Scenic Scenes

Most everyone enjoys a beautiful view. We all hope to have a perspective out the front window of our homes that we can enjoy like a natural painting or photo. In Colorado we like our mountain views and beautiful forests and lakes and streams and grass covered hills. Rolling hills and valleys make an awesome pictorial too.

One thing all these natural beauty views have in common is that they are fairly static. They may change with the seasons, but — in general — they more resemble a still photo than a movie.

But an ocean view is different. It is constantly changing. The surface rolls in and the level ebbs and flows. Plus, there’s a sound track too. The forest has its birds and the wind in the trees, but the ocean has its roar as it comes up on the beach like some amphibian reaching out to dry land. The white noise of the ocean can only be matched by a gurgling stream or a roaring waterfall, and they lack the primordial draw of the sound of the ocean crashing on the beach.

Even the sand changes from day to day and flotsam and jetsam will add new landmarks, modifying the scene overnight.

I’m sitting at the table looking out at the magnificent and fluid view. I spent the last few nights with the windows open allowing access to the timeless sound of the surf. No sleep is more pleasant than the slumber serenaded by the sound of the sea.

My family’s annual trek to the Pacific is a high part of my year. We arrive in September, often better weather on the Oregon coast than the high summer. My brother from Seattle, my sister from Montana, my dad from Portland, and my wife and I from Colorado. All beautiful scenery locations, but none match this awesome ocean views from our house on the cliff overlooking the beach.

This morning we cooperated on breakfast. Dale prepared pancakes that we covered with Huckleberry syrup — a uniquely Montanan sweetener. We had Oregon peaches and a watermelon a friend brought yesterday from her garden. I brewed the coffee, a personal specialty and family favorite. Later we’ll have sea bass caught from a boat a few miles south of here on a leisurely day cruise a couple of days ago. Yesterday, we feasted on salmon caught in Washington by my brother and breads bought at the farmer’s market back near Portland.

It seems we have so much food that we’ll have to have four or five meals a day to finish it off. Some like to go down to the beach and walk on the sand and wade in the water. I’m OK with that. I spent the early hours this morning wandering the beach and picking up shells and other oddities washed up from who-knows-where. But I’m having a great time jus sitting here in the hot tub, drinking wonderful Oregon white wine, and listening to the sea beat on the beach. Besides, those that have sampled the seawater in Oregon know the temperature is not conducive to bathing … unless you are wearing a wet suit. But this hot tub, truly a credit to its name, is so comfy. (I’m actually not in the tub as I type this, but the memory from earlier this week is strong.)

Last night we wandered down to the beach to enjoy a bonfire and play with the neighbor’s dog who sits, lies, and fetches on hand commands. This pup was very well trained. She is owned by a vet assistant and could be a service dog except she was too old when they got her. Still she visits the veterans hospital in Portland and works with the soldiers there. It was great fun to watch the happy dog enjoying fetching a ball and romping in the surf yesterday. It was too dark last night for such play, so she just sat by the fire like the rest of us.

Soon I’m headed south for California. There’s plenty of fine beaches on the way south, but, for now, I’m just soaking up this great scenery … moving pictures and all. Come on in … the water is fine. Today looks like rain, but this week has been the calmest in memory. The wind usually blows on the coast. That’s part of the motion too.

Mathematics -- Part Three: Algebra

Algebra can be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects, often represented by letters or other abbreviations. Initially, these objects represented either numbers that were not yet known (unknowns) or unspecified numbers (indeterminate or parameter), allowing one to state and prove properties that are true no matter which numbers are involved. For example, in the quadratic equation,

ax² + bx + c = 0

a, b, and c are parameters called “coefficients” and x is the unknown. Solving this equation amounts to computing with the variables to express the unknowns in terms of the coefficients. Then, substituting any numbers for the coefficients, gives the solution of a particular equation after a simple arithmetic computation.

You can consider algebra as “abstract” arithmetic where some of the numbers are replaced with letters. You then perform mathematical or arithmetic operations on the letters. This often requires you understand the rules of arithmetic very clearly to perform operations that would be correct for all possible values of the “numbers” including negative values, zero, and even Complex numbers. If given values for the parameters or coefficients, then solve for x. That is algebra.

Another view is that algebra can be used to create a general solution, often called a “formula.” Then all you need to do is substitute actual values for the letters and solve the formula (using arithmetic) to obtain a final result. For example, the solution of the quadratic equation is

This is the “general solution” to the quadratic. This equation and solution is well-known to all graduates of an introductory algebra class. It is part of a group of equations called “polynomials.” Since the highest power in the quadratic equation is 2, this is a second degree polynomial. Less than 2 and it isn’t a “poly.” So the quadratic is the most basic equation of a “non-linear” type. Non-linear means that, if you graph the equation, it is not a straight line. (We will get into graphing equations when we talk about analytical geometry and the connection between the math of numbers and the math of shapes. That’s later in this series.)

There are many ways to solve a quadratic from factoring to completing the square. (Square is where the name “quadratic” came from.) The formula or “general solution” is derived from the latter method. Most remember from that beginning algebra class that there are two solutions or “roots” to a quadratic equation. In the solution formula they come from the “plus-minus” with each sign providing a solution.

There is a proven theorem in mathematics that is so significant it is called the “Fundamental Theorem of Algebra.” It states that, every non-zero, single-variable, degree n polynomial has exactly n roots. (Actually it is a little more complicated than that since the theorem addresses Complex coefficients, but we won’t get into that much detail.)

What this fundamental theorem means is that all second degree polynomial equations (those with the highest power of 2) have 2 roots. In addition, a polynomial equation with cubes will have 3 roots. One with the seventh power will have 7, etc. (It says a little more about the field of complex numbers being algebraically closed, but we REALLY won’t get into that.)

Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. And that is exactly where we want to go. A little history of math that bridges the time from the ancient Greeks to the modern rebirth of math during the enlightenment.

Even though the Babylonians didn't have any notion of what an “equation” is, they found the first algorithmic approaches to problems, which would give rise to a quadratic equation today. Their method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.

The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.

In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was a root of a quadratic equation. Euclid had no notion of equation, coefficients, etc., but worked with purely geometrical quantities.

In his work, Arithmetica, the Greek mathematician Diophantus (ca. 210-290 AD) solved the quadratic equation, but giving only one root, even when both roots were positive. Hindu mathematicians took the Babylonian methods further. Aryabhata around 500 AD gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions.

So the general solution to the quadratic has been known since ancient times, before the concept of an equation had evolved and using somewhat sloppy and unproven methods, at least when compared to modern axiomatic-based math. But the method did work and was well known. It was often used in problems involving rectangles like surveying or planting of crops. It was a beginning.

This knowledge was preserved by the Arabs in the Middle East as civilization in Europe collapsed after the fall of Rome. The Hindus also continued to advance algebra, developing nearly modern methods that found both roots, even negative ones; but the Arabs were responsible primarily for preserving the Greek methods and introducing them to Europe after the end of the Dark Ages.

Even the word “algebra” comes from a bit of a mistranslation of Middle Eastern books. The word algebra is a Latin variant of the Arabic word al-jabr. This came from the title of a book, Hidab al-jabr wal-muqubala ("The Compendious Book on Calculation by Completion and Balancing"), written in Baghdad about 825 AD by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book, Liber embadorum, published in 1145, which is the first book published in Europe to give the complete solution of the quadratic equation.

Next came the general solutions for a cubic equation, a polynomial of power 3. Scipione del Ferro (1465-1526), the Chair of Arithmetic and Geometry at the University of Bologna, is credited with solving cubic equations algebraically, but the story is somewhat more complicated. We believe that del Ferro could only solve cubic equation of the form

x³ + mx = n — no second power term.

However, without the Hindu's knowledge of negative numbers, del Ferro would not have been able to use his solution of the one case to solve all cubic equations. Remarkably, del Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.

Fior was a mediocre mathematician and far less good at keeping secrets than del Ferro. Soon rumors started to circulate in Bologna that the cubic equation had been solved. Nicolo of Brescia, known as Tartaglia meaning "the stammerer," prompted by the rumors, managed to solve equations of the form x³ + mx² = n, a slightly different case, and made no secret of his discovery.

Fior challenged Tartaglia to a public contest: the rules being that each gave the other 30 problems with 40 or 50 days in which to solve them, the winner being the one to solve most but a small prize was also offered for each problem. Tartaglia solved all Fior's problems in the space of 2 hours, for all the problems Fior had set were of the form x³ + mx = n, as he believed Tartaglia would be unable to solve this type. However only 8 days before the problems were to be collected, Tartaglia had found the general method for all types of cubics.

News of Tartaglia's victory reached Girolamo Cardan in Milan where he was preparing to publish Practica Arithmeticae (1539). Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation. This Tartaglia did, having made Cardan promise to keep it secret until Tartaglia had published it himself. Cardan did not keep his promise. In 1545 he published Ars Magna the first Latin treatise on algebra.

Cardan noticed something strange when he applied his formula to certain cubics. When solving

x³ - 15x = 4

he obtained an expression involving √-121. Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation. He wrote to Tartaglia in August 1539 in an attempt to clear up the difficulty. Tartaglia certainly did not understand. In Ars Magna Cardan gives a calculation with "Complex" numbers" to solve a similar problem, but he really did not understand his own calculation which he said is "as subtle as it is useless."

During the 1500s, solving these complex equations became a form of entertainment and mathematicians would travel the country and hold side shows where they challenged the audience to give them a problem and they would solve it quickly. They were using these general solutions and the search was underway for easy ways to solve harder and harder problems. Who knows how many discoveries were made but kept secret to maintain the "magic" of the show.

In 1540, Cardan was given the following problem: Divide 10 into 3 parts: The parts are in continued proportion and the product of the first 2 is 6

This problem led to a quartic polynomial (power of 4) which Cardan was not able to solve. He gave it to his student, Lodovico Ferrari. Ferrari was the first to develop an algebraic technique for solving the general quartic. He applied his technique (which was published by Cardano ) to the equation

x⁴ + 6x² - 60x + 36 = 0

Eventually a more general solution was refined. The general quartic equation is a fourth-order polynomial equation of the form

x⁴ + a₃x³ + a₂x² + a₁x + a₀ = 0

Although this might not appear completely general as there is no coefficient for the first term, it actually is. You may also wonder where are the minus signs since all these polynomials look like they just use pluses. But the coefficients can be negative, which provides the minus signs.

The general solution for the “biquadratic” or “quartic” polynomial was much more complicated and reduced down to a combination of a linear equation (power of 1) and a quadratic. As this general solution was learned, the race was on to solve even higher power polynomials using general solutions.

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Neils Henrick Abel (Abel's impossibility theorem) and by Évariste Galois in the early 1800s.

The theoretical work in devising a general, algebraic solution to these polynomials also provided proof that the mathematicians were at the end of the road. There are no general solutions to any higher power polynomials. The road shows had ended and the work shifted to the university and publications. However, further solutions to the quintic form were added in the late nineteenth century and even in 1994 new and more solutions to certain cases of quintic polynomials were added by Spearman and Williams. So these ancient puzzles are still being solved. More significant, however, was the advancements in algebraic theory that were driven by the old side show competitions and the work to put algebra on solid theoretical grounds.

There are lots of areas of algebra that I’ve skimmed over. There are problems with more than one unknown, systems of equations and problems with complex coefficients and other branches of math such as trigonometry.

As it developed, algebra was extended to other non-numerical objects, like vectors and matrices. Then, the structural properties of these non-numerical objects were abstracted to define algebraic structures like groups, rings, fields and algebras. Eventually, algebra was even extended to Geometry.

And that is our next destination: Geometry.

Mathematics -- Part Two: Sets

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.

Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. Sets are what us older folks that went to school in the days of the one-room school house call “The New Math,” and often struggle helping our elementary school age children with the math because it isn’t how we were originally taught in our day and had these diagrams named after some Swede called "Venn." These days, elementary topics such as Venn diagrams are taught at a young age, while more advanced set concepts are taught as part of a university degree.

You can consider a set as a “bag” that objects are placed in. There is no order in a set (except in an “ordered set”) and the basic concept is membership. An object is either a member of a set or it is not. It’s like a club. (And I’m always reminded of the statement by Groucho Marx that he wouldn’t belong to any club that would allow him to join.) There can be sets within sets, just like a bag within a bag, the so-called subsets.

Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

There are several basic operations that can be performed on a set. Sets can be combined in a number of different ways to produce another set.

Set Union

One operation is called “Union.” In mathematical notation, The Union of sets A and B, denoted by A ∪ B, is the set defined as

A ∪ B = { x | x ∈ A ∨ x ∈ B }

Well, that is the new math indeed. Looks sort of like Greek. Recall from my first article that I said math was a universal language. Well this is actually a very precise mathematical statement using that universal language. You know that the symbol “∪” means “Union.” The vertical bar (“|”) means “such that” and the “V” looking character (“∨”) stands for the english word “or.” The funny looking “e” (“∈”) stands for membership and is read as “is in” or “is a member of.” So now we can translate that the “union of sets A and B equals all x (or members) such that x is in or is a member of set A or x is a member of set B.

It is much more clear if we use a diagram called a Venn diagram. Venn diagrams consist of circles that represent the set. It is sort of the “bag” that all the members are in. Inside a particular circle are all the member objects and outside are all the objects that are not a member. When showing sets, we assume some objects can be members of two or more sets. The set is shown inside a box called the "Universe," which contains all possible objects of the type being considered, such as all Integers or all numbers. For example, the number 12 is a member of the set of all even numbers and it is also a member of the set of all numbers that are multiples of three.

Here is a Venn diagram for two sets and the Union of those two sets is everything inside either the circle for set A or the circle for set B. The Union is sort of like adding the two sets, only remember a given object can only be a member of a given set “once.” That means the Union is all the members that are only in set A plus all the members that are only in set B plus all the members that are in both sets.

Here’s the diagram

My earlier example was mathematical and used numbers, but set theory is so basic and so powerful that objects can be anything from data records in a computer to human beings, animals, and plants. For example, the Union of the set of all members of the Kiwanis and the set of men in the city of Denver. It is both the powerful generality of sets combined with their very specific language and rules that makes them so useful and allows them to be used to construct the very foundation of arithmetic and all of mathematics. It is a modern answer to an age old problem of putting math on a very solid foundation.

As most people know, in the “old days,” houses were not necessarily built on good foundations. Some houses were built on wood that rotted or on soil that shifted. There were no building codes or modern engineering and some old buildings, like the Tower of Pisa, are now suffering from foundation failure. In the Nineteenth Century, set theory was invented to put all of math on a very good foundation. Consider set theory as the concrete footings of modern math.

This set of blog articles (yes, even articles can be in a set) is just intended to be an introduction to math, not an in depth explanation, so I’ll just cover a few more set operations. This will give you a feeling for what modern set theory is all about in case you didn’t go to a school that taught the “new math.”

Set Intersection

Besides Union, set theory defines an “Intersection.” The Intersection of sets A and B, denoted by A ∩ B, is the set defined as

A ∩ B = { x | x ∈ A ∧ x ∈ B }

Translating, Intersection of sets A and B equals all x (or members) such that x is in or is a member of set A AND x is a member of set B. (The symbol “∧” stands for “and.”) In other words, the Intersection includes all the objects that are members of both sets. Do you get a feeling that Union and Intersection are sort of like the left hand and right hand. They are closely related, not exactly inverses, but somehow they are “opposite” or “mirror images” of each other.

Here is the Venn diagram for Intersection.

Subset

I will cover one more set operation. That is the previously mentioned concept of “subset.” That is a set contained within a set. An example would be the set of all citizens of the United States. It contains a subset that is all the women who are citizens of the United States or all citizens of the United States that receive Social Security checks. The subset must be wholly contained within the outer set (called the "superset") to be a subset.

Here’s the Venn diagram.

So if set A is a subset of set B, then every member of set A is also a member of set B. It is logical to think of the subset as a smaller set, but it is possible that both sets are the same size. That is, they have the same number of members. Consider the set of all even numbers which is a subset of all numbers that are divisible by two. Actually, they are the same set. One is a subset of the other and vice-versa, which is the definition of set equality. Usually, though we consider subsets as “smaller” or less members than the superset. A subset that has less members than the superset is called a “proper” subset.

There’s a lot more set operations including set difference, complements, ordered pairs, cartesian products, n-tuples, and equality. All together it is a powerful notation and concepts for describing other mathematical concepts. That’s the point. Prove something with set theory and then you’ve proved it for other branches of math that can be “mapped” to set theory, and that’s about all of mathematics. It is simple enough to teach to elementary school kids and powerful enough to use in a graduate math course. Set theory can also be used to expand mathematics into other areas such as computer database operations and other Computer Science subjects or topics such as anthropology or botany or the theory of games. Sets, in a sense, are the most basic mathematics; even more basic than numbers.

Number Sets

The symbol for subset is “⊂,” although that is actually the symbol for proper subset or the so-called “strict” subset. In the first chapter of this Mathematics series I defined sets of numbers such as the Natural numbers, Integers, Reals, and Complex. We can now use set notation to indicate the relationships between these sets. I’ll use some special symbols for the numeric families.

Natural Numbers (counting numbers) = ℕ
Integer = ℤ
Rational = ℚ
Real = ℝ
Complex = ℂ
Universal Set (all possible values) = U (I couldn’t find the html symbol for “U.” It looks like the others above.)

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ ⊂ U

You can use set notation to define sets. For example:

ℝ = ℚ ∪ ℚ'

where " ' " represents not. That is, the Irrationals are the numbers that are not Rational. (Not exactly true since Complex numbers are also in the Universe. So it can depend on what you define as the Universe, but be careful of circular definitions … just like the caution in grammar. In this case we interpret ℚ' as "Irrational," that is all Real numbers that are not Rational.)

The better formed equation would be ℚ' = ℝ − ℚ. (You may have guessed that there is set "subtraction" and it removes members.)

You may be curious about how these labels were chosen. Obviously, we don't use "R" for Rational since it is used for "Real." You can consider "Q" as "quotient" because the Rational numbers are all formed as the quotient, or division, or fraction with two Integers. The Integers use "Z" from the German word for "number," "Zahlen." These symbols were developed in the '30s and are now accepted in mathematical notations. They started being used in scientific papers and then in text books and now they are the standard.

Of course, using "I" for Integers would be confused with Irrationals and even Imaginary numbers. Due to alphabetic confusion, in electronics engineering, where "I" is used for Current ("Intensity" of Current, "C" was already taken for "Capacitance" … I know … too complicated.) So, in EE you use j instead of i for √−1to keep from confusion with current.

Now let's talk about the two letter abbreviations for the states … just kidding.

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Sets can be used as an abstraction of arithmetic. In reality, all of mathematics is a form of abstraction. Even the numbers are abstractions. What does it mean to talk about “seven”? What is 7? I can explain what seven dollars are, or seven cupcakes, but what is “seven”? It is an abstraction. It is a representation. And … we can use it for keeping track of our dollars and our cupcakes.

The next topic is “algebra.” That’s an abstraction of arithmetic. Instead of “7,” we will let “x” stand for the number of cupcakes. That’s next in Mathematics — Part Three: Algebra.

Mathematics -- Part One: Arithmetic

The term “arithmetic” comes from the Greek word "ἀριθμός," pronounced “arithmos” which means "number." It is the oldest and most elementary branch of mathematics. It is used for counting, balancing checkbooks, and advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. More precisely, this is “elementary arithmetic” in contrast with the so called “higher” or “advanced” arithmetic, which is about number theory.

Besides the basic operations of addition, etc., there are other concepts such as equality and various representations called “notation.” That’s the way that numbers, fractions, exponents, etc. are expressed. This notation has been invented and evolved over the centuries and is a common language in the world. Whether English or French or Chinese or Russian or African or from India, the notation and the symbols for the modern ten digits and the mathematical operations are the same. They may add a slash to the seven in Europe or a stroke through the zero in the Army, but in our world of multiple languages, the language of mathematics is universal. That, in itself, is something to ponder philosophically.

Virtually all ancient cultures understood arithmetic and basic calculations encountered in agriculture and store keeping. Arithmetic is closely related to the system for representing numbers and basic counting. Early math history is more about the advancement of notation with more modern studies of arithmetic focused on proving the correctness of results. The ancients were happy if the math fit common sense, but we now have a complex system of axioms and proofs at the heart of all math including arithmetic.

Various mechanical devices such as the abacus or the more modern adding machine and even digital computers have been developed to “crunch numbers.” These replaced the use of simple fingers for counting which share the term “digits” with these numerical symbols. Some cultures had a base of 60 and even 360, values we still see on clocks and compasses, but most gravitated toward the decimal or “ten-based” system although other concepts such as "dozen" still exist.

As the science of arithmetic advanced, new number types were discovered in addition to the counting numbers from the negative numbers to zero to fractional numbers and even imaginary numbers. The simplest set of numbers is called the counting numbers or “Natural Numbers.” They start with 1 and increase by adding 1 to each preceding number: 1, 2, 3, etc. Formally, these counting numbers are called the Positive Integers. Add zero and the negative values and you have the whole of the Integers.

Negative numbers first appeared as the result of subtraction and were used to represent financial concepts such as debt. Zero was actually a little harder to develop since the concept of nothing was challenging to early societies, but zero soon entered the lexicon. Note that the Zero is neither Positive nor Negative, but represents the boundary between the two sets when shown on the “Number Line.”

Fractional numbers also were an early development, often stated as a fraction with a numerator or “top” and a denominator or “bottom.” They were the natural result of division, and — in fact — a fraction is one way to represent division. The division symbol (÷) is also recognizable as an abstraction of a fraction with the two dots representing the numerator and denominator.

All the numbers that can be formed by making fractions of Integers were called by the Greeks the “Rational” Numbers. We often think of these as “decimal” numbers in modern math where the “decimal point” separates the whole number part from the fractional part.

When the Greeks first leaned there were fractional numbers that weren’t constructed from Integer fractions they were shocked. Legend says that a member of the Pythagorean school who showed that the square root of 2 was not such a number was thrown into the sea to drown for his heresy. Once they became convinced of the existence of such numbers, then the Greeks began calling values like the square root of two “Irrational.” We later leaned that some natural constants such as pi are also Irrational. Therefore, the numbers that could be formed from fractions of Integers were called the Rationals. Those that could not be formed from a division of Integers were called Irrational.

Not only can’t Irrational Numbers be represented as a fraction of Integers, but as decimals they never repeat or terminate. Rationals do. For example, 1/2 = 0.5 and 1/3 = 0.333… to infinity. Note that you could consider 1/2 as repeating without end if you view it as 0.5000…. The numbers in pi, on the other hand, never repeat no matter how many digits you expand the decimal: 3.1415926535897932385…

Now days we consider the Rational numbers plus the Irrational numbers as a set called the “Real” numbers. All the Real numbers can be represented as a point on a number line. That’s sort of like a ruler or scale drawn on paper. Zero is in the center and the positive numbers extend to the right (toward infinity) and the negative numbers extend to the left (toward negative infinity). You can imagine both 2 and 3 as well as 2.5 or 1/3 and even the square root of 2 (about 1.4) as points on the number line or scale.

Mathematicians also considered what appeared to be impossible numbers such as the square root of minus 1, a number that would seem to not exist, and therefore is called “Imaginary.” The square root of minus one is a number that, when squared, equals minus one. But all numbers, when squared or multiplied by themselves, produce positive numbers. Therefore, it would appear that the square root of -1 does not exist except in the imagination of the mathematician. The square root of minus one is often represented by the lowercase letter "i," usually in italics: i. By definition, that is we simply declare it to be true, i = √-1.

Imaginary numbers can’t be represented on a number line, although they can be represented using a Cartesian graph. In fact, a major use of Complex Numbers is to represent multidimensional values and Cartesian coordinates.

“Complex Numbers” consist of the set of Reals and the set of Imaginary numbers combined. They are written as the sum of the real component and the imaginary component: R ± i I.

So, in terms of membership, the largest collection is the Complex Numbers made up of Real + Imaginary. The Reals are made up of Rational plus Irrational Numbers, and the Rational Numbers include both Integers and Fractional numbers. Finally, the Integers include the Natural Numbers plus the Negative Integers and Zero.

It seems like many of these problem numbers came from square roots, and you could question if a square root is a basic arithmetic operation, but square roots are a natural extension of the inverse of multiplication, and multiplication is just accumulated addition, so it is among the basic operations. (An inverse is an operation that “undoes” another operation. Subtraction is the inverse of addition and division is the inverse of multiplication. Repeated multiplication is called exponentiation or “powers” and roots are an inverse of that. ((There is actually another inverse of exponentiation called logarithms. It is part of arithmetic too.)) )

So you can argue that all the trouble started with simple addition and its inverse. Arithmetic is just numbers and these simple and extended operations on numbers. Seems more complicated than just 2 + 2, but it actually grows naturally out of that basic equation or formula.

Usually the purpose of all forms of higher math is to, eventually, get back to arithmetic and numbers. (Not always true, but often true.) Ultimately, arithmetic and numbers give us the answer we usually seek. Saying math is all about numbers is a bit over simplified because geometry, also a part of math, isn’t about numbers directly, although we now realize geometry and algebra can be combined; something called “analytic geometry.” So, one can argue that it all started with numbers. 1, 2, 3, … infinity.

(By the way, infinity is not a number. It is a concept and a destination and a limit, but it is not a number.)

Next we will continue to study numbers and arithmetic, resulting in the modern concept of “sets” and the following article titled “Mathematics -- Part Two: Sets.”

The Three Wise Men

Two things I love to read about are math and history. Both subjects have intrigued me for as long as I can remember. Not only were those topics I was drawn to in school, taking extra classes in both that weren’t required for my degree, but I also searched the bookstores for good books on both areas of study. I was particularly keen on the combination of the two, the history of mathematics. Like regular history, the history of mathematics was a history of the men (and sometimes the women) who made the discoveries and how these ideas built on each other. Regular history is also about people and discoveries and great events and battles. History of math held all of that too, even the battles … on occasion.

I remember when I first found Bell’s “Men of Mathematics” in a college bookstore. I took it home and stayed up most of the night reading since I just couldn’t put it down. I’ve filled my bookshelves and bookcases with stories about mathematics and how it was developed and the personalities involved. It isn’t just math. Other technical topics are interwoven in the advancement as these great mathematicians were also great physicists and great astronomers and great engineers, but mostly great physicists.

Since counting is a natural part of mathematics, and putting things in order is the essence of much math, it is only fitting that I discuss the three greatest mathematicians of all time. This is a pretty settled list. There is no real controversy about which three are in the list I’m about to describe. Oh, some would disagree that an individual is the greatest mathematician because they consider him the greatest physicist. But that’s about the only disagreement.

Aristotle

We will start back in ancient Greece, the cradle of mathematics. Other ancient cultures developed advanced mathematics, at least to a degree, but it is the Greeks and their math that has had the greatest influence on modern mathematics and the development of that math through the twenty plus centuries following the ancients. This is largely due to the development of the axiomatic method by these historical thinkers. Names like Euclid and Pythagorus and many others are at the basic foundation of mathematics and, for over a thousand years, their writings were used as text books by “modern” students up until the sixteen or seventeen hundreds. In fact, you will find Euclid’s Elements still being used at the turn of the twentieth century.

But the first wise man that I will describe in this trio of greatness is Archimedes. He lived in the late part of the Greek empire, around the time that the Romans were conquering. In fact, he was killed by a Roman soldier. If you look him up you will find out that he is listed as a mathematician, physicist, engineer, inventor, and astronomer as he made significant discoveries in all of these fields. Among his advances in physics are the foundations of hydrostatics, statics [mechanics], and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. In so doing, he came within a hairs breadth of inventing The Calculus. He also developed a form of numeric expression that we now know of as “scientific notation.” That is where you give a set of numbers and a power of ten to show the overall magnitude. This method was used by Aristotle to estimate the number of grains of sand on a beach. This was a tremendous advancement since simple number notation of that time was very clumsy and held back advanced numeric thinking. He also defined the spiral bearing his name, formulae for the volumes of solids of revolution, as well as his ingenious system for expressing very large numbers.

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.

Some of his writings have survived to this day, but his fame was also spoken of in other ancient documents and we have a pretty good picture of his work even though he lived a couple hundred years before Christ. In The Sand Reckoner, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×10⁶³ in modern notation.

In his Methods of Mechanical Theorems, Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, another short work by Archimedes, The Method of Mechanical Theorems was written as a letter to Eratosthenes in Alexandria.

There are many reasons that Archimedes is held as the greatest of all of the ancient Greek mathematicians and natural philosophers. His work covered many different areas of study and he came so close to inventing calculus, which is what our next great mathematician is remembered for. He came at the end of a long line of great Greek thinkers and one wonders if the Romans, great engineers but not good scientists, had not concurred Greece, what more inventions would these natural thinkers have discovered. Would there come a Greek even greater than Archimedes?

We can’t change history, so we’ll never know the answer to that question. Instead, following the rule of the Romans, the western world fell into a period of darkness in which little scientific progress was made. For over fifteen hundred years Archimedes was a shining star of invention. Then, in the sixteen hundreds, new lights began to appear. On Christmas day, 1642, this gift to the world of mathematics (and physics and astronomy and …) was born.

Isaac Newton

In June 1661, Newton was admitted to Trinity College, Cambridge as a “sizar” — a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalized binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague.

Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus, optics, and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity.

It was during these few months at home that Newton had his eureka when, as the tale goes, he saw an apple fall to the ground and realized that the moon was constantly falling around the earth, which led to his development of the theory of gravity and his three rules for motion. He perfected his mathematical method we now call calculus in order to solve the equations that he created.

When Newton published his ideas about light and color in Opticks, Robert Hooke, head of the British Royal Society, criticized some of his conclusions. Newton was so offended that he withdrew from public debate. The ensuing controversy turned the shy Newton off to the process of publication. His conclusions about light were eventually shown to be true, as Newton knew they were since he had proven it with mathematics and experiment, but the experience led him to keep the results of his work on gravity and calculus secret for nearly twenty years. At issue was whether light was a particle or a wave. Oddly, both men were right as it was later shown at the advent of quantum physics that light displays both qualities depending on the experiment. However, most of the controversy back in the 1600's had more to do with the poor quality of prisms and optics available to perform experiments, which made it difficult for others to reproduce the results that Newton reported.

Later, Newton’s interest in astronomical matters received stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with John Flamsteed and Edmund Halley, both Royal Astronomers. After the exchanges with Flamsteed and Halley, Newton wrote out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector.

As the story goes, Halley inquired of Newton about a particular mathematical relationship that would result in the elliptical orbits described by Kepler, and Newton instantly responded. When asked how he knew, he said he had worked it out years before. Halley then encouraged his publication of these ideas and even financed the publishing.

Newton communicated his results to Robert Hooke and the Royal Society in De motu corporum in gyrum, a tract written on about 9 sheets which was copied into the Royal Society's Register Book in December 1684. This tract contained the nucleus that Newton developed and expanded to form the Principia, one of the most influential scientific texts ever written. The Principia was published in July 1687 with encouragement and financial help from Edmond Halley. In this work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial Revolution which soon followed and were not to be improved upon for more than two hundred years, and are still the underpinnings of the non-relativistic technologies of the modern world. He used the Latin word gravitas [weight] for the effect that would become known as gravity, and defined the law of universal gravitation.

In the same work, Newton presented a calculus-like method of geometrical analysis by “first and last ratios,” gave the first analytical determination of the speed of sound in air, inferred the oblateness of the spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the moon, provided a theory for the determination of the orbits of comets, and much more.

Newton made clear his heliocentric view of the solar system — developed in a somewhat modern way, because already, in the mid-1680s, he recognized the "deviation of the Sun" from the center of gravity of the solar system. For Newton, it was not precisely the center of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this center of gravity "either is at rest or moves uniformly forward in a right [or straight] line."

Certainly his work in physics would classify him as the one of the greatest of that branch of science, but pure mathematicians memorialize him for this invention of calculus, although his delay in publication allowed another contemporary, Gottfried Wilhelm von Leibniz, to independently discover the important mathematical method. This led to a long fight over priority which actually set British mathematics back since Newton’s notation was not as clear as the notation invented by Leibniz, and the British used Newton’s notation as an act of support.

Even the history of something as dry as mathematics has its controversy and national pride and prejudice.

Perhaps more important than all his fabulous discoveries was his impact as a sort of mentor to the enlightenment. It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one of the seeds for Enlightenment ideology. He received fortune and acclaim and even a civil service job and a knighting in response to his great work.

In his own words, he said, “I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

Carl Freidrich Gauss

Gauss, often called the “Prince of Mathematics” was also busy in other disciplines including physics and astronomy. Born at the time of the American Revolution, Gauss anchored the first half of the Nineteenth Century, a century that formed the foundation for all the wonderful discoveries to come in the Twentieth. He contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics. He is honored in physics by a naming a unit of magnetism, the “gauss,” after him.

Although the mathematics discovered by the other two great mathematicians may be familiar to most high school graduates, even if they never took a course in calculus, Gauss’ work is more advanced and so average students not involved in advanced math may not have be experienced with the areas that he advanced.

Sadly one of Gauss’ principles was “few but ripe.” Therefore he only published ideas that he had fully developed leaving us to wonder at further ground breaking and original ideas he hinted at in letters, but never published. Mathematics would have gained greatly from even the random thoughts of this mathematical prince. He was so significant in the work that he did publish, one can only ponder what other gems of mathematics were not yet ripe in his genius, so he withheld any hint of his insights.

For the benefit of the non mathematical reader, I won’t go into the details of his discoveries except to note the number of modern day prizes, buildings, and astronomical and geological objects that carry his name and reputation. For the mathematical reader, there is no mystery why this modern mathematician is listed with the other two greats.

He supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician. I’ve written about that identity which I call the "most beautiful equation in the world," a view often expressed by other mathematicians. It is most interesting that Gauss thought of that equation as a touchstone for measuring math prowess.

The Most Beautiful Formula in all of Mathematical Thought

I have to admit that I had to study the identity, and I’m still amazed how it shows that an exponential with an imaginary number is somehow equivalent to rotation or how transcendental and irrational numbers can combine with imaginary numbers to produce the most basic integer. I now understand how, but — believe me — it wasn’t, nor is it now, intuitive. I still struggle to understand that damn and wonderful equation, and I run through the transformation in my head from exponential to trigonometric identities anytime I consider it. It is always something I have to go through step by step, rather than leap to the conclusion.

I think Gauss was correct. I will never, ever, ever be a world-class mathematician. All I ever will be is someone scratching the surface of the concepts invented and conquered by these three wise men and all the other wise men and women of science that have come before or since. In all reality, I don’t think I’ll ever get my Ph.D., nor would I have gotten it when in my twenties. I just don’t think I’m good enough. That realization is not going to stop me. I’ll give it my best, using every trick of learning and understanding that I’ve got stuffed up my sleeve. I’ll give it the “old college try.”

These guys are my heroes and my mentors. I will try … try my best … but I don’t guarantee the results. After all, as these three men demonstrate, my sights are set very high. I may miss the target all-together. Yet, I too, feel like a small boy, a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. I may not comprehend that great ocean, but I see it out there … and it excites me to just look out at it. I don’t have what these great, or most great, scientists have, but I do match them in desire. I get it! … I may not understand it, … but I get it!!

Space the Final Frontier

There were so many unknowns and technical problems that had to be solved in the early days of space flight. One thing that no one really knew about was the effects of weightlessness. We’ve all seen the old black and white movies from the fifties … the sci-fi and the thrillers that showed astronauts and pilots spinning around in a centrifuge. That was a way to simulate the acceleration of rockets and jet planes and the extra pressure of acceleration acting as pseudo-gravity, which was measured in “g’s.” Special suits were developed to keep pilots from blacking out as the extreme maneuvers of high performance aircraft would drain the blood out of the brain leading to blackouts and crashes.

But it was much harder to produce weightlessness here on the earth. Scientists and engineers did devise a system of a large plane flying a hyperbolic loop across the sky that could produce weightlessness for about half a minute. The effect was very similar to what a person feels for about half a second on the top of the arc when swinging on a playground swing. This allowed some experimentation and training of personnel, but duplicating the hours and days of weightlessness that would occur in space was not possible without actually entering orbit.

Even the fact that spacecraft would experience weightlessness in an orbit around the earth was poorly understood by the general public. I’ve read several old science fiction novels that didn’t understand the basic physics of weightlessness and assumed the gravity of the earth or the moon or mars would affect the astronauts. So why are people and things weightless when in orbit around the earth? Obviously they are not beyond the reach of the earth’s gravity. After all, that force holds the moon in orbit around our globe.

It is not about being out of the reach of the force of gravity, but rather being in “free fall.” Free fall means falling without a force acting upon your body. If you sky dive from an airplane, you are falling, but the force of the air against the body slows you down and you feel that “acceleration” as weight. If you were falling in open space, with no air to slow you down, then you would not feel gravity. That is what an object in orbit does. It is falling around the earth and there is nothing to give weight.

Reentry into the atmosphere, however, does cause weight because the friction of the air is slowing down the space craft and that is what the centrifuge training was for … that and the launch when gravity is magnified by acceleration instead of canceled out by freely falling. So that is why voyagers in space have no weight except when undergoing changes in velocity, called acceleration, as rocket motors or air friction apply forces to the spacecraft and the occupants.

Since most of a space mission and even a voyage to the moon are done with the engines off and just floating, weightlessness is the normal condition of most of space flight. Not only did weightlessness have an effect on the people, their physiology and movement, but it affected the equipment taken into space too. Besides the often portrayed floating of things in the space craft or the need for squeeze bottles to drink liquids, the lack of gravity had a profound effect on the design of some equipment, and the physics of operation. Weightlessness didn’t have an effect on the flow of electrical currents and electronic circuit components such as transistors or switches, but it had a profound effect on the certain devices.

For example, engineers know that the standard meltable-link fuse is a simple, passive, reliable, and very effective way to protect against damage due to short circuits and overloads; it's normal and wise practice to use these on power and signal lines. Their operating principle is simple: when excess current flows through the fuse, the link heats and melts, and then the molten blob falls away, breaking the circuit and current path.

Whoops … the word "falls" is the key to why a fuse won’t work in the weightless world: there is no force (such as gravity) to cause the blob to go anywhere. It will melt and then stay in place, making and breaking the circuit intermittently. To operate reliably, “spring loaded” fuses were required. That way, when the link melted, the spring would pull the connection apart stopping the current flow.

Another counterintuitive situation has to do with cooling, an omnipresent concern for electronic and other systems. In the vacuum of space, of course, there is no option of conduction or convection cooling; only radiation cooling is possible. This complicates the design of satellites and must be carefully factored into the thermal planning and system design.

But what about the Space Shuttle, Skylab, or the International Space Station, all of which have a "normal" air atmosphere? That should allow convection cooling of the electronics as heating air rises, you might assume.

Whoops, wrong again … another mistaken assumption: the word "rises" has no meaning in this weightless environment. What happens is that the heated air just stays where it is, accumulating around the heat source and acting as a warm — and therefore destructive — blanket. So forced-air or active fluid-based cooling is needed, since passive convection-only cooling does not happen.

Let me explain these last paragraphs and the physics of heat transfer for you non engineers. There are three ways that heat moves from one point to another. The first is “radiation.” That’s how heat gets to us from the sun. The energy that we describe as heat moves across free space via photons. In other words, the heat is in the light (or more accurately, the radiation) from the sun. That’s why shade feels so good on a hot and sunny summer day.

Heat also moves by “conduction.” That’s why a metal handle on a pot on the stove gets hot. Heat is actually the vibrational movement of molecules. The fast moving (or “hot”) molecules bang into nearby molecules making them “hot” too. Metals are very good at conducting heat, while other materials resist this motion more. That’s why some pot handles are made of plastic or wood which doesn’t conduct heat as well. A hot pad is also a poor conductor of heat as is styrofoam. Now you can understand everything form home insulation to coffee cup design is premised on this physical phenomenon called conduction.

The third method of heat transfer is “convection.” It is based on the fact that “hot air rises.” You know … what makes a hot air balloon lift off. The reason is that hot air (or any hot fluid from air to water to oil to …) is less dense than the material when cool. If it is less dense, then gravity forces the more dense fluid “down,” thereby forcing the hot fluid “up.” That’s what is working in your old fashioned steam or hot water radiators. That’s also why it is hotter on the second floor of a two story house and quite cool in the basement. (Convection plus the insulation effects of the ground and solar radiation … it is a combination of physics.)

Convection is what makes the wind blow and thunderstorms and tornadoes and hurricanes and most of the weather effects. Convection is also how a “heat sink” works. Heat sinks are large metal devices that conduct the heat away from the transistor or I.C. or other electronic device. But then the metal fins common on a heat sink heat up the nearby air which floats away by convection, thereby cooling the device. So heat sinks don’t work in weightless air, since the heat is not carried away.

The lack of convection in a weightless condition had some advantages. The lunar spacecraft was a rather cold environment with the hundreds of degrees below zero space just outside the metal walls, the interior was a bit chilly. Yet the astronauts slept very comfortably because a thin layer of warm air from the heat of their body would wrap around a sleeping astronaut like a transparent sleeping bag and keep them warm all night. Normally, convection would disperse this heated air, but in the still air of the weightless spacecraft, the layer clung to the body keeping the sleepers quite comfortable.

Almost every small action, perspective, and operating mode we know on Earth has the presence of gravity effects as an unspoken, "given" assumption — and how its absence in space make these standard operating concepts almost meaningless. It's tough on equipment, and even worse on humans, with bizarre micro- and macro-consequences due to the absence of gravity impact as a pervasive force.

Another example was the simple “fuel gauge” designs on the lunar lander. Just like an automobile gas tank design, the “gas gauge” works with a float in the tank and gravity to force the liquid fuel to settle in the bottom of the tank. As the lunar lander approached the moon’s surface, only when the engines were firing did the fuel gauge work. This led to very stressful moments as the first lunar lander approached the moon’s surface and nearly exhausted the fuel in last minute maneuvering to avoid large boulders. NASA was counting down the fuel supply based on telemetry from the space craft, but they weren’t sure of the amount remaining. They were down to 15 seconds of fuel left when the “Eagle Landed.” At least that’s what their instruments showed, but they were very worried the readouts weren’t accurate due to problems making these devices work in space and during a weightless free fall toward the moon.

To the early NASA scientists and engineers, they had the responsibility to design reliable space craft that would sail into unknown waters. Truly dragons were there. Through a process of “baby steps,” and incremental advances, the staff and crew learned how to build craft and train astronauts that could operate in space. (Even though, as the picture above shows, space is not good for hair styles.) Now we have the perspective of not only the flights to the moon, but the months and years of survival in orbit in the International Space Station. Now we know from experience how to build devices to work in the weightlessness of space. At first we had to figure it out in our heads before our actual experience. The reason that all early astronauts were “test pilots” is that was exactly the job they were taking on.

Ballroom Dancing

When I first moved to Denver, I was working for Channel 9 on Bannock Street. I was hired since I had a First Class Commercial FCC license, but I was really just the station's gopher. “Hey Mickey, why don’t you go-fer coffee.” “Go-fer some RG-8 cable.” “Go-fer some hamburgers.” That’s proof I wasn’t performing a technical job. Only one in three requests was for anything to do with the broadcast industry.

Still, it was a good job and one thing I got to do a lot was “go-fer” the airport and pick up some celebrity or celebrity “wanta-be.” The station had a black Lincoln Town Car … not a stretch limo … but a limo in any case. I would go pick up people that were in town for the morning show or an afternoon interview show or whatever. If it was a large party, I drove a small bus, but mostly I was tooling around town in the black Town Car and they even let me take it home since I only had a motorcycle back then.

One time I picked up sixteen beauty queen finalist or semi-finalist or quarter finalist or whatever. They were on a tour across country traveling to Los Angeles for the pageant's final event and doing interview shows on local TV to increase the contest’s ad revenues. I was in the business. I understood the process. There was one who really caught my eye: her smile. her eyes. her hair; she was a beautiful blond … hello … it’s a beauty contest! Now at that point in my life there was only three things I was truly interested in: girls, girls, and … of course … GIRLS. So I planned my smoothest move on her.

You see, when I was a young lad, my mom used to send me to dance lessons. (How embarrassing.) She was a very classy lady and insisted that I learn BALLROOM DANCING. But now, what had once been a fate worse than death for a young boy only interested in baseball and fishing, I now used to my personal, girl catching, advantage. I had a white tuxedo, all custom fit to my six foot-two, 150 pound frame. (Yeah, I was a lot thinner back then … and a lot taller.)

I had found the greatest ball room dancing ballroom in all of Denver, the Ritz-Carolton. Whether is was the Big Fish Combo or Joey Thomas Big Band, I could knock them dead with my smooth moves. So I asked the young lady if she would like a pleasant night out on the town. She agreed and I told her to wear her grandest gown. (I knew they had these fancy dresses since it was part of the pageant.) I picked her up at eight in the black Town Car and me in my white Tuxedo.

I could tell right away she was impressed by my unique style and attitude and soon we were on the dance floor knocking them dead. She was really a good dancer too, although it seemed all natural to me since I’d be surprised if there was any other soul in the whole United States in the Sixties that had studied the esoteric footwork my mom had forced down my throat.

Now you young bucks out there familiar with the more common method of wooing a lady and those who quote poetry such as “Candy is dandy, but liquor is quicker,” you really don’t get it. Ballroom dancing, fancy clothes, and … champagne. Now that’s a drink straight from France, the loving capital of the world. It tastes like soda pop, but — believe me — it’s got its kick. And the ladies are so unsuspecting. Of course, yours truly imbibed freely too. After all, it was a party.

As we skipped the light fandango and whirled round and round on the floor I soon started to feel the bubbly as my head began to float, the room seemed to fly apart, and I felt decidedly nauseous. I had just got out of the service and my chosen branch was the Navy, so seasickness was something I was very familiar with. She, however, didn’t seem to be affected at all. We were such a striking couple that most the other dancers had stopped dancing and stood in a circle watching us twirling about the floor. They were all clapping and shouting encouragement. The whole room was throbbing, or maybe it was just the drink. No one wanted us to quit, but I had to go sit down.

I took her back to our table and called out for more champagne, and the waiter brought a tray of those classic stemmed glasses. Things were going exactly to plan when an old buddy of mine showed up. His name was Steve Miller and he had also just separated from the military, but he had been a Marine and served in Vietnam, very tough duty. I was quite confident and asked him to join us even though I realized he would be a rival in this carefully planned seduction, but I thought my plan was flawless and working so well I decided to spice up the evening with his presence.

Right away he started talking about his war experiences as I downed another glass. My date seemed enthralled. She had her elbows on the table and her chin in her hands and was staring at Steve intently as he told his stories. It was a gruesome description of death and dying and I could tell she was affected as the blood drained from her face leaving her pale as a ghost. What I didn’t realize was that the stories were connecting with her as the war hero wrought his tale.

I had planned so carefully and it seemed that everything was going so well. Tomorrow the sixteen contestants would leave for L.A. and, if I played my cards right, she would be mine for the night.

She said something about the truth being plain to see. I thought my eyes were wide open, but they might as well have been closed. You can guess the ending. She left that night with the Miller and I wandered home, slightly buzzed, and without a companion. As I sat at my kitchen table drinking coffee to recover my senses, I thought about the evening and penned a short poem, and then laid down on the couch and drifted off to sleep.

Shakespeare said, “If music be the food of love,” I thought, “then laughter is it’s queen.” I rose early the next morning and drove down to the station. Immediately I was sent with the van out to the airport to pick up a British band called the “Paramounts.” They were in town hawking an unsuccessful album. They’d had a small hit with their cover of “Poison Ivy,” and were in town seeking both fans and inspiration.

I had one of those throbbing heads that is the downside of the French grape and suggested we stop at a local watering hole for a Bloody Mary. They’d been on the plane all night and were most willing to join me for a drink or three. The bar was deserted at ten o’clock in the morning and there was a band set up on stage. Soon the rockers were up there playing and singing. I joined them and fired up the Hammond Organ and then thought about the lyrics I had written the night before. As the band took a break to replenish their drinks, I started playing little arpeggios on the organ and singing the lyrics I had written the night before. I started with a simple C - F - G, the simplest progression ever in the easiest key there is on a keyboard. Then I added some related minors, but still stuck with only the white keys. Soon the drummer joined me, but the rest just listened to the cool lyrics and the simple organ sounds.

When I had finished the musical composition before their very eyes and ears, they all laughed, applauded, finished the drinks, and headed for the studio. I had other duties that day and didn’t see them before they left.

Imagine my surprise when, on the radio a few months later, I heard … not only my words and melody, but even the simple organ centric production. It became a big hit for them, although they had changed their name to something they got from a cat. Most people misspell their name to this day and they went on to great success playing with large orchestras. Who knew that their start was a young man in a tux trying to woo a gal with ballroom dance. The truth is strange to see.