RIDDLE: HOW IS A CONCERT HALL "LIKE" A STEALTH BOMBER?
(Adapted -- with my emendations for neophytes -- from Ch. 13, Number Theory in Science and Communication, M. R. Schroeder, Springer-verlag, 1984.)
ANSWER: BOTH HAVE SURFACES DESIGNED TO SCATTER "WAVES" (OF SOUND OR OF RADAR).
Think about it. If sound in a concert hall ricochets directly back to it point of origin -- as a billiard ball, hit dead center, bounces off the table-rail -- then few people in the auditorm will obtain "a good hearing".
If radar ricochets directly back to the radar device, then the plane can be easily located and shot at.
But, if the sound waves SCATTER, many hearers will derive "a good hearing"; if the radar SCATTERS, the plane may not be detected.
OK. How DESIGN this? ANSWER: START WITH "HIGHER ARITHMETIC", known as "NUMBER THEORY".
ARITHMETIC FOR CONCERT HALLS, STEREO-SPEAKERS, AND STEALTH FIGHTERS AND BOMBERS
How do you construct a concert hall ceiling which "bathes the listener in sound"? You make of the ceiling a "diffraction-grating" that reflects phonons (pulses) of sound in every direction except downward (known, technically, as "the specular direction"). For "good listening", you need every possible phonon of sound. A phonon bounced downwards (specularly) is lost to nearly all of the hall. You want a ceiling that "wastes not" phonons!
And "remainder" arithmetic tells how to do so.
You can understand "diffraction-grating" by realizing that this process deals with an analogous but more familiar processing of light.
You've seen a prism diffract light into various colors. The longer wavelength, red, is separated out at the bottom; the shorter, violet, is separated out at the top.
The same process is achieved by grooves (known as "rulings") in glass. Such a piece of prepared glass -- with many grooves or rulings -- is a diffraction grating.
The varied projection of the ceiling section illustrated in the Figure below is analogous to this. One purpose is to exclude any reflection of sound perpendicular to the surface, all others "bathing" the hall. Another purpose is to prepare reflected waves of sound so that each is approximately the same as any other, providing a rich uniformity of effect.
A similar surface makes possible the stealth bomber or fighter. A plane can be detected by radar pulses that bounce perpendicular to the surface back to the source. A submarine can be detected by sonar pulses that reflect similarly from the surface of a "target" sub back to the "searching" sub. A plane or submarine surface that does not bounce or reflect pulses "directly back" will be difficult to detect.
"REMAINDER" ARITHMETIC "Remainder" arithmetic tells us how to do that. Students with interests in architecture, music, electronics, military equipment, etc., can be motivated by this material to develop their comprehension of multiplication and division.
Such motivation is greatly needed in our schools today as the demands of "high-tech" increase in our economy. MIT economist Lester Thurow has warned, "Americans are not used to a world where ordinary production workers have to have mathematical skills." As I note elsewhere:
Ashes to ashes, Dust to dust, If Math doesn't grab you, CATCHUP must!
Near the beginning of this century, Max Planck (x-y) and Albert Einstein (x-y) showed that energy seems to be emitted or absorbed in quanta: discrete ("digital") units or gnomons, similar to the units or gnomons of arithmetic.
In 1925, Einstein adapted this model to show thatsound seems to be emitted or absorbed in quantized units called "phonons".
Phonons bouncing off surfaces are modeled by "remainder" arithmetic.
This "remainder" arithmetic was created by a poor German teen-ager, Carl Friedrich Gauss (1777-1855). At age 19, Gauss used it to construct (for the first time!) a regular ("all-sides-equal") 17-sided polygon only by ruler and compass, extending the work of ancient mathematicians.
Gaussian "remainder" arithmetic was further extended by a provincial French teen-ager, Évariste Galois (1811-1832), whose "Galois fields" now provide the most efficient codes for signals between ground stations and satellites, providing us with news, sports, musical or dramatic programs from distant places of the globe. You can see this "remainder" arithmetic operating in the mileage gauge of an automobile.
Look at such a mileage gauge, ignoring the right-most slot, which records tenths of miles. Suppose the remaining five slots read "99999", meaning that the car has traversed 99,999 miles since the gauge was set or reset. Suppose you now travel two more miles. Instead of reading "100001", the gauge now reads "00001". Having no slot to record that "new" 1, the mileage gauge has reset after 100,000 miles.
Arithmetically, 99999 + 2 = 100001, and 100001 = 1 x 100000 + 1 (which is of the form D = d x Q + R: dividend D equals divisor d times quotient Q plus remainder R.) The gauge "keeps only the remainder". The mileage gauge operates by a remainder arithmetic. (Mathematicians call this "a modular arithmetic" and call the remainder "a residue".)
To discover the standard notation for these systems, let's write:

100001 "leaves remainder or residue" 1 ("when divided by" 100000).
Next, replace "leaves remainder or residue" with "is congruent to": 100001 "is congruent to" 1 ("when divided by" 100000).
Next, replace "when divided by" with "modulus": 100001 "is congruent to" 1 ("modulus" 100000).
Next, replace "is congruent to" with the mathematical symbol,º : 100001 º 1 ("modulus" 100000). (Note: "equals", =, is symbolized by two parallel line segments or bars; congruence, º, by three.)
Finally, replace "modulus" simply by mod: 100001 º 1 (mod 100000).
Modular notation can be understood in terms of the DIVISION ALGORITHM: D º R (mod d) IF, AND ONLY IF, D = d X Q + R, for some QUOTIENT Q, and 0 £ R < d.
Those "Galois" systems, cited above, all involve divisors (moduli) which are prime numbers, for reasons explained below.
INFINITE SETS MODULATED INTO "SKINNY" FINITE SETS
So I'll illustrate a "remainder" or congruence system by using, as modulus (divisor), the prime number 5. Then, for two reasons, I'll show how a "diffraction grating" ceiling (for "good listening") can be constructed using the prime number 11 as modulus:

11 has 2 as its least "primitive root" ("generator"), generating one of the dimensions of the "building blocks" of that ceiling-"grating".
11 - 1 = 10, our decimal base, with 10 = 2 x 5. Those 10 factors, namely, 2 and 5, tell us another dimension of the "building-blocks" of that ceiling-"grating".
I start with a promised simpler case.
When we divide by 5, how many remainders or residues result? Answer: five residues: 0, 1, 2, 3, 4.
Now, 0 º 0 (mod 5); 5 º 0 (mod 5); 10 º 0 (mod 5); 15 º 5 (mod 5); 20 º 0 (mod 5); etc.
The first congruence implies (by ignoring) quotient Q = 0; the second, (ignoring) Q = 1; the third, Q = 3; the fourth, Q = 4; etc.
Equivalently, note that we are shifting from 1 by d = 5 units, by twice 5 or 10 units, by thrice 5 or 15 units, by four steps of 5 or 20 units, etc.
In computer jargon, the first interpretation, involving "quotients" and such, is a "software" interpetation; the second, involving "shifts", is a "hardware" interpretation, since hardware devices exist for shifting stored bits.
Just as "numbers equal to the same number are equal to each other" so "numbers congruent to the same number are congruent to each other". Hence, we have found: 0 º 5 º 10 º 15 º 20 (mod 5); etc.
All multiples of 5 are congruent to 0 (mod 5), that is, leave remainder (residue) 0 when divided by 5.
Now, consider all the numbers leaving remainder 1 when divided by 5: 1 º 1 (mod 5); 6 º 1 (mod 5); 11 º 1 (mod 5); 16 º 1 (mod 5); 21 º 1 (mod m); etc. That is, starting from 1, we shift 5, 10, 15, 20, etc. 1 º 6 º 11 º 16 º 21 (mod 5); etc.
Similarly, 2 º 2 (mod 5); 7 º 2 (mod 5); 12 º 2 (mod 5); 17 º 2 (mod 5); 21 º 2 (mod 5); etc. That is, starting from 2, we shift 5, 10, 15, 20, etc. 2 º 7 º 12 º 17 º 22 (mod 5); etc.
And 3 º 3 (mod 5); 8 º 3 (mod 5); 13 º 3 (mod 5); 18 º 3 (mod 5); 23 º 3 (mod 5); etc. That is, starting from 3, we shift 5, 10, 15, 20, etc. 3 º 8 º 13 º 18 º 23 (mod 5); etc.
Finally, for the only other possible remainder (residue), 4 º 4 (mod 5); 9 º 4 (mod 5); 14 º 4 (mod 5); 19 º 4 (mod 5); 24 º 4 (mod 5); etc. That is, starting from 4, we shift 5, 10, 15, 20, etc. 4 º 9 º 14 º 19 º 24 (mod 5); etc.
What just happened? Ah! The INFINITE set of natural numbers (or positive and negative integers) has shrunken to or collapsed into exactly five ("skinny") residue classes of numbers.
MODULAR ("MILEAGE GAUGE") ARITHMETIC
Mathematicians denote these five residue classes of numbers as: C₀, C₁, C₂, C₃, C₄.
Modular arithmetic achieves an effect analogous to the biologist's classifying animals into various species or the botanist classifying plants into various classes.
But more is achieved than mere classification. For we can do arithmetic with these residue classes! Actually, the addition mirrors the subtraction; the multiplication mirrors the division.
Here's the residue class addition rule: C_a + C_b º C_a+b (mod m).
Thus, for m = 5, C₃ + C₄ = C₇ º C₂ (mod 5). Etc.
The addition table of residue classes mod 5 reads:
+ || C₀ | C₁ | C₂ | C₃ | C₄ -------------------------- C₀|| C₀ | C₁ | C₂ | C₃ | C₄ C₁|| C₁ | C₂ | C₃ | C₄ | C₀ C₂|| C₂ | C₃ | C₄ | C₀ | C₁ C₃|| C₃ | C₄ | C₀ | C₁ | C₂ C₄|| C₄ | C₀ | C₁ | C₂ | C₃
Notice this:

the rows and columns run through the cycle, 0, 1, 2, 3, 4.
ignoring the zero-column and the zero-row, look at the C₀'s in the other rows and columns. Thus, C₁ + C₄ = C₄ + C₁ = C₀. Note: C₁ and C₄ additively cancel each other -- as in (+2) + (-2) = 0, for +2 and -2 as additive inverses, giving the effect of subtraction.
Then, C₁and C₄ are additive inverses. So are C₂ and C₃, since C₂ + C₃ = C₀.
Behold! Under CONGRUENCE, residue classes of natural numbers behave in the manner of positive and negative integers!
What happens under multiplication?
The residue class multiplication rule reads: C_a x C_b º C_{a x b} (mod m).
Thus, C₃ x C₄ = C₁₂ º C₂ (mod 5).
The multiplication table of residue classes mod 5 reads:
x|| C₁ | C₂ | C₃ | C₄ ---------------------- C₁|| C₁ | C₂ | C₃ | C₄ C₂|| C₂ | C₄ | C₁ | C₃ C₃|| C₃ | C₁ | C₄ | C₂ C₄|| C₄ | C₃ | C₂ | C₁
In this table:

Please look at the nonunitary rows and columns, that is, the second, third, and fourth rows and columns. Each of these rows and columns contains C₁, the residue class that plays the role in residue classes that 1 plays for numbers. Thus, C₂ x C₃ = C₃ x C₂ = C₁.
That resembles the product of rational numbers: 2 x 1/2 = 1. In the product, 2 and 1/2 are mulitiplicative inverses -- one balances out the other, resembling division.
Similarly, C₂ and C₃ are multiplicative inverses.
And C₄ is its own inverse -- as with C₁, C₄ is self-inversive -- sinceC₄ x C₄ = C₁.
Lo and behold! Residue classes of natural numbers modulo 5 behave in the manner of integers under addition, and in the manner of rational numbers ("fractions") under multiplication! We gained powerfully in STRUCTURE by MODULATION!
We'll later see what OTHER powerful applications arise from this "collapsing" of the INFINITE set of natural numbers (or the integers) into a FINITE few of residue classes.
But, here, I'll briefly refer to a vocational use of the concept of modulus. Architects, in planning a wooden house, designate some modulus for the length of boards that are cut for construction. Suppose the modulus chosen is 6 feet. Then carpenters cut boards to lengths 6 feet or 3 feet or 2 feet or 1 foot. (The numbers 3, 2, 1, are, of course, the factors of 6.) But the carpenters do not cut boards to lengths of 4 or 5 feet. A 4-foot section can be spanned by combining a 3-foot board and a 1-foot board; a 5-foot section, by 3-foot and 2-foot boards; a 9-foot section, by 6-foot and 3-foot boards; etc.
To be CONTINUED