11

A Kaleidoscope of Ingredients

Many things of interest to mathematicians or engineers are very complex and have several complicated structures layered on top of one another. As we do in other things in life, we like to take things apart into more elementary building blocks so we can unravel this rich structure.

To do such unravelling, we view these complex things as a superposition of much simpler ones.… Decomposing into simple building blocks can be done in many different ways in mathematics.

—Ingrid Daubechies, interview for The Intrepid Mathematician

Elementary algebra turns out to contain, in the small, examples of ideas that in the large have played central roles in mathematics and science. Though many of these roles have their historical origins much further up the mathematical and scientific food chains (and certainly are not close to being contained in entirety in these small examples), the kernel of some of the ideas, upon revisit, are present or reflected in basic algebra. The aim of this chapter is to discuss a few of those kernels.

SEPARATING OUT NUMERICAL INTERACTIONS (REPRISE)

In 2002 University of Alberta mathematician David Pimm wrote:

Transformation is a key power of algebra, the most important means for gaining knowledge… In algebra, we can use symbols to express a perceived relationship and then manipulate the result to obtain new information and insights. Indeed, these transformations embody the essential power of algebra because, due to their concise and non-physical nature, symbols are much more easily manipulable than the things they represent.¹

This has certainly been on display throughout the book. One of the best examples of this occurred in Chapter 5, where we encountered word problems that were initially presented to us as a mix of pure English and verbalized numerical relationships. Our task there was to distill the quantitative essence out of such presentations.

Algebra provided the symbolic language for this process, which in that chapter I termed the “jump for joy stage”—not an official mathematical phrase, of course, but somewhat popular with many of my algebra students who were living the experience in real time. Once this stage is reached, a wide host of options became available to us.

The most important and dramatic part of it all is the entire track of transformation from the presentation of the word problem in original form to the “jump for joy stage” (where the equation describing the situation is obtained), on through the symbolic maneuvers that reduce the equation to its simplest form, yielding the numerical value for the unknown and then, where applicable, cascading it back to find other unknowns as well.

But there are also other interesting items of note along that track, if we care to take a gander.

One occurs during the simplification phase of the equation, the point just after we have obtained the equation in its raw form (jump for joy stage) and simplified it, yet before the point where we start to move objects from one side of the equation to the other.

Let’s go back and revisit Word Problem 7 in Chapter 5, which states:

A total of $17,252 is divided up into one-, five-, and ten-dollar bills. If the number of fives is seven times the number of ones and the number of tens is quadruple the number of ones, how many of each type of bill are there?

For this problem, the regime of interest is from

1(x) + 5(7x) + 10(4x) = 17252

to

x + 35x + 40x = 17252

to

76x = 17252.

Now, instead of going on through and solving the equation, let’s stop and ask the question: Can the simplified expression 76x on the left-hand side of the equation tell us anything in the context of the problem?

If you remember, in the original problem, there are three unknowns that harmonize according to the interrelationships specified—“the number of fives” is “seven times the number of ones” and “the number of tens” is “quadruple the number of ones.” In a sense, these unknowns are really three moving targets that vary or dance in concert together, and along with them the total amount of money we have varies:

	Number of Bills	Amount of Money (in Dollars)
One-dollar bills	x	1x
Five-dollar bills	7x (seven times the number of ones)	5(7x) = 35x
Ten-dollar bills	4x (quadruple the number of ones)	10(4x) = 40x

Our ability to tag the three unknown dollar bill amounts as relationships in x demonstrates that for this problem each of the three unknowns varies in the same fundamental way. Or put algebraically, the variations that describe each amount are of the same fundamental type—each has a part that is some multiple of x (some number times x).

What the simplified expression essentially tells us is how all of that coordination and variation collectively mix together. That is, after combining the like terms together, the net result is equivalent to getting a bang of $76 in value times the number of one-dollar bills we have. This means that we can momentarily, if we want, switch how we look at the problem, viewing it now not as having three separate denominations of paper money but instead as a problem involving one new enhanced denomination that consists of hypothetical 76-dollar bills (whose internal composition breaks down according to the originally prescribed relationships between the three denominations). And now the question becomes, how many 76-dollar bills are needed to give a total of $17,252?

In this momentary viewpoint, it is important to note that the number of hypothetical 76-dollar bills matches the number of actual one-dollar bills. One of our 76-dollar bills contains 1 one-dollar bill plus 7 five-dollar bills and 4 ten-dollar bills. The following diagram illustrates the components:

If x = 10 (there are ten 76-dollar bills), then the total amount of money we have is 76(10) = $760. Now, if we wanted to see how this breaks down via the internal structure of the hypothetical enhanced bill, we simply multiply each of the constituents by ten. This yields 10 one-dollar bills, 70 five-dollar bills, and 40 ten-dollar bills for a total of $760. Another way to think about this is that if we had ten physical 76-dollar bills, we could take them to the bank and have the teller give us change in the following amounts as 10 ones, 70 fives, and 40 tens:

If x = 75 (there are seventy-five 76-dollar bills), then the total amount of money we have is 76(75) = $5700. Internally the money actually breaks up into the following amounts: 75 one-dollar bills, 525 five-dollar bills, and 300 ten-dollar bills:

Try this interpretation out on the actual solution scenario for Word Problem 7 and see if you can organize the number of bills into 227 groups of “76-dollar bills,” each consisting of 1 one-dollar bill, 7 five-dollar bills, and 4 ten-dollar bills according to the prescribed interrelationships.

This hypothetical viewpoint wasn’t what we were particularly interested in at our initial reading of the problem, as we just wanted to use the expression to find out what value of x makes 76x yield $17,252, and from this obtain the breakdown into the combination of one-, five-, and ten-dollar bills needed to satisfy the stated interrelationships. But this particular interpretation emerges from the algebraic maneuvers and is there for the taking if we so desire. It also yields a couple of observations that are worth mentioning.

INTERPRETATIONS OF THE MEANING OF AN EQUATION OR ITS SOLUTIONS

For the problem just discussed, a much clearer view of what is happening was obtained by coalescing together into one combined object the individual pieces of information related to the three denominations of money. Moreover, that combined object (76x) had an interesting interpretation all on its own.

Once the jump for joy stage was reached, we could have, before combining the values, tried to find a value of x all at once such that x + 35x + 40x added up as three individual components to equal 17,252. But that would be a harder thing to do versus first forming the combined object to yield the singular 76x and then dividing 17,252 by 76 to obtain 227. Afterward, it is a simple matter to go back and find out how many of each of the constituent denominations there are.

This notion of combining a bunch of interacting objects into a new transformed object is very much present in higher mathematics and science. And it can work in reverse as well, in that we may want to break up a bigger object into meaningful subcomponent pieces, study these smaller pieces first, and then reconstitute what is found to gain new and impactful information about the larger object (see the discussion later in this chapter).

This type of interpretation can also be used in a similar manner on the combined objects in the other word problems in Chapter 5. For those problems, however, there is a slight difference in that they consist of two fundamental components instead of one component, as demonstrated by the single term 76x for the bills problem. In looking at those other problems, it can be observed that adding up all of the relationships for the various unknown parts of information in each respective problem ultimately simplifies to an expression of the form (some number) times x + (a constant value).

This can be seen with Word Problem 3 in Chapter 5:

A 570-foot rope is cut into three pieces. The second piece is four times as long as the first, and the third piece is ten feet longer than twice the first piece. How long are the three pieces?

The regime of interest for this problem is

x + 4x + 10 + 2x = 570

to

7x + 10 = 570.

Before going any further, let’s ask what types of interpretation can be given to the simplified expression 7x + 10 on the left-hand side of the equation. We will give a couple.

One way to think of this is to imagine the full 570-foot-long uncut rope that we want to subdivide according to the stated relationships between the three pieces. We can guess a value for the length of the first piece, mark it, and then measure out seven such lengths (including the first piece). We then see if the remaining length of rope after the last mark is 10 feet, to yield the entire 570 feet of rope.

Of course, going ahead and solving the equation will give us the desired value of 80 feet in length (x = 80). Once known, we can measure out 80 feet of rope and cut it; this gives the first piece. Then, we can measure out 4 times 80 or 320 more feet of rope and cut it to get the second piece. This would then leave the third piece, which should measure 2 × 80 = 160 feet plus the additional 10 feet to give 170 feet, which of course matches our earlier determination.

Another, more radical way to interpret the situation—described by 7x + 10 = 570—is to look at the three individual unknowns in the problem as combining to give a collective bang of 7 feet. If the rope lengths were money as in the last problem, the equivalent would be to think of each of these 7-foot lengths as a seven-dollar bill. In order to include all of the information in the problem, however, we must hold back in reserve 10 feet of rope to be added to the third piece at the end.

In this viewpoint, we can then ask, how many 7-foot-long rope segments must we have such that at the end, when we add the 10-foot segment of rope held in reserve, we have a combined total of 570 feet of rope? Solving this equation tells us that we will have 80 such segments—which gives 560 feet—plus the 10 feet of rope.

Once we find the number of rope segments, we can imagine taking each 7-foot segment and breaking it apart according to the initially given interrelationships between the pieces: the second piece is four times as long as the first and the third piece is twice the first. This will yield a breakdown for a given 7-foot segment into the first piece being 1 foot long, the second piece being 4 feet long, and the third piece being 2 feet long:

[Image provided courtesy of William Hatch]

Continuing on with this hypothetical situation, we can now reimagine the three unknowns as being represented by three containers. We cut and collect in the container for the first piece 1 foot of rope from each of the eighty 7-foot segments. This yields 80 individual 1-foot-long segments of rope that sum to a collective total of 80 feet. We cut and collect in the second container the 4-foot segments from each of the 80 segments to yield a total of 320 feet of rope. In the third container, we collect the remaining 2 feet of rope from each of the 80 segments to obtain 160 collective feet of rope, to which we also add the additional 10 feet of rope held initially in reserve to give a final value of 170 feet of rope. Check that the total amount of rope assigned for each of the three containers given here matches the correct values for the three pieces discussed earlier and in Chapter 5.

This time, however, unlike the first interpretation presented, there is a big difference between the problem as initially stated and this second hypothetical interpretation. In the former, three continuous pieces of rope that satisfied certain specified conditions were asked for and then discovered to be of lengths 80 feet, 320 feet, and 170 feet. But now, in this second interpretation, we are dissecting the rope into a great many segments and then aiming to find the sum total length of rope that we must place into each of three containers. This construction ends up yielding 80 individual 1-foot-long segments of rope in the first container, 80 individual 4-foot-long segments of rope in the second container, and 81 segments in the third container (80 individual 2-foot-long segments plus 1 segment 10 feet long). This is quite a different physical situation.

Thinking of the problem in either of these vastly different physical interpretations presents no issue, as long as all we ultimately want is to use algebra as an aid in solving the problem as originally stated. It would be quite a different matter, on the other hand, if we were trying to instruct someone on how to physically perform the desired construction.

One may skeptically question the realism of either of these problems. But as mentioned in Chapter 6, whether or not the problems themselves are realistic is not as important as the educational illumination that they can sometimes cast on the relationships and situations that do occur with great force and relevance elsewhere.

Science and engineering offer many examples where symbolically maneuvering equations, expressions, or data or changing the viewpoint or interpretation of their solutions (or forms) can yield pay dirt. One of the most dramatic examples of this occurred in the 1860s in the field of electromagnetism. The star theoretician James Clerk Maxwell pieced together and wrote the foundational narrative of the subject by synthesizing together a group of principles and equations much of which had been discovered and articulated earlier in the century, in a piecemeal fashion by many prominent figures (some of whom were mentioned late in Chapter 10). To this he added his own essential contributions and extraordinary insight, an insight often compared with those of Newton and Einstein themselves.

Maxwell then manipulated the equations, looking at them from various viewpoints, and in the process discovered that they predicted the existence of hypothetical electromagnetic waves. From this, he proceeded to compute the speed of these mathematical waves and found that their predicted speed matched that of the known speed of light, which had been calculated through other independent means starting with Danish astronomer Ole Rømer’s successful approximation in 1676 (and known now to be about 670+ million miles per hour or 186,000+ miles per second).

Maxwell realized that the shock of this coincidence was too much to be accidental and hypothesized that visible light itself must be a particular version of these electromagnetic waves. Moreover, his analysis revealed that there should be other electromagnetic waves with different wavelengths than that of light.

At the time, through earlier experiments that had extended information to just beyond both ends of the visible light spectrum (red and violet), infrared light and ultraviolet light had already been discovered. Maxwell hypothesized that these too must be electromagnetic radiation, and furthermore predicted the existence of other electromagnetic waves. These further predictions were confirmed in the late 1880s, 1895, and 1900 with the discovery of radio waves, X-rays, and gamma rays, respectively.

So, though the simple re-interpretations we engaged in with our simplistic word problems here are quite trivial in terms of direct meaning, imagine for a moment if they were actually different viewpoints to some scientific analysis, and that the alternate way of looking at the money problem—as 76-dollar bills—or the rope problem—as 80 segmented pieces—opened up a whole new way to look at some phenomenon; and you will get a small taste of what can and sometimes does occur in science and mathematics.

It goes without saying that Maxwell’s interpretations regarding the existence of electromagnetic waves is about as revolutionary an insight as you will find in any science, and the subsequent applications of such waves—radio, television, X-rays in medicine, and microwave ovens—have literally transformed our world.

CLOSURE

Let’s now look a bit closer at the fact that the word problems in Chapter 5 all simplified to expressions of the same form: namely, (a number) times x + (a constant value).

If we look at each of these simplified problems—from jump for joy stage to simplified form—as a different scenario, we can use parameters to represent them all by expressions of the form Ax + B. (We use uppercase letters here to distinguish these “dual-acting parameters,” as will become clear later.) For the time being, we are only looking at the left-hand side of those equations.

Thus, for the rope lengths situation we would have A = 7 and B = 10, which would give the form 7x + 10 on the left-hand side; and for the money situation we would have A = 76 and B = 0, yielding 76x + 0 or simply 76x. For Word Problem 4 from Chapter 5, we have that A = 5 and B = 20, and for Word Problem 5 we would have A = 8 and B = –302, corresponding to 5x + 20 and 8x – 302, respectively. All five of the problems in Chapter 7 simplify to the situation 16x + 10, which corresponds to A = 16 and B = 10.

The fact that the individual pieces of information in the various word problems all combine to an expression of the same form is representative of an important mathematical property called closure under an operation. Before discussing this a little more in the context of the word problems, let’s look at closure in a few other scenarios:

1. The even numbers are closed under the operation of addition (under addition): Two even numbers added together always yield another even number.

2. The odd numbers are not closed under addition: Two odd numbers added together don’t yield another odd number but an even number.

3. The integers (…, –3, –2, –1, 0, 1, 2, 3,…) are closed under both addition and multiplication:

° Two integers added together always yield another integer.

° Two integers multiplied together always yield another integer.

4. The nonzero integers (…, –3, –2, –1, 1, 2, 3,…) are not closed under division: Dividing one integer by another integer does not always yield another integer. For example:

° is an integer.

° is not an integer.

° is not an integer.

5. The common fractions (or rational numbers), which can be written as (or using alphabetic symbols as where both x and y are integers with y ≠ 0), are closed under addition, subtraction, multiplication, and division (where defined): Two rational numbers added to, subtracted from, multiplied by, or divided by one another (where defined) always yields another rational number. For example:

°

°

6. Expressions of the form Ax + B (where A and B are real numbers) are closed under addition.

° Take two such expressions 5x + 70 and 11x + 20, and add them together to obtain (5x + 70) + (11x + 20) = (5 + 11)x + 70 + 20 = 16x + 90, which is still of the same form Ax + B with A = 16 and B = 90.

° This holds in general. Take two general expressions ax + b and cx + d, where a, b, c, and d are real numbers. Adding them together yields ax + b + cx + d, which can be simplified to (a + c)x + b + d. This is still of the same form Ax + B with A = a + c and B = b + d. In the previous example, we have a = 5, b = 70, c = 11, and d = 20.

What, if anything, is to be gained by knowing that expressions that can be written in the form of Ax + B are closed under addition? Two things immediately come to mind.

Firstly, a general method of solution can be implemented for all of the word problems posed in Chapter 5, as well as thousands more similarly posed problems. That is, if you know how to solve equations such as 76x = 17292 or 7x + 10 = 570, then the general equation Ax + B = F can be solved in a similar manner. We have introduced the parameter F on the right-hand side of the equation to complete the picture. In the two equations just shown, F is equal to 17,292 and 570, respectively.

The general solution to this equation can be found as follows:

Once we find x, we can of course cascade it to find other unknown pieces of information according to the relationships specified in each particular problem. For instance, for Word Problem 4 in Chapter 5, the simplified form is 5x + 20 = 180. This gives A = 5, B = 20, and F = 180. Thus, putting these values into the simplest form for x in the reduction diagram gives

From this we can cascade it according to the given relationships to find the two other angle values.

This general form can now be used in any similar type of problem whatsoever; thus, in one grand gesture we can represent the solutions to all of these problems by this all-encompassing symbolic maneuver.

We see this in the brief description of quadratic equations in Appendix 1, where it is mentioned that all quadratic equations in any form are ultimately of the same type. They can be described by the one super-equation ax² + bx + c = 0 and then solved by the same technique, which ultimately yields the quadratic formula.

The second thing that comes to mind is that similar types of interpretations as we have given for the money and rope piece problems can also be given to problems expressible in the Ax + B form.

Let’s pose a more general version of Word Problem 7 involving the money denominations discussed previously, complete with parameters: A total of F dollars is divided up into one-, five-, and ten-dollar bills. If the number of fives is a times the number of ones and the number of tens is b times the number of ones, how many of each type of bill are there?

We can set this problem up as before with the following breakdown:

	Number of Bills	Amount of Money (in Dollars)
One-dollar bills	x	1x
Five-dollar bills	ax (a times the number of ones)	5(ax) = 5ax
Ten-dollar bills	bx(b times the number of ones)	10(bx) =10bx

We know that adding up the money from all three bill types must yield the total amount of money (now given by F dollars). Comparing side by side the progression from the jump for joy stage to the simplified form for both problems yields the following:

Steps in Original Concrete Word Problem	Steps in General Word Problem
1(x) + 5(7x) + 10(4x) = 17252	x + 5(ax) + 10(bx) = F
x + 35x + 40x = 17252	x + 5ax + 10bx = F
(1 + 35 + 40)x = 17252	(1 + 5a + 10b)x = F
76x = 17252	Ax = F

Note that for the form Ax + B = F, we have here that A = (1 + 5a + 10b) and B = 0.

Using a similar interpretation as we did for the hypothetical 76-dollar bills, we now have a hypothetical (1 + 5a + 10b)-dollar bill that breaks down as follows:

Note that for a = 7 and b = 4, this becomes a 76-dollar bill, and this diagram becomes equivalent to the original diagram we presented. However, this new diagram is far more general. So, for instance, if we let a = 9 and b = 11, we get a hypothetical 156-dollar bill that breaks up into 1 one-dollar bill, 9 five-dollar bills, and 11 ten-dollar bills.

We are not completely free to choose the parameters at will in this money scenario, however, as they have to all be positive values and we additionally have to make sure that (1 + 5a + 10b) divides evenly into F. For example, F = 1248 will work in the 156-dollar bill case, yielding 8 one-dollar bills, 72 five-dollar bills, and 88 ten-dollar bills. This requirement is based on the fact that the problem has to both be logically consistent and describe possible situations regarding money in US currency—we can’t have a negative number of bills or a bill denomination that corresponds to one-fifth of a dollar. But that still leaves quite a bit of variety in the types of word problems we can analyze—differing in the details but similar in type and interpretation.

Similar setups could be done for the other word problems in Chapter 5, with additional modifications for when B is nonzero. We are assured of this because of the fact that we know all such problems are of the same type and can ultimately be simplified to and represented by the same type of equation (the closure property): Ax + B = F. Closure allows us to corral them all under a single tent.

Closure under an operation turns out to be a crucial component in many other more advanced kinds of algebra, along with other types of important properties. These systems are part of one of the most important regions of mathematics, abstract algebra, which includes such fields as group theory, ring theory, field theory, module theory, representation theory, and linear algebra. Astonishing applications both within and outside of mathematics have been uncovered for some of these areas, particularly group theory, representation theory, and linear algebra. Such applications have occurred in a wide variety of places, including geometry, topology, quantum physics, particle physics, relativity, solid state science, chemistry, some forms of spectroscopy, data science, internet search algorithms, cryptography, error correcting codes, combinatorics, and so on.

A mighty river indeed this abstract algebra is, with many powerful tributaries feeding it, yet the symbolic innovations in the development of basic algebra occurring in the sixteenth and seventeenth centuries remain a primary direct source. The simplest renditions in this groundbreaking achievement live on in perpetuity in the schoolroom algebra of today.

QUANTITATIVE COCKTAILS AND ATOMIC SPECTRA

Let’s reverse gears and consider situations where complex encoded information can be broken down into meaningful subcomponents for great gain. It is an especially general and cross-disciplinary idea with spectacular materializations across the breadth of human experience. We begin with the highly interdisciplinary subject of spectral analysis.

In the late 1850s, famed German chemist Robert Bunsen—of Bunsen burner fame—wrote to a friend:

At the moment I am occupied by an investigation with Kirchhoff which does not allow us to sleep. Kirchhoff has made a totally unexpected discovery, inasmuch as he has found out the cause for the dark lines in the solar spectrum and can produce these lines artificially intensified both in the solar spectrum and in the continuous spectrum of a flame, their position being identical with that of Fraunhofer’s lines. Hence the path is opened for the determination of the chemical composition of the sun and the fixed stars with the same certainty that we can detect [strontium chloride], etc., by our ordinary reagents. By this method the chemical elements occurring upon the earth may also be detected and separated with the same degree of accuracy as upon the sun…²

A landmark moment in the history of astronomy—indeed all of science—that opened to the world in a truly systematic way the vast field of spectroscopy. Astronomer Carl Sagan described it thus in 1980: “Astronomical spectroscopy is an almost magical technique. It amazes me still.”³

What is it about the subject that so excited Bunsen and Sagan in their respective eras as well as legions of other scientists up to the present? Undoubtedly, much owes to the subject’s clever ability par excellence to extract detailed information in signals from unbelievably remote and inaccessible sources. The manner in which it has been used to reveal chemical and physical knowledge of the heavens, as well as so much here on Earth, is reminiscent of the abilities of mathematics as a diagnostic instrument in analyzing a broad array of questions, with algebra and arithmetic being essential tools in these investigations.

The discipline stands up well against the very best that science has to offer, with nearly two dozen individuals having received Nobel Prizes for discoveries, revolutionary perceptions, and inventions that have been related to some aspect of spectroscopy.⁴ Here, we try to give just a tad bit of insight into this capital and highly relevant field by coming at it tangentially through a few of the examples discussed in this text—using them as conceptual fuel.

In 1666, Isaac Newton opened the door to the world that spectroscopy would become by passing white light through his prism and uncovering the fact that it was not at all the single ingredient that it appears to be, but rather a diverse cocktail of the major colors of the rainbow. The prism naturally takes the combined object, white light, and splits it into its constituent colors or ingredients. Later it was revealed that these different ingredients can be distinguished and identified by their wavelengths, which determine the differences in how they pass through the prism. The following numerical scale shows the sizes of some of these wavelengths, which are extremely small, in Angstroms (1 nanometer = 10 Angstroms or approximately 0.00000000328 feet), abbreviated Å:

Wavelengths given in Angstroms [Image from UT Austin–Principles of Chemistry (Li et al. 2018)]

From this spectral reading, we can see that the colors visible to the human eye have wavelengths that range from violet (around 3900 Angstroms) to red (around 7400 Angstroms).

Can we do this with the quantitative cocktails we have discussed? That is, can we look at the blended information obtained there and definitely determine the exact contributions from each of the constituent ingredients that make it up?

Specifically, in the language of course averages, given an overall course average for a student of 80%, can we figure out the exact scores that this student made on all of the homework assignments, each of their tests, and the final exam? Is there a mathematical prism that will allow us to unambiguously break up that 80% score back into its component parts?

Our discussion in the first part of Chapter 10 bore critically on this question. There, we saw that, even if we dial it back from every individual assignment to just the averages on each of the three categories of homework, tests, and the final exam—at contributions of 20%, 60%, and 20%, respectively—the answer to this question is still in the negative. We found that the equation 0.20x + 0.60y + 0.20z = 80 was indeterminate, and that there were many possible varying combinations of scores that could all coalesce to yield the same course average of exactly 80%. So, without more information, we can’t recapture exactly how the individual component scores combine to yield the 80% score.

What about the number of days and age problem and the data generated there?

Consider the situation with the same criteria as discussed at the beginning of Chapter 8, with the date being June 30, 2021. For those who satisfy the criteria of the problem, being younger than 100 years of age and having had a birthday by this date, can their number (without knowing the person) be parsed out into the two ingredients that combine to make up its value—namely, the number of days they like to eat out in a week and their year of birth?

The answer this time is in the affirmative, due in part to the fact that though these ingredients combine to form a three-digit number, they don’t interfere with each other in a way that loses the information. They combine in a block-like fashion such that their tracks in the formation of the final number can be uncovered when algebra is employed.

For example, if in 2021 someone running through the procedure generates the number 647, it is straightforward to show how this breaks down into the two unknown components that make it up. We saw in Chapter 2 that the algebraic representation of the entire procedure reduces to the expression 100x + (2013 + z – y). For the year 2021, z is 8, so the expression becomes

Here, x represents the number of days the person likes to eat out in a week and y represents the calendar year the person was born.

Our knowledge of how this problem works means we can partition 647 as 600 + 47, and then solve the following two equations (often in our head): 100x = 600 and 2021 – y = 47. Solving both gives us that the person likes to eat out six days in a week and that their year of birth was 1974.

The recovery succeeds here because the relevant items, 600 and 47, are retained whether we write them separately or add them to obtain 647. In general, for a given year, we can break down all such magical three-digit numbers this way as long as the person is younger than 100 years old and we know whether or not they have had a birthday.

For a person who is 100 years or older, we have a three-digit number for the age that will interfere with the digit in the hundreds place. For example, for a person 105 years old, we would have 600 + 105, which becomes 705. This number fails to properly decompose using the simple solving procedure for this problem.

HOT GAS CLOUDS

Amazingly, in the case of light from distant sources in the universe, the situation is more analogous to resolving the magical three-digit numbers back into their component unknowns. The light, which in some cases comes from trillions of miles away, contains data that is encoded in such a way—as a cocktail of rays of different wavelengths—that it can yield useful diagnostic results. Often the components that make up this complex mix of information retain enough of their individuality to allow them to be selected back out using the right equipment and analysis. This often yields valuable information about both the nature of the cosmic sources that produced the light and the physical matter with which the light has interacted along its journey to Earth.

Visible light can be produced from chemical elements whose atoms have been heated up or excited, but they don’t produce a continuous rainbow-like spectrum like the white light that Newton observed; instead, they produce a spectrum that shows individual lines of varying colors. This line spectrum turns out to be unique for each particular element. The following diagram displays the visible emission spectra (with the more prominent lines) of the first 99 elements of the periodic table:

Spectral lines for the elements (presented vertically) in black and white; the shorter wavelengths (blue) are at the bottom for each element’s spectrum, and the longer wavelengths (red) are at the top [Reprinted with kind permission from Julie Gagnon at umop.net/spectra]

The line configurations of each of the elements are distinct from each other, meaning that such patterns can work like fingerprints to identify the presence of particular elements in the sources that produced the light. Scientists labored hard in the time following Kirchhoff’s and Bunsen’s discovery to generate the spectra of the many elements. In fact, such methods even allowed for the discovery of new elements.

Here are the prominent visible emission line spectra of the elements sodium (Na), hydrogen (H), calcium (Ca), and mercury (Hg) for comparison.

Line spectra of the visible light from excited sodium, hydrogen, calcium, and mercury atoms (presented horizontally) [Image from UT Austin–Principles of Chemistry (Li et al. 2018)]

Thus, when light from a source such as a hot gas in a nebula reaches Earth as a cocktail of light rays of different wavelengths, we can be fairly certain that these wavelengths have been produced or affected by the various elements present in the gas. The question is, can we tell which elements?

The way to separate out the light cocktail into the characteristic light rays of the elements present is through the use of specialized instruments, such as a spectroscope, which employ devices that are far more sophisticated at dispersing a light beam into its component beams than the simple prisms such as the one that Newton used. The line spectrum so generated may be a mix of many spectral lines. If an astronomer notices in the pattern prominent lines characteristic of a particular element, such as the distinct double lines just less than 6000 Å (and bright yellow in color) characteristic of sodium or the quartet of lines just above 4400 Å (and blue in color) characteristic of calcium, they can feel fairly justified in the conclusion that excited sodium or calcium is present in the gas cloud.

The next diagram gives an example of what an observed spectrum may look like. Below it are some of the brighter lines from the characteristic spectra of five important elements:

Observed spectrum (wavelengths given in nanometers) and spectra of some potential component elements [Image provided courtesy of Nagwa Limited]

(Recall that 1 nanometer = 10 Angstroms or approximately 0.00000000328 feet; thus, 400 nm = 4000 Angstroms, 700 nm = 7000 Angstroms, and so on.) In this particular observed emission line spectrum, many lines matching those in hydrogen, helium, and carbon are present. The lines for oxygen and boron don’t match lines in the observed spectrum. Based on this, a reasonable conclusion would be that hydrogen, helium, and carbon are producing this observed light coming from the nebula, whereas oxygen and boron are not.

To give another view of what is happening, the next diagram provides a visual of a hypothetical observed spectrum from an imaginary nebula and also hypothetical elements whose lines are distinguished by shapes rather than colors. In the top diagram, notice that lines matching the spectra for Elements A, C, and E are present in the observed spectrum. We can conclude from this matching that these elements are most likely present in the nebula.

Hypothetical line spectrum where Elements A, C, and E are present in the observed spectrum [Artwork provided courtesy of William Hatch]

In the following diagram, we can see lines in the observed spectrum matching lines from Elements B and D, meaning those elements and not Elements A, C, and E are likely producing the observed light in this case.

Hypothetical line spectrum where Elements B and D are present in the observed spectrum [Artwork provided courtesy of William Hatch]

ABSORPTION SPECTRA: STARLIGHT

Light from stars, including the sun, share spectral information in a different way than do the emission nebulae just discussed. Their spectra are not bright lines but rather a continuous rainbow band like the one that Newton observed when performing his prism experiments on sunlight. However, on closer examination through a spectroscope, we find that such spectra are not completely continuous, but rather can show certain dark lines peppered throughout the rainbow band of colors. These dark lines are called absorption lines and form what is called the absorption spectrum of the star. Just like the bright emission lines of nebulae, they too contain information.

Stars consist of a hot, dense interior of gases surrounded by a layer of gases called the photosphere (often called the surface of the star). Many stars, like the sun, also have additional atmospheric layers called the chromosphere and the corona. It is the light from the photosphere that we observe when we see the sun and other stars. By the time a particle of light or photon reaches and exits the photosphere, it has traveled a long way from the deep stellar interior where the nuclear fusion that produced it occurs.

By this time, such a photon may have been involved in billions of interactions, continually being absorbed and re-emitted by atoms in the interior. In reality, the photon at the surface represents the last generation of a cascade of energy-matter interchanges. It is estimated that this process can take tens of thousands of years and these continual interactions have an averaging effect on the energy of the photons that reach the photosphere and then travel to Earth, meaning that a large number of the wavelengths of visible light in the range from 3900 to 7400 Angstroms (390 to 740 nanometers) are present in the observed spectrum. This accounts for the continuous-appearing rainbow band spectrum observed in Newton’s prism experiment, which was not sensitive enough to detect the dark lines.

The dark lines in the spectrum come from elements that are present in the photosphere. In experiments performed on Earth, it was observed that elements can absorb light of the identical wavelengths in which they emit light. In terms of quantum theory, this corresponds to an electron in an atom going from a lower energy level to a higher energy level: Emission of light occurs when it goes from a higher energy level to a lower energy level, but the energy difference is the same either way. This means that, on Earth, if you placed relatively cooler sodium vapor in between white light and a spectroscope, the sodium atoms would absorb light of the same wavelength that they emit light in the excited state, and the observed spectrum would be a continuous rainbow band peppered by dark lines that would be in the exact locations—by wavelength—that the bright lines appear in the emission spectrum of sodium. This was one of the big findings in Kirchhoff’s and Bunsen’s great epiphany regarding the solar spectrum, whose dark bands were first observed by William Wollaston in 1802 and more intensely analyzed and reported on by Joseph von Fraunhofer beginning in 1814.

Another way to think of it is that just like a tree can block or absorb sunlight to create a shadow, so too could gas surrounding a star that contained sodium vapor absorb or block out light. The difference here is that the tree blocks out all of the visible light when it creates a shadow, whereas in the case of sodium, the vapor acts as a filter due to absorption, and only blocks out a certain type of light—exactly the same type of light that excited sodium emits. So, the observed spectrum of the star contains sodium shadows.

In the following diagram of the sun’s spectrum, the shadow of the prominent sodium lines can be clearly seen at position D at approximately 589.0 and 589.6 nanometers (5890 and 5896 Angstroms). The C line corresponds to a hydrogen shadow at 656.3 nanometers (6563 Angstroms). The A and B lines correspond to oxygen shadows. Many of the other shadow lines in this spectrum correspond to lines from iron.

The sun’s spectrum [Image is in the public domain]

From these readings, astronomers have been able to conclude that there is iron, sodium, oxygen, and, of course, hydrogen in the photosphere of the sun. The same techniques can be used to identify elements that are present in more distant stars and other astronomical objects. These days, however, ever more sophisticated instruments and computer software are employed to greatly enhance the analysis, even allowing information from the atmospheres of planets around other stars (exoplanets) to be obtained through their atmospheric absorption of the light from the primary star that they orbit.

Moreover, what can be done with atoms can also be done with molecules (chemical combinations of two or more atoms). The emission and absorption processes in molecules are different and more varied than they are for individual atoms. This consequently translates to different types of spectra, often outside the visible wavelength range, but the general idea of identification of substances remains and can be extremely useful both astronomically as well as here on Earth.

The visible spectrum is only one portion of the vast electromagnetic spectrum. Observations made on these other types of non-visible electromagnetic radiation—including radio wave, infrared, ultraviolet, and X-ray wavelengths—can tell us even more about the composition and activities of many objects in the cosmos. Using the results from spectral analysis, astronomers have come up with entire classification schemes for stars, nebulae, and other astronomical objects. Spectral characteristics can even be used to reveal the temperatures of astronomical objects and whether or not they are moving toward us or away from us. These are not small things, as most such objects in the nighttime sky are trillions of miles away from us, and for astronomers to be able to identify many of the substances that make them up, as well as other key indicators from the coded information in light, remains one of the landmark collective achievements of science. It truly is the gift that simply keeps on giving.

A GRAND IDEA

As fundamental and groundbreaking as spectral studies in the heavens have proved, they are just one of the most visible manifestations of a powerful idea that can be found throughout mathematics, science, and engineering (see this chapter’s epigraph by Ingrid Daubechies).

Hearkening back to our early discussions in Chapter 4, we mentioned that one could learn basic things about arithmetic just from playing with pebbles in the dirt of our backyard. We further likened the idea of learning some arithmetic this way, to the materialization—in our backyard—of the spirit of the vast and broad subject of arithmetic. Similarly, though we have discussed spectral analysis in a specific and foundational context—“our backyard”—it reveals itself in other areas such as theoretical quantum mechanics, nuclear physics, radar, sonar, and signal processing in general.

For instance, with sonar, sound waves from various activities in the ocean are received by a detector on a submarine, ship, or underwater device. The task of sonar technicians is then to take such signals and try to extract information from them by decomposing the received sound into subcomponent pieces for analysis. The idea rests in the knowledge that different objects should give off different sounds or acoustical fingerprints. A humpback whale makes a sound that is different from the sound made by a fin whale, which is different from the sound made by an enemy submarine (or surface ship going through the water), which is different from the sounds made by a large iceberg running aground or a hydrothermal vent.

Like astronomers who have in their possession the characteristic spectral lines for the many elements, the sonar technicians undoubtedly have a database of various sound signatures for different objects. Their desire is to break down the observed acoustical spectrum into information that can help identify what objects or entities are producing the sounds received by the detector. However, sound waves are not electromagnetic waves, and the breakdown of information from the audio spectrum is not as clean and definitive as the chemical information that is obtained from the electromagnetic spectrum. Consequently, mathematics and computer science weigh heavily in this diagnostic analysis, which falls under the important electrical engineering field of signal processing.

Another type of sonar called active sonar can involve a ship or submarine sending out its own sound signal as a probe and then analyzing the reflected signal on return. Such sonar allows technicians to obtain more detailed information—such as the distance to the object or its speed—but it also has the side effect of giving away the vessel’s position to a listening enemy ship or submarine. This obviously can be highly disadvantageous in the case of military actions, where the need for stealth is at a premium. But there are many seagoing operations that are of a scientific or exploratory nature where the need for secrecy is not so important—mapping the ocean floor, locating navigational hazards, finding missing aircraft, ships, or submarines, or tracking underwater geological activity; and active sonar can be very useful in such cases.

Radar, used to detect airplanes or measure weather-related information in the atmosphere, works on a similar idea to sonar except that its devices detect and emit electromagnetic waves—radio waves and microwaves—instead of sound waves. Radio and microwaves have wavelengths much longer than the wavelengths of visible light. These wavelengths range roughly from 10 million to 1 trillion Angstroms (1 million to 100 million nanometers) depending on the specific type of radar used.⁵ Radar can be used in an active or passive manner as well. Active radar is the more familiar of the two.

CONCLUSION

Famed linguist Noam Chomsky stated in 1970:

Language is a process of free creation; its laws and principles are fixed, but the manner in which the principles of generation are used is free and infinitely varied. Even the interpretation and use of words involves a process of free creation.⁶

The same can be said of mathematics, science, and engineering. There is a great deal of freedom to roam in all of these areas, yet the logic of mathematics and the known laws of nature are always there to be reckoned with and respected.

Hopefully, this chapter has hinted just a bit at the additional treasure that can sometimes be gleaned in looking at mathematical and scientific results from different yet complementary viewpoints. In the first half of the chapter, we gave a different look to a word problem from Chapter 5, which then served as a gateway to briefly touch on the topic of closure under an operation.⁷

From using closure, we saw that it became possible to generalize both the method of solution as well as the new interpretation we gave to the problem, as involving a hypothetical larger bill denomination. Parameters continued to play a key role in allowing for the generalization of these efforts.

One may well ask, did these additional interpretations really assist us in better solving these word problems? Probably not, as the solutions are more efficiently obtained through the processes already discussed in Chapter 5. What these interpretations granted us was the ability to shed a tad bit of illumination on something that can often occur to far greater effect elsewhere in mathematics and science. That is, by simply reanalyzing a problem from a different viewpoint, whole new vistas may open wide to us—vistas that are so deep and vast, in some cases, that they can lead to groundbreaking insights into mathematics or into physical phenomena.

A case in point is the ancient Indians’ decisive reimagining of numeration away from the much more common additive systems, such as the Egyptian hieroglyphic system or Roman numerals, to far more potent and extendable systems where positional values play a featured role. It is no stretch to say that this switch to the Hindu-Arabic numerals is one of the most important changes in operational viewpoints in the history of human thought. An in-depth discussion of the environment of this reorganization can be found in my earlier book, How Math Works: A Guide to Grade School Arithmetic for Parents and Teachers.

The second half of the chapter saw us continue in this vein by looking at course averages and the number of days and age problem from slightly different points of view, now seeing if the consolidated information they contained could be unambiguously unpacked into the basic elements involved in their construction. We found that we couldn’t definitively do this in the case of course averages, but in the number of days and age problem it was indeed possible.

This latter unpacking was then used in a tangential way to mildly season the truly meaty subject of spectroscopy, where such unpacking is elevated to the level of one of the great wonders of science. We further saw that the grand idea that underwrites spectroscopy has other materializations in the analysis of phenomena much closer to home.

In the case of astronomical spectroscopy, the physical world has been extremely generous in the tools it naturally grants astronomers in the identification of substances. This allowed the field to progress far even before the advent of the computer. However, in the quest to extract more of the information contained in the electromagnetic signals received from space, this identification becomes a far messier operational challenge. This is also true for the operational challenges in other areas such as the acoustic signals received in sonar, the radio and microwave signals received in radar, the seismic signals received in geology, and the diverse electrical signals needed in a host of engineering applications, including communications and image processing. Here, sophisticated mathematical tools combined with high-powered computers to implement fast algorithms become critical features in handling many of these challenges.

In the early 1800s, a French mathematician—and one of Napoleon’s scientific advisors and administrators—named Joseph Fourier made a remarkable discovery. He realized that you could take certain primary objects and break them up into radically new subcomponents that individually were totally unlike the main object. Yet, if you took enough of these new subcomponents together, their collective mass could behave like the big object, at least over a certain region. This decomposition was anything but natural. Yet it has proven to be extraordinarily powerful, with a reach that extends all of the way up to our current time. Along with its subsequent developments, it forms one of the most important tools in the analysis of all sorts of signals, including the sound signals in sonar and the radar and microwave signals in radar.

Imagine if the methods employed in the simplistic yet radical re-interpretation of the rope problem—involving the dissection of the 570-foot-long rope into 80 individual 7-foot segments—ultimately turned out more than 200 years later to be useful in a wide array of physical real-world problems—problems and challenges that were not even a glint in the eye of the original dissecting mathematician, at the time of the genesis conception of the idea—and you will get perhaps a glimpse of the miracle spawned by Fourier’s original great analysis.