Numbers: Natural and Unnatural

Nick spots an Unnatural Number: Graphic by August O'Connor

God made the integers; all else is the work of man.
  Leopold Kronecker (1823-1891)

It's customary to call the "counting numbers" 1, 2, 3, 4, 5 ... the NATURAL NUMBERS which somehow implies that all the other numbers are (as Kronecker suggested) artificial or unnatural.

Some thinkers include zero in the set of natural numbers and others do not. The Roman numeral system had no symbol for zero (Romans used the word nulla instead. On the other hand, in the Arabic numerals that we use today, zero plays an essential role.

In the 6th century BC, Greek philosopher Pythagorus and his followers declared that All is Number, an opinion largely echoed by today's theoretical physicists. To the NATURAL NUMBERS, Pythagorus added fractions, numbers that can be expressed as the ratio a/b of two natural numbers. Derived from primordial integers, these so-called RATIONAL NUMBERS were considered by the Pythagoreans to be the basic building blocks of the physical world.

An impressive triumph of the Pythagorean view was a discovery that linked RATIONAL NUMBERS to the human mind. The Pythagoreans discovered by experiment that the human sense of musical concordance was stimulated most strongly by pairs of tones whose wavelength ratios are the rations of small natural numbers. The musical unison is a 2/1 ratio of tones; the perfect fifth is a 3/2 ratio, the perfect fourth is the ratio of 4 to 3 and so on. In the intervening 8 centuries, humans have made no further discovery comparable to the remarkable Pythagorean musical scale that solidly links human subjectivity to the properties of rational numbers.

This ideal Pythagorean paradise was shattered by the discovery of IRRATIONAL NUMBERS, such as the square root of 2, which cannot be expressed as the ratio of two integers. Rumor has it that revealing the fact that the SQUARE ROOT OF TWO is irrational (a proof that is taught today in every high school) was punishable by death. A mathematician named Hippasus was supposed by some to have been drowned at sea by the Pythagorean Mafia for sharing this dark mathematical secret.

I'm currently reading An Imaginary Tale by Paul Nahin which tells the story of the IMAGINARY NUMBER "i" defined as the SQUARE ROOT OF -1. Physicists routinely use "i" in their calculations but few are aware of how long and difficult was the process involved in bringing this bizarre new number into the charmed circle of conventional math.

Nahin's tale involves dozens of famous and not so famous mathematicians who were baffled by the concept of the square root of a negative number. Judging from his recountings of obscure mathematical contests, long forgotten rivalries and obscure misunderstandings, Nahin has done a lot of research for this book. One of the facts that impressed me was that even at the time of Newton and beyond, mathematicians were not entirely comfortable with the notion of a NEGATIVE NUMBER. What is the true meaning of a number that is "less than nothing"?

When the NEGATIVE NUMBERS (both rational and irrational) are added to the POSITIVE NUMBERS plus ZERO, the result is called the REAL NUMBERS. The REAL NUMBERS can be considered to lie on a REAL LINE that stretches from minus infinity at the far left to plus infinity at the far right. For a very long time, it was believed that the REAL NUMBERS were the only numbers that existed -- hence the term "real"

The concept of the negative square root occurs in the theory of algebraic equations, most starkly as the solution to the simple equation: x^2 +1 = 0. The names that various mathematicians gave to the alleged solutions to such an equation are indicative of their attitude to the existence of the negative square root. They called it "unacceptable", "sophistic", "impossible" or just plain "wrong". To the French philosopher Rene' Descartes goes the honor of calling such numbers "imaginary" but he meant it in a dismissive way. Later when such numbers were finally welcomed into the canon, Swiss mathematician Leonard Euler resurrected Descartes' slur and christened these numbers IMAGINARY NUMBERS with no harm intended.

The crucial breakthrough towards making sense of IMAGINARY NUMBERS was achieved not by a mathematician but by a Danish surveyor Caspar Wessel (1745-1818) who postulated that imaginary numbers represented a distance at right angles to the REAL LINE. If the REAL LINE represents locations in the East/West direction, then according to Wessel the IMAGINARY LINE can represent locations in the North/South direction. No doubt from his experience in making maps, Caspar Wessel had invented what we today call "the complex plane", the mathematical country where real and imaginary numbers can dwell together in perfect harmony

Complex Plane: Red Line maps the Reals; Green Line maps the Imaginaries

Wessel's new geometric scheme literally put IMAGINARY NUMBERS on the map and opened up a flood of research into these previously dubious and mysterious quantities. Once IMAGINARY NUMBERS had been tamed, amazing calculations could be carried out and previously impossible tasks became easy.

For instance, what is the value of i to the ith power? Turns out this is a REAL NUMBER with the value of 0.2078... And easily calculated from equations derived from Wessel's construction.

With the introduction of Wessel's map (also called the Argand plane after a Parisian book-keeper who independently made the same discovery) one more kind of number has to be added to the list of man-made UNNATURAL NUMBERS. When one adds a REAL NUMBER (such as 2) to an IMAGINARY NUMBER such as 2i) one obtains a new number which is neither real nor imaginary. Numbers such as z = 2 + 2i have been given the name COMPLEX NUMBER. And the flat map on which COMPLEX NUMBERS enjoy their existence is accordingly called the complex plane.

Many remarkable discoveries have been made in the COMPLEX NUMBER realm. The theory of quantum mechanics uses COMPLEX POSSIBILITIES to represent Nature rather than REAL PROBABILITIES, a situation which still puzzles most physicists. And in Einstein's relativity, time can be viewed as an IMAGINARY quantity in contrast to the three REAL spatial dimensions.

Dozens of new mathematical formulas emerged from the study of the complex plane, including Euler's Identity which connects the sine and cosine function with the number e, the base of the natural logarithms.

e^ix = sin x + i cos x       Euler's Identity

This equation is enormously useful in many fields, especially in electrical engineering where the author Paul Nahin made his mark. When x = π, the Euler Identity reduces to:

e^iπ +1 = 0

This impressive little equation brings together in one simple statement 5 of the most important constants in mathematics. At age 15, the physicist Richard Feynman wrote this formula into his notebook with the caption: THE MOST REMARKABLE FORMULA IN MATH.

Since his specialty is electrical engineering, Nahin gives an example of the usefulness of COMPLEX NUMBERS in the analysis of electrical circuits. In the space of a few pages headed "A Famous Electronic Circuit That Works Because of Square Root of -1" Nahin describes the inner workings of a device called the phase-shift oscillator.  

Why is this device so famous? Turns out it was the first product manufactured in the legendary Palo Alto garage of William Hewlett and David Packard. Their variable-frequency audio oscillator became the basis of a billion-dollar industry. That's a lot of bang for a purely imaginary buck.

Hewlett-Packard 200A Audio Oscillator