Friday, January 22, 2016

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


27 comments:

Jack Sarfatti said...

Nice

Wes Hansen said...
This comment has been removed by the author.
Wes Hansen said...

According to Kevin Knuth and Philip Goyal, the complex field is required by Quantum Mechanics because of complementarity: http://knuthlab.rit.albany.edu/papers/PhysRevA.81.022109.pdf

It seems to make a lot of sense to me; more so than other arguments I've read . . .

You know, it seems to me that the Virtual Reality Hypothesis is at least as viable as Copenhagen or Relative State; Sean Carroll, a leading proponent of the relative state interpretation, is putting out a new book in which,according to his blog, he discusses ethics; but how could ethics apply in a relative state? If everything possible happens in a different perpendicular universe, there's no need to speak of free-will, hence, no need to bother with ethics. In one universe you help the little old lady carry her groceries across the busy intersection; in another you push the bitch into traffic and steal her groceries . . . Just where, exactly, is the room for ethics?

It's not that I believe in free-will, I most certainly do not, rather, I just think complementarity is a function of resource optimization, things are a bit fuzzy until some character in the VR happens to look . . .

You know, it just goes back to the old wisdom saying, "In every tree there roosts two birds . . . " So, you know, when we look so too does the Big Bird! Attention is focused . . .

nick herbert said...

One of the most astonishing facts about quantum theory is that there exist several very different (some insanely preposterous -- like Many Worlds) but no way to experimentally decide between interpretations. Seems to me that this situation must be temporary and that the next big advance in quantum theory (perhaps as a consequence of attempts at unifying QM with gravity will be testable consequences for some of the interpretations.

Anonymous said...

"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. "

Real numbers are unreal. Rather, as counting takes energy - particularly the energy used by the brain - then only a certain set of the infinite number of real numbers will ever be expressed in a finite universe by finite minds. The rest will never be realised.

How real are real numbers that can never be counted ?

Unless you have infinite universes, and can continue counting in one where you left off counting in the other. But even then - this is infinity we are talking about here, and the mind cannot conceive true infinity, it can only talk about it. Signifier without a signified. Horseshoes for unicorns.

Anonymous said...

Physics class.

Teacher: "...now, if imagine a line to infinity..."

Student 1: "Sir, can't"

Teacher: "uh?"

Student 1: "It would take forever. I have biology in twenty minutes time, we're doing metabolism. Hey wouldn't imagining an infinite line take an infinite amount of glucose, like brain metabolism, sir ?"

Teacher: "ah..."

Student 2: "Sir, why are you asking us to imagine an infinite line when it's impossible ? Sir, will this be in the exam ? "

Teacher: "err.."

Student 1: "I don't think these infinite lines exist."

Student 2: "Yeah, how do we falsify this claim ? If things can't be falsified they're not supposed to be true. he said that last week. Sir can't we scientifically test numbers and throw them out if they're not real ?"

Teacher: "Numbers aren't objects, physics doesn't apply to them..."

Student 1: "Brain is an object though, isn't it sir, and numbers come from the brain don't they ?"

Student 2: "He'll be going on about god next, you watch..."

Kurt Hanson said...

Anonymous …, the concept of infinity can be taught and expressed as a term or a finite number at any given y(∞) when ∞ is within a continuous state or system of addition; as the term, ∞ –> ∞ + 1 . The concept can be imagined now.

Nick, if you would be kind enough to entertain this question of mine I'd appreciate your thoughts: other than because the equation doesn't "work" that way, why wouldn't've been more correct to have the speed of light cubed in the equation, e=mc^2 ?

Anonymous said...

Well Kurt, I agree it can be taught and expressed. At least, the attempt is made. But I gave up saying I can imagine it. I can use the symbols, I can say the words, and write down definitions, but I can't imagine it. I used to nod and and say I could when people asked, but then I realised I was kidding myself. I'm quite strict in that I expect an image in the imagining, but as many people have found (back to Plato etc) - it's impossible to truly imagine mathematical entities, let alone draw them. I can fudge a square or triangle in my minds eye, sort of, but I definitely fail at anything infinite.

I can nod and say "sure, yeah" when someone asks me to imagine an infinite thing, and I've done it many times. But the truth is I'm lying and after poking the thought around for a bit, and asking around, I've come to the conclusion that I'm expected to lie.

It's fun to see people getting tangled up in word spaghetti !

Anonymous said...

..in other words, and I may be overstepping my competence here, maths just seems like an extension of language - and you can talk nonsense in language. So I can talk about weigh an eight sided triangle, and you can go on about the definition of infinity - but neither of us have, or will, see or imagine those things. Unless you're mind is very different.

But if your mind IS significantly different then it might be a good idea to go to your local psychology department and tell them that you can imagine something infinite using a finite amount of glucose, which I think would be a revolution in our understanding of brain metabolism.

So what people are saying is that they have the ability to set up an infinite image - which necessitates an infinite space in which to imagine. That's quite a claim. People seem to think they are like the TARDIS, bigger on the inside than the outside.

And for some reason THAT never gets discussed in physics classes. Talk about missing the wood for the trees.

Yet all these imaginings come to an end, usually within the space of a couple of seconds. Was an infinite line that was imagined for a few seconds really infinite ?

The only way for that to be true would be if it existed for real somewhere and was merely being glimpsed momentarily It's out there still, Platonically, like the great line server in the nether world.

But I doubt it.

Anonymous said...

typo - [weighing an eight sided triangle]

Anonymous said...

[typo - unless your mind is etc etc]

Advantage of Facebook - one can edit one's comments.

Anonymous said...

So.... DOES imagining an infinite line take an infinite amount of energy and brain fuel ? If so, how ?

http://www.scientificamerican.com/article/thinking-hard-calories/

"Extending the logic of such findings, some scientists have proposed the following: if firing neurons require extra glucose, then especially challenging mental tasks should decrease glucose levels in the blood and, likewise, eating foods rich in sugars should improve performance on such tasks. Although quite a few studies have confirmed these predictions, the evidence as a whole is mixed and most of the changes in glucose levels range from the miniscule to the small."

So if you are a physicist, and you claim to have just imagined something infinite, or even imagined something moving at the speed of light, you have just produced that from the energy you got from your breakfast.

That's a real bargain !

So what's really going on when people say these things, and when did it start ? What are the origins of talking about infinity ?

Evolutionary theory tells us we are evolved from primates experiencing a finite environment of solid and limited objects. Even the sky, as a simple blue covering, doesn't have to be seen as going on forever if there is no compelling reason to do so.

I can only guess this is tied to the development of myth and abstract reasoning. Or funny mushrooms growing on cow pats. But nobody seems to know.

The gods can do ten impossible things before breakfast, and what does the Earth rest on ? Elephants all the way down....

Anonymous said...

...did everyone just move away from me by a couple of feet ? I was hoping for a response... :-)
Glad to have got that off my chest anyway.
Infinity. Pah!

Kurt Hanson said...

Places to go. People to see. Things to do. I live in New Yawk, and so it goes. I'm back.

Anonymous, you have to let the equation stand alone, and not attempt to picture a finite conclusion to infinity. Similar to limits or progressions which can be finite, infinity is not static but a dynamic equation: f(∞) = ∞ + (∞+1) The equation is a running engine of continuous addition, as opposed to an equation which terminates with, or becomes static with a term, or result.

Infinity is a dynamic equation a better mathematician than I could devise to scratch upon the blackboard, as opposed to an equation such as 1 + 1 = 2, which is static.

Anonymous said...

:)

Anonymous said...

So, all the times in maths and physics classes, when I was asked to imagine something like a straight line, or an angle or an asymptotic graph going to infinity, or a beam of light.... what was I actually seeing ?
Someone (psychologist?) must have surveyed a class at one some time to see what was going on up top.
Beams of light are easy. They look like lasers from Star Wars, or neon strip lights, hanging in blackness.
Number lines tend to look like something drawn on paper, maybe going up to a double digit and stopping but it's all rather blurry. Same with angles, and shapes, sort of, but sometimes also floating against a plain dark background.

I have no idea what other people see in their minds. Big fat fuzzy triangles ? Wobbly circles. Incomplete beams of light because attention is always wandering ?

Maybe some autistic geniuses CAN picture immense number lines in one go, and other marvelous feats, but I doubt most people can.

And then where is infinity in all this ? Nowhere, seems to me.

Anonymous said...

Kurt, what do you see when asked to imagine an infinite line ?

And if I say, "Imagine this rocket moving at the speed of light...." what do you see ?

And how do you see an infinite number line - how many numbers does it have, what is it written on, does it have a colour, what is the background, how long does it take to vizualise ?

Kurt Hanson said...

The infinite line is represented as an equation of additive terms: ∞ = (∞+x) I am not a mathematician who can scribe the proper notation to this concept and blog post. The symbol, or term, ∞ represents a system of continuous counting, whether addition, or subtraction, multiplication, or division by x. This shouldn't be difficult to conceptualize, and to "picture" such a concept in the mind is an exercise in futility.

Futility unlike such wrought by certain physics teachers today smugly vocalizing absurdities to their students, "Many today hypothesize the universe may have originated from nothing," and then scratching the numeral zero upon the blackboard, thus leaving a few of the best and brightest students to wonder how, through multiplication or division of zero, a term of one or greater could possibly occur.

To realize the physics departments of many Western universities are filled with so many of these types, perhaps is why a monkey wrench has been thrown into the logic faculties of many good students' minds 🖖🏻

Anonymous said...

"This shouldn't be difficult to conceptualize, and to "picture" such a concept in the mind is an exercise in futility"

That's the bunny. Now why don't they ever tell you that ?

Kurt Hanson said...

Picture a continuous sequence of division to an irrational number. The tiny bits to a divided number line can be imagined magnified to continue the thought of division for a time but how long does one care to further doing so? I can imagine infinity would be the same and, to begin picking my nose, or something would distract me from the futility of doing so in time.

Anonymous said...

"how long does one care to further doing so? " That's what I'm asking - what do you actually see when you do this stuff ?
What do you see when I ask "imagine an infinite line" ?
Maybe some people are more visual than others. I was talking to someone with advanced degree in logic, brilliant with symbols, but when I said how I imagined all sorts of things as I listened to music it was plain that he didn't.
I thought the personal internal cinema was part of the deal when listening to music, for everyone, but it turns out it's not a universal thing.
Maybe it's like that with maths and physics- very variable.

Anonymous said...

"I can imagine infinity would be the same and, to begin picking my nose, or something would distract me from the futility of doing so in time."

Indeed.

J said...

re: number lines

"Tammet visualizes numbers in their unique forms and then melds them together to create a new image for the solution. When asked to multiply 53 by 131, he explains the solution in shapes and textures: "Fifty-three, which is round, very round...and larger at the bottom. Then you've got another number 131, which is longer a little bit like an hourglass. And there's a space that's created in between. That shape is the solution. 6,943!""


Various surveys indicate that as many as 10-15 percent of people report some kind of graphic mental representation of numbers. Francis Galton, a psychologist and cousin of Charles Darwin, carried out the first of these surveys back in 1880. The responses he obtained offer a fascinating glimpse into the sheer variety of mental number representations, though many number lines also displayed consistent patterns: about two-thirds were left-to-right and ran more often upwards than downwards. Some of the number lines had twists and bends, some turned upside down or back on themselves. A physicist replying to Galton's questionnaire described seeing numbers in the form of a horseshoe, with 0 at the bottom right, 50 at the top and 100 at the bottom left. Another respondent, a barrister, described visualising the numbers 1-12 as though on the face of a clock, with the following numbers tailing off afterwards into an undulating stream with the tens – 20, 30, 40, etc. – at each bend."

http://abcnews.go.com/2020/autistic-savant-daniel-tammet-solves-problems-blink-eye/story?id=10759598

J said...

...so you go, not all number lines are the same :-)

Ta

Kurt Hanson said...
This comment has been removed by the author.
Kurt Hanson said...

J …, "Fascinating, Captain," a focused, pensive Spock would indubitably say 🖖

Kurt Hanson said...
This comment has been removed by the author.