out here that the original expression and the simplified expression are equal for all values of x.
So in this case, the expression 3x + 4x and the result 7x are interchangeable because the final numerical value obtained from evaluating 3x + 4x can be faithfully preserved. We’ve added apples to apples.
Our task now is to figure out a more efficient way to determine when we can combine terms and when we cannot. What are the criteria? This will be obvious to some of you, but to many it may not be so obvious, and it is worth a bit more discussion: so they too may acquire a firmer grasp of the essential principle.
One way to conceptually sift out the key ingredients is by envisioning the variable terms as more familiar objects. Imagine a set of coins that use variables as their face values, represented as
and
Looking at the expressions 3x^{2} + 4x and 3x + 4x in this way gives
and
This perspective shows that we have two types of coins: and . As with the fruit, we can simplify an expression involving coins of the same type, whereas with coins of a different type we cannot:
and
These results match our earlier conclusions using tables. From this, we can immediately surmise that the value of the exponent is an important factor in determining whether we can combine two terms to become one. Although the variables involved both contain the letter x, it appears that the exponent must be the same in both terms or we will have different types of coins.
Let’s now consider the case of 6x^{3} + 7y^{3}. Here, the exponents are the same, but rendering them as coins still shows them to be two independent and different types of objects (when included in the same expression):
So, though it is true that the exponents have to be the same, it seems that the type of variation (represented by letters) needs to be the same, too.
Let’s add a little variety to the mix by looking at the following two expressions each containing two terms: (a) 12x^{2}y^{3} + 17x^{3}y^{2} and (b) 12x^{2}y^{3} + 17x^{2}y^{3}. Converting the variable parts to coins and simplifying where possible yields
and
Thus, we can simplify the second expression, but not the first. Based on these results, we are ready to make the following assertion: Two terms represent the same “fundamental type” of variation if they have the same variables each raised to the same powers respectively (and can therefore be simplified).
We see this at play in (b), where x and y are raised to the same respective powers in both terms and thus we are able to combine them to a single term.
Circling the variable works well as a visual guide in cases like these where we are dealing with exponents with whole number values, and where the letters are written in alphabetical order; however, it is just that—a guide.
Now that we understand the rule, we can make the following simplifications:
6x + 4y + 12x + 25y simplifies to 18x + 29y;
30x^{2}y^{5} + 40y^{4}z^{8} + 60x^{2}y^{5} – 25y^{4}z^{8} simplifies to 90x^{2}y^{5} + 15y^{4}z^{8}.
These simplifications are symbolic maneuvers that improve readability and clarity. They are the most standard of the fare in elementary algebra, but another essential type of simplification has so far been left out—a real gamechanger.
It comes from a property that is truly one of the unsung heroes of elementary arithmetic, one whose tracks are often cleverly masked in elegant algorithms such as long multiplication and long division. However, in algebra there is no more denying this property its place in the sun.
A HERO UNMASKED
It is a fact of arithmetic that
(where we interpret 3(4 + 2) to mean 3 times the sum 4 + 2).
This is straightforward to verify because on the lefthand side we have 3(4 + 2), which after adding the numbers inside the parentheses becomes 3(6) or 18, and on the righthand side we have 3 · 4 + 3 · 2, which becomes 12 + 6 or 18. Here is the long form:
• The expression 3(4 + 2) means three copies of 4 + 2 added together or (4 + 2) + (4 + 2) + (4 + 2).
• Dropping the parentheses gives 4 + 2 + 4 + 2 + 4 + 2.
• Rearranging the values gives 4 + 4 + 4 + 2 + 2 + 2 or
We have attached or distributed the 3 to both the 4 and the 2, and you might recall from arithmetic that this property is often called the distributive property of multiplication over addition, or put more simply the distributive property. It also holds if we replace the addition by subtraction, which yields
Both sides give the value of 6.
This property holds for all real numbers, meaning that another ensemble of infinitely many numerical expressions is thrust upon us. This ensemble includes the following expressions:
230(18 + 99) = 230 · 18 + 230 · 99,
where both sides equal 26,910 [having two hundred thirty (18 + 99)’s means that we individually have two hundred thirty 18s added to two hundred thirty 99s], and
12(11 – 7) = 12 · 11 – 12 · 7,
where both sides equal 48.
As in Chapter 1, we can capture the essence of the phenomenon and store the varied information algebraically by using a different letter (x, y, and z) to represent each of the three varying numbers that we have in these expressions like so: x(y + z) = x · y + x · z. Conceptually when x is a whole number, having x number of (y + z)’s means that we individually have x number of y’s added to x number of z’s. For instance, if we let x = 98, y = 115, and z = 345, this expression becomes 98(115 + 345) = 98 · 115 + 98 · 345, both sides of which equal 45,080.
You might ask: Why we would ever want to explicitly use this property? It seems quicker to add the two numbers in the parentheses first and then multiply (e.g., 3(4 + 2) → 3(6) → 18) instead of distributing first, multiplying each pair, and then adding (e.g., 3(4 + 2) → 3 · 4 + 3 · 2 →
12 + 6 → 18). Now if we were only adding apples to apples, such as we do in elementary arithmetic, you might be right, and this property could possibly stay hidden in the background. But in algebra sometimes we can’t simplify what is in the parentheses first, because we’re frequently dealing with different terms (apples to oranges).
For example, how do we simplify 3(x^{2} + x) + 7x^{2}? We can’t combine the terms inside the parentheses first as they are different types, so the usual arithmetic route will fail us here. However, we can still simplify this expression by using the distributive property:
After combining the two terms with x^{2}, this simplifies to
10x^{2} + 3x.
This simplified expression is easier on the eyes than the original expression. Remember that this simplification simultaneously signifies, in a single expression, infinitely many arithmetic simplifications all at once—one for each numerical value that x can represent.
KEEPING TRACK OF SIGNS
Engaging the distributive property also introduces another hidden difficulty from arithmetic. Once negative numbers enter the picture, the ideas get a little more complicated.
The symbol “–” serves a dual role as a minus sign and a negative sign, indicative of both an arithmetic operation and a part of a number (e.g., appearing when we subtract 6 – 4 and as a negative sign in –4 + 6). The language equivalent would perhaps be to use the same symbol both as a punctuation mark and as a letter in the construction of a word.
This is important because we can think of positive and negative signs as interacting with each other as well as with the operations of addition and subtraction. That is, just like two or more numbers can combine to form one number (e.g., 5 + 8 becomes 13), so can two or more signs combine to form a single sign. This is demonstrated in the case of multiplication here:
• Positive times positive becomes positive, or more compactly (+)(+) = +.
• Negative times positive becomes negative, or more compactly (–)(+) = –.
• Positive times negative becomes negative, or more compactly (+)(–) = –.
• Negative times negative becomes positive, or more compactly (–)(–) = +.
Some positive sign interactions are often implicit and not always traceable by symbols.
These interactions are on full display in algebra when we utilize the distributive property to simplify expressions like the following:
–6(x + 9) + 12(x – 5) + –4(x – 30).
Applying the distributive property like so
yields
(–6 · x + –6 · 9) + (12 · x – 12 · 5) + (–4 · x – –4 · 30).
Applying the rules of signs given above as well as those from addition/subtraction results in
–6x + –54 + 12x – 60 + –4x – –120,
which, after two “(+)(–) to –” interactions and one “(–)(–) to +” interaction, becomes
–6x – 54 + 12x – 60 – 4x + 120.
Collecting together and combining like terms (x’s to x’s and numbers to numbers) gives
which simplifies to
or
2x + 6.
Correctly simplifying this expression requires that we keep track of the distributive property and the rules for addition, subtraction, and multiplication of positive and negative numbers as well as the algebraic rules for combining variable terms of like types.
We can eliminate working through several of the steps in the previous calculations if we remember all of the interactions possible between the + and – signs, where the – sign simultaneously serves its dual roles. This is a skill worth developing and becomes a huge advantage once mastered, but it can be very confusing and frustrating until such mastery has been achieved.
A REMARKABLE CANCELLATION
Sometimes all of the variations in a problem can align themselves in ways that lead to remarkable cancellations. Let’s consider the simplification of the following expression:
5(x + 3) + 6(x^{2} – 3x) + –2(x – 5) + 15x + 3(–2x^{2} – 8),
where the values will distribute as indicated:
Multiplying these out and using the rules for positive and negative signs gives
5x + 15 + 6x^{2} – 18x – 2x + 10 + 15x – 6x^{2} – 24.
Collecting like terms together (x^{2}’s together, x’s together, and numbers together) yields
6x^{2} – 6x^{2} + 5x – 18x – 2x + 15x + 15 + 10 – 24.
Simplifying (including combining 5x + 15x to 20x and –18x – 2x to –20x) yields
0 + 20x – 20x + 25 – 24,
which gives
0 + 0 + 25 – 24
or
1.
This simplification may look routine at first, but it reveals a surprising result: All of the infinite variation in this problem no matter the value substituted for x will always combine and arrange in such a way to ultimately yield the number 1. Let’s demonstrate this for a few possible values of x.
For x = 10, the original expression becomes
5(10 + 3) + 6(10^{2} – 3 · 10) + –2(10 – 5) + 15 · 10 + 3(–2 · 10^{2} – 8).
We can simplify inside the parentheses first because we are working exclusively with numbers, and this results in
For x = 20, the expression becomes
We can see in each case that we ultimately end up with the same final result of 1. This will happen for the billions and trillions of other values (infinitely many in fact) that we can substitute for x. Each case uses a unique path to get there, but they all wind up simplifying to 1. Try it for a few more values yourself to get a better feel for what is happening.
Consequently, in doing the routine simplification of this single innocentlooking algebraic expression, we are not just doing the simplification for the two paths that the expression takes when x = 10 or x = 20. Rather, we are effectively doing, in one fell swoop, the simplifications for all of the infinitely many paths that the expression can ever take on!
Put another way, an infinite ensemble of individual arithmetical moves surprisingly all still have enough in common that the result always winds up yielding the value 1; and the basic rules of algebra allow us to easily communicate this grand fact.
This is an example of algebra making fascinating results look very routine.
We are now ready to put algebra to work to make sense of the number of days and age problem discussed in Chapter 1.
SEPARATING OUT NUMERICAL INTERACTIONS
In January of 1897, the great German chemist and discoverer of the element germanium, Clemens Alexander Winkler (in probably a take on the famous Shakespeare quote), stated: “The world of chemical reactions is like a stage, on which scene after scene is ceaselessly played. The actors on it are the elements.”^{8}
Related statements can be made of quantitative variations, with one possible adaptation being that the world of numerical variations is like a stage, on which scene after scene is ceaselessly played; the storytellers of it are the algebraic expressions we can construct.
We now want to use this stage to orchestrate an algebraic script that provides insight into the number of days and age problem from the previous chapter. As a refresher, you may want to try it again for a few different values:
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 threedigit 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.
Someone running through this problem at a normal pace will generate a fluid sequence of numerical calculations on their way to obtaining their very personal number at the end. A second and third person will do the same thing and so on, each producing a number related to them, but different from hundreds of others. It can seem somewhat magical how all of the various instructions, in the haystack of the many possible outcomes, wind up delivering that very special threedigit number for each individual.
If you read the problem carefully, however, you can see that there are numbers that stay the same in everybody’s calculations (constants) and values that can potentially differ from person to person (variables). That is, stability and variation are both present. See the two examples given for this problem in the previous chapter that generated the numbers 580 and 237, and compare them with your own.
When the problem is performed in the usual manner of simply picking a number and going through the procedure, we are unsuspectingly mixing together different types of behavior (namely, stability and specific instances of different types of variation): dissolving them collectively into the symbolic cauldron and losing critical information. That is fine if producing a specific value is all that we are interested in. However, it is the algebraic way of thinking that we are trying to apply to the situation now rather than the arithmetic way, which means that we want a more comprehensive and global understanding of what is happening in the problem—particularly why it generates threedigit numbers that are so personal to the many individuals participating in the process, all of which requires that we be more studied in how and what we mix.
In other words, we need to slow down or separate out the numerical interactions in order to isolate the values that are truly constant and unchanging in the problem from the values that are specific instances of a varying quantity. The question is, how do we do this?
Let’s begin by first identifying the constant parts of the problem and then separating them from the varying parts for the case of someone who had a birthday during the current year. The processes that are the same for every person engaged in the problem are multiplying by 4, adding 17, multiplying by 25, and adding 1588.
By contrast, the number of days and the year of birth will vary based on the specific participant. There is also a third value that stays the same for each person in a given year, but varies from year to year: the number of years away we are from 2013.
Thus, there are three quantities that can change from person to person and year to year in this problem, meaning that we can think of each of them as a different type of variation (or metaphorically as a different type of fruit). To indicate this, we will tag each variable with a different letter. To remain consistent with standard algebraic notation, let’s set x to represent the number of days the participant likes to eat out, y to represent the year of the participant’s birth, and z to represent the number of years it is past 2013. The algebraic rules for combination will then automatically ensure that the different types of variation retain their individuality throughout our analysis of the process and won’t dissolve with each other or the stable numbers. Let’s see what, if anything, we can learn by reframing the problem in this way.
The following table breaks up the problem into various steps and shows the accompanying expression in symbols, assuming that the person has had a birthday:
Step of the Procedure 
Expression at That Step 

Step 1 
Pick the number of days you like to eat out in a week (choose from 1, 2, 3, 4, 5, 6, 7) 
x 
Step 2 
Multiply this number by 4 
4x 
Step 3 
Then add 17 
4x + 17 
Step 4 
Multiply that result by 25 
25(4x + 17) 
Step 5 
Add the number of calendar years it is past 2013 
25(4x + 17) + z 
Step 6 
If you have had a birthday this year, add 1588 
25(4x + 17) + z + 1588 
Step 7 
Subtract the year you were born from this 
25(4x + 17) + z + 1588 – y 
All of the content of this problem is now neatly stored away in the final algebraic expression in Step 7. Look for yourself to see if you can find evidence of the four constant processes that stay the same for each person and the three variable processes that can differ from person to person and year to year. This expression visually captures, in algebraic language, the crux of all of the quantities and processes involved. However, in its present form, it is still difficult to see exactly how the procedure generates the personalized threedigit numbers that it does.
The good news is that this algebraic form is now operational and thus far more clearly possesses something that its language form lacks: maneuverability. This was one of the major goals we set out for ourselves, and we have achieved it! Let’s now leverage this maneuverability through applying the basic rules of algebra discussed in this chapter to see what is going on in this problem.
Simplifying from the original expression:
25(4x + 17) + z + 1588 – y
becomes by way of the distributive property
25 · 4x + 25 · 17 + z + 1588 – y,
and multiplying by 25 yields
100x + 425 + z + 1588 – y.
Combining the numbers (425 and 1588) gives
100x + 2013 + z – y.
The last expression gives us the look that we need. To more clearly illustrate this requires only that we rewrite it as Let’s examine how this expression looks for the example of Abu Kamil given in the last chapter. He likes to eat out five days a week, the current date is November 5, 2030, and since his birthday is February 6, 1950, he has already had a birthday this year. This information allows us to set x = 5 (days of the week), z = 17 (2030 is 17 years after 2013), and y = 1950 (year of birth). Plugging these into our algebraic expression yields that
becomes
or
500 + (2030 – 1950)
or
500 + 80.
Let’s take a closer look at 500 + 80 before completing the addition. The value 500 comes from multiplying the number of days Abu Kamil likes to eat out, in this case 5, by 100. This gives zeros for the last two digits. The number 80 represents his age and, once added to 500, gives us his magic number: 580. Thus, the first digit (reading from left to right) corresponds to the number of days he likes to eat out in a week (5) and the last two digits (80) correspond to his age.
This situation will always work for anyone less than one hundred years old. If you satisfy that criterion and have had a birthday in a given year, set the variables in the formula for your own circumstances and see how the scene plays out.
Let’s now analyze the formula to see why it works for all cases. Because z is the number of years since 2013, the expression 2013 + z gives us the current year. For instance, if the current year is 2021, then it is 8 years since 2013 (meaning z is 8), and 2013 + z becomes 2013 + 8 or 2021, which matches.
Because y corresponds to the year of birth of the participant, the expression with words becomes
or more clearly (current year – year of birth).
For someone who has had a birthday in the current year, this gives the age of the person for that year. For someone less than one hundred years old and 10 years of age or older, this will always be a twodigit number.
Finally, the term 100x from the formula will generate the numbers 100, 200, 300, 400, 500, 600, or 700, corresponding respectively to the number of days of the week (x) being equal to 1, 2, 3, 4, 5, 6, or 7. Each of these seven numbers has “memory space” for accommodating the addition of two digits in their final two slots.
Because the ages for persons 10 years of age or older but less than 100 years old are two digits long, their value is preserved perfectly in this addition (just as the 500 preserved the 80 in 580); and the final result satisfies the condition that the first digit will equal the number of days the person said they like to eat out in a week and the last two digits will be their age.
This will also be true for those less than 10 years of age as well if we interpret 00, 01, 02, 03,…, 09 as 0, 1, 2, 3,…, 9 years, respectively.
Why won’t the problem work for someone who is at least one hundred years old? It fails in such a case because the participant’s age will then correspond to a threedigit number, which will force a change in the first digit of the round numbers 100, 200,…, 700.
For instance, let’s say that a 102yearold person likes to eat out six days in a week. Plugging in their values for x, y, and z and simplifying will lead to 600 + (the age of the participant) or 600 + 102. The 1 in 102 will change the 6 in 600 to a 7, thus giving the threedigit number as 702. This implies that the person likes to eat out seven days a week and is two years old, which we know to be false, and so the conclusion of the problem is no longer accurate. This scenario of a change in the first digit will always occur for anyone one hundred years of age or older.
You’ll notice that we’ve focused our analysis on cases where the participant has already had their birthday in the year in question. A slightly different expression is yielded for someone who has not had a birthday yet. The only difference is that 1587 is added instead of 1588. Our expression then becomes Rewriting the part of this expression in parentheses in words yields
or (previous year – year of birth). This is the age of someone who has yet to have a birthday in the current year. This will still be a twodigit number for someone younger than 100 years of age, from which the rest of the analysis follows verbatim.
CONCLUSION
What are we to make of all of this?
A relatively simple application of a foundational principle of elementary algebra has allowed us to crystallize a process whose explanation on first sight is anything but obvious to most. This illustrates the potential gamechanging ability of algebra to be an effective tool in revealing the hidden structures in numerical procedures in ways that enable us to better understand them.
Before moving on, let’s analyze just a bit more this idea of mixing and not mixing instances of variation and stability.
Consider the following sets of information:
• First Set: [5 · 2 = 2 · 5]; [6 · 3 = 3 · 6]; [8 · 5 = 5 · 8]; [(–16) · 4 = 4 · (–16)]; [25 · 40 = 40 · 25].
• Second Set: [5 + 5 = 2 · 5]; [9 + 9 = 2 · 9]; [20 + 20 = 2 · 20]; [–32 + –32 = 2(–32)]; [500 + 500 = 2 · 500].
If we complete the additions and the multiplications for both cases, we obtain the following information:
• Third Set: [10 = 10]; [18 = 18]; [40 = 40]; [–64 = –64]; [1000 = 1000].
It is clear that the information in the third set can come from completing the calculations in either of the first two sets. Ask yourself: Which of the three sets of data conveys more information?
The first offers several instances of a fundamental property of multiplication—namely, the order in which we multiply two real numbers does not change the result we obtain. The second set gives us examples of another fundamental fact of real numbers: If you add a number to itself, this is equivalent to multiplying the number by 2. The third set of data gives several instances of the fact that a number is equal to itself.
Though all three sets are true, the first and second show what path we take to get to the result, therefore conveying to us more structural information about real numbers and how they work. In the third set, all of the detail in the previous two sets gets lost by reducing to the simple statement that a number equals itself. This is obvious information that doesn’t tell us anything new. Thus, whereas all three sets of data are true, there is something to be gained by keeping the information in a form that shows the bare relationships, or the true nature of the variation, as opposed to only seeing the ultimate result.
The equalities in the original two sets are, of course, representative of infinitely many similar expressions. In both cases, algebra can be used to show what is changing as well as what is not.
In the first set of information, the things that can vary are the two values being multiplied, while the idea that remains constant is the fact that switching the order of the multiplication doesn’t change the result. By simply tagging the two variations with the letters x and y, we are able to capture and store it all with a single expression: x · y = y · x. From this form, the commutative property of multiplication is transparent, and the entire ensemble of occurrences of this property can be generated.
In the second set, the things that are changing are the values being added to themselves on the lefthand side of the equals sign and being multiplied by 2 on the righthand side. The items that remain constant are the addition and the number 2. In this case, we simply tag the single varying number by the letter x and everything else is captured and stored by writing x + x = 2 · x or x + x = 2x. To get the specific occurrences in the second set, all we need to do is set x equal to 5, 9, 20, –32, and 500.
These are basic examples of how information can be lost in mathematics by simply performing calculations to the end without thinking about how they work. Sometimes if we want to better understand a phenomenon in a more fundamental way and capture its essence, it is best if we preserve the trail of how some of that information was obtained.
This is a critical difference between the way we think in algebra and the way we usually think in arithmetic, and it explains why algebra is taught subsequently to arithmetic. Algebra is more interested in preserving and analyzing the relationships between numbers and operations, rather than only focusing on the calculation of specific values—it is in those relationships that lie the keys to comprehending the inner workings of the processes and configurations that we seek to better understand.
With its ability for tagging and holding different types of variation in suspended animation (amidst a tsunami of numerical data), algebra is tailormade for this activity. It is the subject’s “default setting,” if you will.
As originally presented, the result of the number of days and age problem appears to work like magic, but by using a little bit of algebra, the procedure has now been tamed through a relatively straightforward and simple process. One of the requirements is that we preserve the different variations in the problem by tagging them with letters (creating an algebraic expression), followed by simple maneuvers according to straightforward algebraic rules. This allows us to mix together objects of the same type and keep separate the objects of different types. At the end of it all, we obtain a form of the expression that allows us to completely explain what is going on. It is somewhat remarkable that it all works in such a routine way—so routine, in fact, that the subtleties of what is truly happening can be easily missed.
This is but a basic use of algebraic machinery. There are much more powerful ways to demonstrate and program it. In physics, for example, we can perform experiments analyzing a certain type of behavior observing how it changes as we vary properties such as temperature, mass, or velocity and, in the process, generate an ensemble of values interconnected around a central theme. Physical theory can then be used, with algebra as an essential ally, to find an expression that generates the given values or provides effective estimates of them (i.e., mathematically explains the results of the experiments and makes new predictions).
The same may occur in many of the natural sciences and in other important areas, where statistics, operations research, and data science may be called upon to play key roles.
Thus, algebra is indeed “electrical soil.” Once something is placed into algebraic form, it enters the mathematical network and acquires the potential to be maneuvered in ways that does remind one a bit of the way that electricity is used to facilitate the transmission of information.
Think of it, a camera phone can capture a complex physical scene as an electrical pattern (via a digital image) and then transmit this image at breathtaking speed to another phone 1000 miles away—in what amounts to a maneuver via technology that elegantly overcomes the difficulties of communicating at great distances. Similarly, algebra can be so used to first capture a complicated numerical variation as a symbolic pattern, then maneuver this pattern into a repackaged, more readable form to great advantage.
This concludes the first movement of the book, where our goals have been to illustrate how to give a mathematical shape to numerical variations, and then learn how to perform basic maneuvers on these symbols to gain additional information and insight. Hopefully the narrative up to this point has made the purpose of these fundamental ideas a little clearer and has nontrivially hinted at their significance.
Now the time has come to transition to the next phase of our journey, where we will come upon what has traditionally been considered the core of elementary algebra, its very heart and soul even: the art and science of solving equations. Historically, it is in this arena where the maneuvering of expressions to acquire new information became something truly spectacular—ultimately reaching such commanding heights that even to this very day they come darn close to obscuring in many people’s minds everything else about the entire subject.