The Space Between the Stars

This image was selected as a picture of the we...

This image was selected as a picture of the week on the Farsi Wikipedia for the 8th week, 2011. (Photo credit: Wikipedia)

Space is big. The Voyager spacecraft (Voyager I and II) were launched in 1977 and are, forty years later, only just entering interstellar space. Though the exact point at which space becomes “interstellar” is debatable.

The Voyagers will take 40,000 years or so to reach one of the stars in the “local” group. That’s about one fifth of the time that humans have existed as a separate species. Or 400 times as long as the length of time that a human is able to live. If a generation is around 20 years long, that is about 2,000 generations. It is a long, long time, and we may well be extinct as a species by then, for one reason or another.

English: Diagram of the Voyager spacecrafts wi...

English: Diagram of the Voyager spacecrafts with labels pointing to the important instruments and systems. (Photo credit: Wikipedia)

The “local group” of stars is an arbitrary group of stars which are (relatively) close to the Sun. I’m unsure whether they really constitute a group of bodies bound by gravity or whether they are close to the sun by chance. Of course if any of the stars in the local group are bound by gravity, then the stars would form a binary or multiple star system.

Of course, our star and all the others in the local group are part of our galaxy, the Milky Way. Specifically we are part of one of the arms of the Milky Way, which is a spiral galaxy. All stars in the Milky Way are bound by gravity, with the possible exception of stars which are merely passing through the Milky Way at this time.

English: Using infrared images from NASA's Spi...

English: Using infrared images from NASA’s Spitzer Space Telescope, scientists have discovered that the Milky Way’s elegant spiral structure is dominated by just two arms wrapping off the ends of a central bar of stars (Photo credit: Wikipedia)

Just like stars, galaxies seem to form groups, which then form super-groups and so on. All these structures are several orders of magnitude larger than the prior smaller ones, are more complex and contain more matter.

The majority of space however is just space. The gaps between the bits of matter, stars, systems, galaxies, groups and so on contain almost nothing, or a seething sea of virtual particles depending on how you look at it.

Map of the Local Group of Galaxies

Map of the Local Group of Galaxies (Photo credit: Wikipedia)

The “almost nothing” consists of a very small number of particles (usually hydrogen atoms or nuclei, protons) in a cubic metre. For comparison the best vacuum that can be created on Earth may contain several million atoms in that volume. This of the same order of magnitude as molecular clouds as observed by astronomers. Molecular clouds are among the densest clouds observed in space.

The stars, planets, asteroids and similar bodies comprise only a very small part of the Universe and the average density of the Universe is much the same as the density of empty space. In other words, the Voyagers are heading into areas where the conditions are more typical of the Universe than those around our star.

Voyager 1 is currently within the heliosheath ...

Voyager 1 is currently within the heliosheath and approaching interstellar space. (Photo credit: Wikipedia)

I mentioned virtual particles earlier. While virtual particles show up as short-lived particles that briefly come into existence in some particle interactions, the virtual particles that I refer to come into existence in a vacuum as pairs and almost immediately mutually annihilate. Though they do not interact with other matter, they do have an effect which can be measured.

Mathematicians have a different concept of space. In mathematics space is a (usually) three dimensional construct that serves merely to separate and give structure to such things as points, lines, planes, volumes and shapes. Point A is distinguished from point B by the distance between them and also the orientation of a line joining them.

In a simple case every point has (usually) three coordinates which define its position relative to some fixed point or origin and fixed coordinate system. The coordinate system can be any system that locates the point.

For instance, you can describe the point A’s position as “Face along a given axis, rise up until you are level with point A. The distance moved up is one coordinate. Rotate left through an angle until a line parallel to the plane you rose up from passes from you through the point A. The angle you turned through is the second coordinate. The third coordinate is the distance along the line from you to the point.

English: 3D spherical polar coordinates

English: 3D spherical polar coordinates (Photo credit: Wikipedia)

I’ve described a cylindrical coordinate system, but the coordinate system may be any system that gives three unique (in that system) coordinates for point A. A common system is the Cartesian system of three mutual perpendicular axes. Another is the spherical system, defined by two angles and a distance.

Of course, such systems can be generalised to more dimensions or fewer, depending on the needs of the mathematician. Most people can understand simple two dimensional graphs which are usually drawn using a two dimensional Cartesian coordinate system.

English: Diagram showing relationship between ...

English: Diagram showing relationship between polar and rectangular coordinates (Photo credit: Wikipedia)

Of course scientists use mathematical models for various purposes. For instance the scientist may wish to know the probability of one hydrogen atom in the interstellar space meeting another such molecule. Since we have only about one such atom in every cubic metre, the probability is going to be small, but, of course, we can assume as lone a time period as we wish.

Gravity has to be figured in, to be sure, but a long time will be required for such atoms to collide. If the atoms are by chance moving slowly relative to each other, they may stick together and form the basis of a particle of matter. Such a clump might attract other atoms and before long (well actually after literally an astronomical length of time) a star will form.

The trajectories that enabled Voyager spacecra...

The trajectories that enabled Voyager spacecraft to visit the outer planets and achieve velocity to escape our solar system (Photo credit: Wikipedia)

It must have happened otherwise we would not be here. We are the result of matter aggregating and then exploding. All the atoms in our bodies that are not hydrogen were made in the centres of stars. The stars have to have exploded to allow these atoms to end up in our bodies.

A long time has passed since the birth of the Universe. In that time matter has crept together hydrogen atom by hydrogen atom until great collections of atoms have compressed in the centre to the point where nuclear reactions have occurred. Hydrogen fused to helium, then to heavier elements all the way up to Uranium.

Ball-and-stick model of the haem a molecule as...

Ball-and-stick model of the haem a molecule as found in the crystal structure of bovine heart cytochrome c oxidase. Histidine residues coordinating the iron atom are coloured pink to distinguish them from haem a. Colour code: Carbon, C: grey-black Hydrogen, H: white Nitrogen, N: blue Oxygen, O: red Iron, Fe: blue-grey Structure by X-ray crystallography from PDB 1OCR, Science (1998) 280, 1723-1729. Image generated in Accelrys DS Visualizer. (Photo credit: Wikipedia)

At which point the stars have exploded throwing all the elements out into the Universe. These elements then crept together again to produce new stars like our sun and gaseous and rocky planets orbiting them. Prior to this there were no rocky planets and no life. We live in the Universe Mark II.

NGC 1531

NGC 1531 (Photo credit: Wikipedia)

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Capitalism

Labor supply and demand in a perfect competiti...

Labor supply and demand in a perfect competition labor market (Photo credit: Wikipedia)

A free market is one in which there is no government, monopoly or other authoritative interference in the workings of a market. However there is in practise no such thing, as there are always constraints on a market from one or more of those sources.

For instance, in a small country there may be only two or three organisations which are involved in the whole supply chain, and if they are much the same size there is no drive to compete strongly. If one large competitor decided to drive another large competitor out of the market, it would be expensive and difficult, and would more likely than not trigger monopoly prevention legislative mechanisms.

An example of a cover from a Monopoly video game

An example of a cover from a Monopoly video game (Photo credit: Wikipedia)

On the other hand, a small competitor might be worth an aggressive approach as an attack could be targeted and localised. It would be cheaper and while it might raise a few worries about lack of choice (in an area), it would not trigger any monopoly laws.

An open market goes hand in hand with the laws of supply and demand. Generally these are expressed as graphs showing the intersection of the supply curve (an upwards trending line) with the demand curve (a downwards trending line). Any change in conditions is shown by other lines more or less parallel to the first.

Fig5 Supply and demand curves

Fig5 Supply and demand curves (Photo credit: Wikipedia)

These curves can only be illustrative as they are almost never drawn with quantified axes, and the curves are drawn without the use of any measured data. They are arbitrary. Nevertheless they purport to show the effect of market changes on the equilibrium or balance point where the curves cross.

While the laws of supply and demand may be true in the sense that if either the price or demand changes the other also changes, the graphs are of little practical use, and they are only marginally mathematical, as definite mathematical conclusions cannot be made from them. It is impossible to quantify the effect on demand of raising the price of a can of beans by 10c, for example.

Curried Beans

Curried Beans (Photo credit: Wikipedia)

Nevertheless one could probably use the graphs to suggest that if the price changes in one direction the demand will move in another direction, and these guesses may be used to decide on price changes. It’s definitely a guess, though as the opposite may happen – if you put up a sign saying “Beans now $1.55 per can”, having raised the price by $0.05, you may sell more as you have drawn the customers’ attention to the beans.

The “Free Market”, the “Laws of Supply and Demand”, and the principle of “Laissez Faire” are part of the backbone of Capitalism. Capitalism is a robust economic system which has achieved immense feats and advances. It has harnessed science and sent men to the Moon, given us a computer and communication devices in our pockets. There is no doubt that Capitalism has been hugely successful.

A capitalism's social pyramid

A capitalism’s social pyramid (Photo credit: Wikipedia)

In spite of its amazing successes, there have always been drawbacks to Capitalism. The trend of prices is to rise continually, though at times, they do fall, as demand reduces in recessions and market collapses. These recessions and collapses hurt the poor much more than the rich, as the poor have fewer resources to cope with these setbacks.

Capitalist markets lead to concentration of resources, especially money, in the hands of the rich, and a scarcity of resources in the hands of the poor. It leads to the growth of large market dominating firms, as one firm succeeds while others fail. The successful firm often widens its control of the market by purchasing up and coming smaller firms or older firms who themselves may control a smaller market niche.

Capitalism fosters the growth of the gap between the very rich and the very poor. It is often argued that, in countries where the economic system is Capitalist in nature, the “poor” have much more in the way of consumer items than their parents could have imagined. Most people have a car. Most people have a television. Most have a cellphone.

This is all true, but that is only because these items are both essential and relatively cheap. At the same time, health care is becoming unaffordable for many of the new poor. Schooling is also a huge drain on the poorer families. Many poor people work at multiple jobs to bring up their children and pay for the operations that their parents are coming to need.

As a result, many of the new poor live from day-to-day, with no real opportunity to save for retirement or to lay by a little money to allow for the vicissitudes of life. A small accident that requires time off work and consequently reduction of income becomes a disaster in such a situation.

Capitalism stratifies society and the bottom strata, often those with a lack of education or intelligence, lags behind those who are in higher strata. Those at the highest levels tend to outstrip those at lowest levels until their wealth, to those in lower strata, appears as meaningless numbers. What the difference between $100 and $1000 to those at the bottom? It’s a huge amount. What about the difference between $10 billion and $100 billion? It’s irrelevant.

English: Memorial to a wealthy benefactor

English: Memorial to a wealthy benefactor (Photo credit: Wikipedia)

Capitalist market forces tend to favour those who already have over those who don’t and the barriers that prevent those in the lower strata from moving up are immense. Those few who make are the lucky ones. Yes, luck plays almost as big a part in entrepreneurial success as luck does in winning the lotto.

Capitalism is the best economic system that we have ever had, without a doubt. It is however not without its flaws. Socialism is not a good economic system, but purports to deal with the issues of poverty by redistribution of wealth. (Maybe I’ll do a piece on socialism’s flaws at some time).

Karl Marx (1818-1883)

Karl Marx (1818-1883) (Photo credit: Wikipedia)

Capitalism however does not deal with poverty or the poor. Some effects do trickle down and today’s poor appear rich in comparison with the corresponding strata in the past, but the fundamental poverty still exists.

It would be nice to think that there is some other system, waiting for someone to discover it. The odds are probably good, as no system lasts forever. What it would look like I’ve no idea. We would need to get a much better scientific view of the so-called social sciences to really solve this fundamental problem.

Iconic image for social science.

Iconic image for social science. (Photo credit: Wikipedia)

 

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Simple Arithmetic

Addition, division, subtraction and multiplica...

Addition, division, subtraction and multiplication symbols (Photo credit: Wikipedia)

There periodically appears on the Internet an arithmetic type of puzzle. Typically it will be a string of small natural numbers and a few arithmetic operations, such as “3 + 7 x 2 – 4” and the task is to work out the result.

The trick here is that people tend to perform such a series of calculations strictly from left to right, so the sequence goes:

3 + 7 = 10, 10 x 2 = 20, 20 – 4 = 16. Bingo!

Most mathematicians, and people who remember maths from school would disagree however. They would calculate as follows:

7 x 2 = 14, 3 + 14 = 17, 17 – 4 = 13. QED!

Why the difference? Well, mathematicians have a rule that states how such calculations are to be performed. Briefly the calculations is performed from left to right, but if a multiplication or division is found between two numbers, that calculation is performed before any additions or subtractions. In fact the rule is more complex than that, and a mnemonic often used to remember it is “BODMAS” or “BEDMAS” (which I’m not going to explain in detail here. See the link above).

This rule is only a convention and so is not followed everywhere, so there are various “correct” answers to the problem. Also, there are still ambiguities if the conventions are applied which could cause confusion. However, most people with some mathematical training would claim that 13 is the correct answer.

Interestingly, computer programming languages, which are much stricter about such things, codify the precedence of operations in a calculation exactly, so that there can be no ambiguity. It is the programmers task to understand the precedence rules that apply for a particular language.

example of Python language

example of Python language (Photo credit: Wikipedia)

In most cases the rules are very, very similar, but it is the documentation of the language which describes the rules of precedence, and wise programmers study the section on operator precedence very closely.

There are ways of specifying an arithmetic problem uniquely, and one of those (which is sometimes of interest to programmers) is “Reverse Polish Notation“. Using this my original puzzle becomes “3 7 2 x + 4 -” which looks odd until you understand what is going on here.

Illustration of postfix notation

Illustration of postfix notation (Photo credit: Wikipedia)

Imagine that you are traversing the above list from left to right. First you find the number “3”. This is not something you have to do, like “+”, “-“, “x” or “/”, so you just start a pile and put it on the bottom. The same goes for “7” and “2”, so the pile now has “3” on the bottom and “2” at the top and “7” in the middle.

Next we come across “x”. This tells us to do something, so we pull the last two things off the pile and multiply them (7 x 2 = 14) and stick the result, “14” back on the stack which now contains “3” and “14”. The next thing we find is “+” so we pluck the last two things off the pile “3” and “14” and add them, putting the result “17” back on the (empty) pile.

Next up is “4” which we put on the pile, and finally, we have “-“, so we pull the two last elements from the pile (“17” and “4”) and subtract the second from the first, giving “13” (Yay!) and that is the answer which we put back on the stack. The stack now contains nothing but the answer.

This looks confusing, but that may be because we are used to the conventional left to right way of doing things. It is actually easier for a computer to understand the RPN version of the puzzle and there are no ambiguities in it at all. Technically, it’s a lot simpler to parse than the conventional version.

Image for use in basic articles dealing with p...

Image for use in basic articles dealing with parse trees, nodes, branches, X-Bar theory, linguistic theory. (Photo credit: Wikipedia)

Parsing is what happens when you type a command into a computer, or you type something complex, such as a credit card number into the checkout section of a web site. The computer running the web site takes your input and breaks it up if necessary and checks it against rules that the programmer has set up.

So, if you type 15 numbers or 17 numbers into the field for the credit card number, or you type a letter into the field by mistake, the computer will inform you that something is wrong. Infuriatingly, it may be not be specific about what the trouble is!

Español: Un Guru meditation en una amiga

Español: Un Guru meditation en una amiga (Photo credit: Wikipedia)

Anyway, back to the arithmetic. It grates with me when people make simple arithmetical errors and then excuse themselves with the phrase “I never was much good at maths at school”! That may well be true, but to blame their problems with arithmetic of the whole diverse field of mathematics.

It’s like saying “I can’t add up a few numbers in my head or on paper because I missed the class on elliptic functions“! It’s way over the top. For some reason people (especially those who can’t get their head around algebra) equate the whole of mathematics with the bit that they do, which is the stuff about numbers, which is arithmetic.

English: Weierstrass p, Stylised letter p for ...

English: Weierstrass p, Stylised letter p for Weierstrass’s elliptic functions from Computer Modern font (obtained by TeX command \wp) Deutsch: Weierstrass p, stilisierter Buchstabe p für die elliptische Funktion von Weierstrass in der Computer-Modern-Schrift (generiert durch das TeX-Kommando \wp (Photo credit: Wikipedia)

As we evolved, we started counting things. It’s important to know if someone has got more beans than you or that you have enough beans to give everyone one of them. We invented names for numbers and names for the things (operations) we did on them.

We did this without much thought about what numbers actually are. We as a species have only relatively thought deeply about numbers fairly recently, and we only discovered such things as real numbers and geometry in the last couple of thousand years so it is not surprising that the average brain has yet to expand to cope with the more advanced mathematical concepts.

Graphic showing the relation between the arith...

Graphic showing the relation between the arithmetic mean and the geometric mean of two real numbers. (Photo credit: Wikipedia)

This could be why so many people these days equate fairly simple arithmetic with mathematics as a whole – our brains are only now coming to grips with the concept that there is more to maths than simply manipulating numbers with a very few simply operations.

It may be that the average human brain never will get to grips with more advanced maths. After all, people can survive and thrive in the modern world with on a rudimentary grasp of mathematics, the arithmetic part.

English: the arithmetic sequence a_n=n Deutsch...

English: the arithmetic sequence a_n=n Deutsch: die arithmetische Folge a_n=n (Photo credit: Wikipedia)

Some human brains however do proceed further and much of modern society is the result of mathematics in its wider sense applied to the things that we see around us. For instance,  we could not have sent men to the moon without advanced mathematics, and technology relies heavily on mathematics to produce all sorts of things. It’s a good things that some brains can tell the difference between the field of arithmetic and mathematics as a whole.

English: A .gif animation of the vibration cor...

English: A .gif animation of the vibration corresponding to the third smallest eigenvalue of the electric pylon truss problem from EML 4500 HW 6. (Photo credit: Wikipedia)

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Here be Dragons

Dragon Green

Dragon Green (Photo credit: Wikipedia)

Dragons, big scaly fire breathing reptiles. So many of our folk tales and even many modern tales include dragons as an important component, usually as a hostile force. Of course in many tales the dragon is merely a device to give the hero some seemingly impossible difficulty to overcome.

Sometimes dragons are mere beasts, but in some tales they are intelligent, if malevolent, beasts. Smaug, in “The Hobbit” by J R R Tolkien is of the latter kind. He sits on a pile of treasure and is furious when Bilbo Baggins steals a golden cup. He later accuses Bilbo of trying to steal from him (which is true).

Smaug as he appears in the animated film.

Smaug as he appears in the animated film. (Photo credit: Wikipedia)

The patron saint of England is Saint George, who was an early Christian martyr. Saint George is noted for slaying a dragon to save a princess. The princess was intended as a sacrifice to the dragon who was causing sickness in the inhabitants of the local town.

In legends, once a dragon has been killed, it’s body, blood and teeth could be used for various purposes. Sometimes the blood was beneficial to humans, conferring invincibility or other virtue, or it could be poisonous. The teeth could be sowed to raise armies, sometimes of skeletons.

Most fictional and mythical dragons are scaly reptiles, but one of the odd ones out is the furry creature called the “Luck Dragon” in the film “The NeverEnding Story”. This dragon had a head resembling that of a dog, front limbs and a tapering furry body which merged into a tail.

Most fictional dragons are noble creatures, but the “Swamp Dragons” created by Terry Pratchett in his discworld series of books which are altogether baser than the “Noble Dragons“. Swamp dragons are small creatures, are almost always ill (because of their diet) and are prone to explode if very ill or excited.

English: The Nine Dragon Wall in the Beihai Pa...

English: The Nine Dragon Wall in the Beihai Park, a large imperial garden in central Beijing (Photo credit: Wikipedia)

On one occasion one exploded after being enraged at the sight of itself in a mirror, imaging that it was in the presence of a rival. It does appear though that the fraught gastric processes may have a reason – a swamp dragon is described as flying on its stubby wings by emitting gasses created by its digestive processes.

Some dragons can apparently be tamed. In Anne McCaffrey’s Pern of books series, a partnership has developed between the flying dragons and humans to deal with the threat of “thread” which comes from a companion planet and is inimical to all life forms on Pern.

All the Weyrs of Pern

All the Weyrs of Pern (Photo credit: Wikipedia)

These dragons, which similar the standard dragon type from mythology, are large enough to be ridden by humans, and breath fire to kill the thread on being fed a particular type of rock. The dragons are genetically modified from the much smaller fire lizards and communicate with their riders by telepathy.

One unique ability of these dragons is to teleport from place to place carrying their riders with them. It also becomes apparent that they can also time travel while teleporting, Unsurprisingly, Terry Pratchett created a cameo parody or homage to the Pern books and their dragons in the first book of his discworld series, “The Colour of Magic”.

The Discworld as it appears in the SkyOne adap...

The Discworld as it appears in the SkyOne adaptation of The Colour of Magic. (Photo credit: Wikipedia)

Where there are dragons, the untamed variety, The more I think about them, the more I remember cases of dragons in literature and in films. A fairly recent example is the film “How to Train Your Dragon“. The dragons are at first treated as hostile, but the aspiring dragon killer, Hiccup, finds an injured dragon, it transpires that the dragons are friendly creatures and only attack humans because the humans are attempting to exterminate them.

The modern dragon is built along the same physical plan, whatever the media they are described in. Dragons are reptiles, usually lay eggs, mostly have four legs or limbs and a pair of wings. Mostly they breathe fire, and where this is touched on, it is usually implied that the fire is generated internally by ingesting and digesting rocks.

Saint George and the Dragon at Casa Amatller

Saint George and the Dragon at Casa Amatller (Photo credit: Wikipedia)

However, early myths about dragon describe dragons as more akin to large serpents, even to the extent of having no limbs. Indeed, in early texts, the word used for dragons also means serpent.

Interestingly, although England’s patron saint is a dragon killer, the red dragon has come to symbolise Wales. “Y Ddraig Goch” is a red dragon and can be found on the Welsh national flag. He attains ascendency over an invading white dragon who symbolises the Saxons, after a long battle and an interval when both dragons were imprisoned in a hill in Snowdonia.

English: Welsh Dragon

English: Welsh Dragon (Photo credit: Wikipedia)

Dragons are associated power. Having scales and claws, and being able to breath fire, are attributes that give them strong defensive and offensive capabilities. Their size gives them strength and they are a very great challenge to any heroes who take them on. Often they can only be defeated by trickery or luck, such as when Smaug was killed because he had a small unprotected area on his belly which allowed the hero to shoot fatally in that one spot.

Dragons are associated with magic, with wizards, witches, princes and princesses and supernatural items and events of all sorts. “Dungeons and Dragons” melds all these factors into a table top and role playing game which was popular in the 1970s and 1980s.

Dungeons & Dragons game in IV Getxo Comic Con.

Dungeons & Dragons game in IV Getxo Comic Con. (Photo credit: Wikipedia)

However, dragons as such do not appear to be a large factor in the game, which revolves more around the characters who may be clerics, fighters or magic users. Dungeons and Dragons does have monsters and while some may be dragons, there are many other types of monsters, which may or may not be controlled by other players taking part in the game.

Finally, to bring this ramble through the topics dragons to a close, I will mention one other dragon that I recall from films, and that is the one which appeared in film and book “Doctor No” by Ian Fleming.

From http://www.33rdinfantrydivision.org/archi...

From http://www.33rdinfantrydivision.org/archivesphotos/may5_flamethrower.jpg source information from 33rdinfantrydivision.org : S/Sgt Bill Seklscki fires a flame thrower at a Japanese position near Manacag, Luzon, P.I. Jan 25 1945. Photo: National Archives. Webmaster note: 33rd did not land on Luzon until Feb 1945. Date of picture could be a mistake. (Photo credit: Wikipedia)

In the book and film James Bond is sent to Crab Key to investigate Doctor No. Rumours abound about the “dragon” which roams the island, deterring anyone from visiting. In the end the fire breathing dragon turns out to be a vehicle fitted with a flamethrower. This goes to show that while fictional and mythical dragons may be common, real dragons are scarcer than hen’s teeth.

Dr. No as seen in the James Bond Jr. animated ...

Dr. No as seen in the James Bond Jr. animated series. (Photo credit: Wikipedia)

 

 

 

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Let’s be Rational – Realer Numbers

Symbol often used to denote the set of integers

Symbol often used to denote the set of integers (Photo credit: Wikipedia)

Leopold Kronecker said “God made the integers, all else is the work of man”. (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”). However man was supposedly made by God, so the distinction is logically irrelevant.

I don’t know whether or not he was serious about the integers, but there is something about them that seems to be fundamental, while rational numbers (fractions) and real numbers (measurement numbers) seem to be derivative.

English: Note: The irrational and rational num...

English: Note: The irrational and rational numbers make the set of real numbers. (Photo credit: Wikipedia)

That may be due to the way that we are taught maths in school. First we are taught to count, then we are taught to subtract, then we are taught to multiply. All this uses integers only, and in most of it we use only the positive integers, the natural numbers.

Then we are taught division, and so we break out of the world of integers and into the much wider world of the rational numbers. We have our attention drawn to one of the important aspects  of rational numbers, and that is our ability to express them as decimal fractional numbers, so 3/4 becomes 0.75, and 11/9 becomes 1.2222…

Parts of a micrometer caliper, labeled in Engl...

Parts of a micrometer caliper, labeled in English. Someone can replace this with a prettier version anytime. (Photo credit: Wikipedia)

The jump from there to the real numbers is obvious, but I don’t recall this jump being emphasised. It barely (from my memories of decades ago) was hardly mentioned. We were introduced to such numbers as the square root of 2 or pi and ever the exponential number e, but I don’t recall any particular mention that these were irrational numbers and with the rational numbers comprised the real numbers.

Why do I not remember being taught about the real numbers? Maybe it was taught but I don’t remember. Maybe it isn’t taught because most people would not get it. There are large numbers of rational accountants, but not many real mathematicians. (Pun intended).

Square root of two as the hypotenuse of a righ...

Square root of two as the hypotenuse of a right isosceles triangle of side 1. SVG redraw of original work. (Photo credit: Wikipedia)

In any case I don’t believe that it was taught as a big thing, and a big thing it is, mathematically and philosophically. It the divide between the discrete, the things which can be counted, and the continuous, things which can’t be counted but are measured.

The way the divide is usually presented is that the rational numbers (the fractions and the integers) plus the irrational numbers make up the real numbers. Another way to put it, as in the Wikipedia article on real numbers, is that “real numbers can be thought of as points on an infinitely long line called the number line or real line”.

Collatz map fractal in a neighbourhood of the ...

Collatz map fractal in a neighbourhood of the real line (Photo credit: Wikipedia)

Another way to think of it is to consider numbers as labels. When we count we label discrete things with the integers, which also do for the rational numbers. However, to label the points on a line, which is continuous, we need something more, hence the real numbers.

Real numbers contain the transcendental numbers, such as pi and e. These numbers are not algebraic numbers, which are solutions of algebraic equations, so are defined by exclusion from the real numbers. Within the transcendental numbers pi and e and a quite large numbers of other numbers have been shown to be transcendental by construction or argument. I sometimes wonder if there are real numbers which are transcendental, but not algebraic or constructible.

A rather sexy image of Pi from the german wiki...

A rather sexy image of Pi from the german wikipedia. (Photo credit: Wikipedia)

The sort of thing that I am talking about is mentioned in the article on definable real numbers. It seems that the answer is probably, yes, there are real numbers that  are not constructible or computable.

Of course, we could list all the constructible real numbers, mapped to the real numbers between 0 and 1. Then we could construct a number which has a different first digit to the first number, a different second digit to the second number and so on, in a similar manner to Cantor’s diagonal proof,  and we would end up with a number that is constructed from the constructible real numbers but which is different to all of them.

English: Georg Cantor

English: Georg Cantor (Photo credit: Wikipedia)

I’m not sure that the argument holds water but there seems to be a paradox here – the number is not the same as any constructible number, but we just constructed it! This reminds of the “proof” that there are no boring numbers.

So, are numbers, real or rational, just labels that we apply to things and things that we, or mankind as Kronecker says, have invented? Are all the proofs of theorems just inventions of our minds? Well, they are that, but they are much more. They are descriptions of the world as we see it.

Apollonius' theorem

Apollonius’ theorem (Photo credit: Wikipedia)

Whether or not we invented them, numbers are very good descriptions of the things that we see. The integers describe things which are identifiably separate from other things. Of course, some things are not always obviously separate from other things, but once we have decided that they are separate things we can count them. Is that a separate peak on the mountain, or is it merely a spur, for example.

Other things can be measured. Weights, distances, times, even the intensity of earthquakes can be measured. For that we of course use rational numbers, while conceding that the measurement is an approximation to a real number.

Tape ruler

Tape ruler (Photo credit: Wikipedia)

A theorem represents something that we have found out about numbers. That there is no biggest prime number, for example. Or that the ratio of the circumference to the diameter is pi, and is the same for all circles.

We certainly didn’t invent these facts – no one decided that there should be no limit to the primes, or that the ratio of the circumference to the diameter of a circle is pi. We discovered these facts. We also discovered the Mandlebrot Set and fractals, the billionth digit of pi, the bifurcation diagram, and many other mathematical esoteric facts.

Mandlebrot Fractal made with Paint.NET

Mandlebrot Fractal made with Paint.NET (Photo credit: Wikipedia)

It’s like when we say that the sky is blue. To a scientist, the colour of sunlight refracted and filtered by the atmosphere, peaks at the blue wavelength. The scientist uses maths to describe and define the blueness of the sky, and the description doesn’t make the sky any the less blue.

The mathematician uses his tools to analyse the shape of the world. He tries to extract as much of the physical from his description, but when he uses pi it doesn’t make the world any the less round as a result. Mathematics is a description of the world and how it works at the most fundamental level.

English: Adobe photoshop artwork illustrating ...

English: Adobe photoshop artwork illustrating a complex number in mathematics. (Photo credit: Wikipedia)

[I’m aware that I have posted stuff on much the same topic as last time. I will endeavour to address something different next week].

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Rational versus real

English: Dyadic rational numbers in the interv...

English: Dyadic rational numbers in the interval [0,1] (Photo credit: Wikipedia)

(My last post was very late because I had taken part in a 10km walk on the Sunday and spent the week recovering.)

There’s a fundamental dichotomy at the heart of our Universe which I believe throws some light on why we see it the way we do. It’s the dichotomy between the discrete and the continuous.

A rock is single distinct thing, but if you look closely, it appears to be made of a smooth continuous material. We know of course that it is not really continuous but is constructed of a mesh of atoms each of which is so small that we cannot distinguish them individually, and which are connected to each other with strong chemical and physical bods.

An early, outdated representation of an atom, ...

An early, outdated representation of an atom, with nucleus and electrons described as well-localized particles on well-localized orbits. (Photo credit: Wikipedia)

If we restrict ourselves to the usual chemical and physical processes we can determine to a large extent determine what the atoms are which comprise the rock, and we can make a fair stab at how they are connected and in what proportions.

We can explain its colour and its weight, strength, and maybe its magnetic properties, even its value to us. (“It’s just a rock!” or “It’s a gold nugget!”) We have a grab bag of atoms and their properties, which come together to form the rock.

English: Gold :: Locality: Alaska, USA (Locali...

English: Gold :: Locality: Alaska, USA (Locality at mindat.org) :: A hefty 63.8-gram gold nugget, shaped like a pancake. Very beautiful and classic locality nugget. 4.5 x 3 x 0.6 cm Deutsch: Gold :: Fundort: Alaska, Vereinigte Staaten (Fundort bei mindat.org) (Photo credit: Wikipedia)

The first view of atoms was that they were indivisible chunks with various geometric shapes. This view quickly gave way to a picture of atoms as being small balls, like very tiny billiard balls. Then the idea of the billiard balls was replaced by the concept of the atom as a very tiny solid nucleus surrounded by a cloud of even tinier electrons.

Of course the nucleus turned out not to be solid, but to be composed of neutrons and protons, and even they have been shown to be made up of smaller particles. Is this the end of the story? Are these smaller particles fundamental, or are they made up of even smaller particles and so on, “ad infinitum”?

English: "Ad Infinitum" Oil in Canva...

English: “Ad Infinitum” Oil in Canvas 109 x 152.5 by peruvian painter Ricardo Córdova Farfán (Photo credit: Wikipedia)

It appears that in Quantum Physics that we have at least reached a plateau, if not the bottom of this series of even smaller things. As we descend from the classical rock, through the smaller but still classical atoms, to the very, very small “fundamental” particles, things start to get blurry.

The electron, probably the hardest particle that we know of, in the sense that it is not known to be made up of smaller particles, behaves some of the time as if it was a wave, and sometimes appear more particle like. The double slit experiment shows this facet of its properties.

Diagram of the double-slit experiment

Diagram of the double-slit experiment (Photo credit: Wikipedia)

The electron is not unique in this respect, and in fact the original experiments were performed with photons, and scientists have performed the experiment even with small molecules, showing that everything has some wave aspects, though the effect can be very small, and is for all normal purposes unnoticeable.

A wave as we normally see it is an apparently continuous thing. As we watch waves rolling in to the beach we don’t generally consider it to consist of a bunch of atoms moving up and down in a loosely connected way that we call “liquid”. We see a wave as distributed over a breadth of ocean and changing in a fairly regular way over time.

Wineglass with blue liquid

Wineglass with blue liquid (Photo credit: Wikipedia)

At the quantum level particles are similarly seen to be distributed over space and not located at a particular point. An electron has wave like properties and it has particle like properties. Interestingly the sea wave also has particle like properties which can be calculated. Both the sea wave and the electron behave like bundles of energy.

You can’t really say that a wave is at this point or that point. A water may be at both, albeit with different values of height. If the wave is measured at a number of locations, then by extension it has a height in between locations. This is true even if there is no molecule of water at that point.  The height is in fact the likely height of a molecule if it were to be found at that location.

English: A particle motion in an ocean wave. A...

English: A particle motion in an ocean wave. A=At deep water. B=At shallow water. The elliptical movement of a surface particle flattens with increasing depth 1=Progression of wave 2=Crest 3=Trough (Photo credit: Wikipedia)

By analogy, and by the double slit experiment, it appears that the smallest of particles that we know about have wave properties and these wave properties smear out the location of the particle. It appears that fundamental particles are not particularly localised.

It appears from the above that at the quantum level we move from the discrete view of particles as being individual little “atoms” to a view where the particle is a continuous wave. It points to physics being fundamentally continuous and not discrete.

The Continuum

The Continuum (Photo credit: Wikipedia)

There’s a mathematical argument that argues against this however. Some things seem to be countable. We have two feet and four limbs. We have a certain discrete number of electrons around the nucleus of an atom. We also have a certain number of quarks making up a hadron particle.

Other things don’t appear to be countable, such as the positions a thrown stone can traverse. Such things are measured in terms of real numbers, though any value assigned to the stone at a particular instance in time is only an approximation and is in fact a rational number only.

Stonehenge sulis

Stonehenge sulis (Photo credit: Wikipedia)

At first sight it would appear that all we need to do is measure more accurately, but all that does is move the measurement (a rational number) closer to the actual value (a real number). The rational gets closer and closer to the real, but never reaches it. We can keep increasing the accuracy of our measurement, but that just gives us a better approximation.

It can be seen that the set of rational numbers (or the natural numbers, equivalently) maps to an infinite subset of the real numbers. It is usually stated that the set of real numbers contains the rational numbers. I feel that they should be kept apart though as they refer to different domains of numbers – rational numbers are in the domain of the discrete, while the real numbers are in the domain of the continuous.

Particles by fundamental interactions

Particles by fundamental interactions (Photo credit: Wikipedia)

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Numbers are fascinating

A little image of aleph_0, smallest infinite c...

A little image of aleph_0, smallest infinite cardinal (Photo credit: Wikipedia)

Numbers fascinate me. What the heck are they? They seem to have an intimate relationship with the “real world”, but are they part of it? If I heave a rock at you, I heave a physical object at you. If I heave two rocks at you, I heave real objects at you. It’s a different physical experience for you, though.

If I heave a third rock at you, again, it’s a different qualitative experience. It’s also a different qualitative experience from having one rock or two rocks thrown at you.

Glyder Fawr

Glyder Fawr (Photo credit: Wikipedia)

Numbers come in three “shapes”. There are cardinal numbers, which answer questions like “How many rocks did I throw at you?” There are ordinal numbers, which answer questions like “Which rock hit you on the shoulder?” Finally there are nominal numbers, which merely label things and answer questions like “What’s you phone number?”

As another example, in the recent 10km walk which I took part in, I came sixth (ordinal) in my age division. That sounds good until I admit that there were only seven (cardinal) entrants in that division. Incidentally, my bib number was 20179 (nominal).

Cardinal numbers include the natural numbers, the integers and the rational numbers and the real numbers (as well as more esoteric numbers). For instance the cardinal real number π is the answer to the question “How many times would the diameter fit around the circumference of a circle?”

It’s a bit more difficult to relate ordinal numbers with real numbers, but the real numbers can definitely be ordered – in other words a real number ‘x’ is either bigger than another real number ‘y’, or vice versa or they are equal. However, there are, loosely speaking, more real numbers than ordinals, so any relationship between ordinal numbers and real numbers must be a relationship between the ordinal numbers and a subset of the real numbers.

English: Example image of rendering of ordinal...

English: Example image of rendering of ordinal indicator º Italiano: Immagine esemplificativa della resa grafica dell’indicatore ordinale º (Photo credit: Wikipedia)

Subsets of the real numbers can have ordinal numbers associated with them in a simple way. If we have a function which generates real numbers from a parameter, and if we feed the function with a series of other numbers, then the series of other numbers is ordered by the way that we feed them to the function, and the resulting set of real numbers is also ordered.

So, we might have a random number generator from which we extract a number and feed it to the function. That becomes the first real number. Then we extract another number from the generator, feed it to the function and that becomes the second real number, and so on.

The random map generator provides a limitless ...

The random map generator provides a limitless supply of colourful terrains of various themes. Open island maps, like this one, allow players to use airstrikes. Cavern maps have an indestructible roof which cannot be passed. (Photo credit: Wikipedia)

What we end up doing is associating a series of integer ordinal numbers with the generated series of real numbers. These ordinal numbers are associated with the ordered set of real numbers that we create, but the real numbers don’t have to be ordered in terms of their size.

Nominal numbers such as my bib number are merely labels. They may be generated in an ordered way, though, as in the case of my bib number. If I had registered a split second earlier or later I would have received a different number. However, once allocated they only serve to show that I have registered, and they also show which event I registered for.

On the occasion that I took part there were two other events scheduled : a 6.5km walk and a half marathon. My bib number indicated to the marshals and officials which event I was taking in and which way to direct me to go.

I’m not a mathematician, but it seems to me that ordinal numbers are more closely aligned to the natural numbers, the positive integers, than to any other set of numbers. You don’t think of someone coming 37 and a half position in a race. Indeed if two people come in at the same time they are conventionally given the same position in the race and the next position is not given.

English: Selby Apartments, located on 37th Str...

English: Selby Apartments, located on 37th Street, 37th Avenue, and Marcy Street in Omaha, Nebraska. The view is from 37th Avenue and Marcy, looking northeast. At left is 825 S. 37th Ave; at right is 3710 Marcy. (Photo credit: Wikipedia)

There’s a fundamental difference between natural numbers or the integers, or for that matter the rational numbers and the real numbers. The real numbers are not countable : they can’t be mapped to the natural numbers or the integers. The rational numbers can, so can be considered countable. (Once again, I’m simplifying radically!)

Natural numbers and integers are related to discrete objects and other things. The number of dollars and cents in your bank account is a discrete amount, in spite of the fact that it is used as real number in the bank’s calculations of interest on your balance. If I toss two rocks at you that is a discrete amount.

English: Causal loop diagram (CLD) example: Ba...

English: Causal loop diagram (CLD) example: Bank balance and Earned interest, reinforced loop. Diagram created by contributor, with software TRUE (Temporal Reasoning Universal Elaboration) True-World (Photo credit: Wikipedia)

Even I tip a bucket of water over you, I douse you in a discrete number of water molecules (plus an uncertain number of other molecules, depending on how dirty the water is). However the distance that I have to throw the water is not a discrete number of metres. It’s 1.72142… metres, a real number.

At the level at which we normally measure distances distances don’t appear to be broken down into tiny bits. To cover a distance one first has to cover half the distance. To cover half the distance one must first cover one quarter of the distance. It is evident that this halving process can be continued indefinitely, although the times involved are also halved at each step.

This seems a little odd to me. Numbers are at the basis of things, and while numbers are not all that there is, as some Greek philosophers held, they are important, and, I think, show the shape of the Universe. If the Universe did not have real numbers, for example, then it would be unchanging or perhaps motion would be a discrete process, like movements on a chess board.

If the Universe did not have any integers, the concept of individual objects would not be possible, since if you could point at an object you would have effectively counted “one”. In other words we need the natural numbers so that we can identify objects and distinguish one from one another, and we need the real numbers so that we can ensure that the objects don’t all exists at the same spot and are, in fact separated from one another.

Visualisation of the (countable) field of alge...

Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic…). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, +i near top. (Photo credit: Wikipedia)

 

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