Miscellaneous – Page 7 – Me on the net

Simple Arithmetic

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).



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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.

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 (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.



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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 (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.

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.

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.

Here be Dragons

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).

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.



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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 (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 (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.

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.

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.

Let’s be Rational – Realer Numbers

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…

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).

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.

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.

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.

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.

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].

Rational versus real

(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”?

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 (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.

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.

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.

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.

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.

Numbers are fascinating

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.

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).



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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.



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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.

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.



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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)

Mainframes were different

IBM System 360/65 Operator's Panel The IBM Sys... — IBM System 360/65 Operator’s Panel The IBM System/360 Model 65 first shipped in 1966 This photo was taken by Mike Ross of corestore.org . He has given permission via email “Feel free to make use of … my pictures under the GNU license. All I ask is that they are credited & linked…”. –agr 19:07, 17 Aug 2004 (UTC) (Photo credit: Wikipedia)

Mainframes were … different. Today we have devices that have orders of magnitude more processing power than the old mainframes. The old mainframes were big machines that performed business tasks for large organisations and that cost millions of dollars. The phone in your pocket shows you your email, let’s you Facebook or tweet your friends where ever you might be and what ever you might be doing. You can even use it to phone people!

Arguably we fritter away most of the enormous amount of processing power and storage that our hand held devices provide. We watch videos on them, yes, even porn, and play endless silly games on them. And cats. Cats have taken over the Internet and thereby our connected devices.

The fundamental concept of the mainframe is one single powerful(!) computer with all devices, such as card readers, terminals (screen and keyboard, no mouse), printers, tape units and other devices, more or less directly and permanently connected to the computer.

Interestingly, when you typed something into a terminal using the keyboard, it wasn’t sent immediately to the mainframe but was recorded in a buffer in the terminal. When one of a number of keys was hit the whole buffer was sent to the mainframe. These special keys were “Return”, which is similar to the “Enter” key, a “PF Key”, which is similar to a function key, and a few others.

This meant that you could type and edit stuff at your terminal and it would only be sent when you were finished. That is different from the model used by PCs and other modern devices, where every single key press that occurs is sent to the target computer, including typos and correction to typos.

Of course, when you press a key on your PC keyboard, the computer that is the target is the one that the keyboard is connected to, and what you type in goes into a buffer, but the principle still applies. The effect is more obvious if a lot of people are connected to a multi-user computer and are using it heavily, when the response to hitting a key takes a second or so to echo back to the screen.



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Multi-user computers are not common these days as they are not trivial to set up, and computers and networks have become so fast that it is generally easier to change the model and access applications over the network rather than use the direct connect model that was used in the early days.

A lot of the features of modern computing devices originated in mainframes. Mainframes originally ran one job or task at a time, but soon they became powerful enough to run many jobs at the same time. Mainframe operating systems were soon written to take advantage of this ability, but to achieve the ability to run multiple jobs, the operating systems had to be able to “park” a running job while another job got a slice of the processor.

To do that the operating system had to save the state of the process, especially the memory usage. This was cleverly achieved by virtualising memory usage – the job or task would think that it was accessing this bit of memory but the memory manager would make it use that bit of memory instead. The job or task didn’t know.

For instance the job or task might try to read memory location #ff00b0d0 (don’t worry about what this means) and the memory manager would serve up #ffccbod0 instead. Then a moment of so later another job or task might try to read that memory location. It would expect to find its own data there not the first task’s data, and the memory manager would this time serve up, say, #ffbbb0d0.

SVG Graph Illustrating Amdahl's Law — SVG Graph Illustrating Amdahl’s Law (Photo credit: Wikipedia)

The key point is that the two tasks or jobs access the same address, but the address is not real, it is what is known as a virtual address, and the memory manager directs the request to different real addresses. This allows all sorts of cunning wheezes – a job or task can address more memory than the machine has installed and memory locations that have not been used in a while can be copied out to disk storage allowing the tasks in the machine to collectively use more memory than physically exists!

Exactly the same thing happens in your phone, your tablet, or your PC. Many tasks are running at the same time, using memory and processors as if these resources were dedicated to all the tasks. (Actually not all tasks are running at the same time – only as many tasks as there are processors in the processor chip can be running at the same time, but the processors are switched between task so fast it appears as if they are. The same is also true of mainframes).

Of course, there’s a downside to the mainframe model and that is that if the mainframe goes down, everyone is affected. In the early days of the PC era, every PC was independent and if it went down (which they often did) only one or two people, those who actually used the computer were directly affected. So if the Payroll computer crashed it didn’t affect Human Resources.

Soon though the ability to connect all the computers over a network became possible and computing once again became centralised. Things have changed, but the corporate server or servers now fulfil the role that once belonged to the mainframe.



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All the advantages of centralisation have been realised again. Technical facets of the operation of computers have been removed from those whose job was not primarily computing, much to the relief of most them I’d suspect. Backups and technical updates are performed by those whose expertise is in those fields, rather than by reluctant amateurs in the field.

However, the downside is that a centralised computing facility is never as flexible as the end users would like it to be and, somewhat ironically, that as an outage of the old mainframe used to affect many people, so will an outage of a server or servers in the current milieu.

Ganglia report showing editing outage, when Wi... — Ganglia report showing editing outage, when Wikipedia server srv156 stopped responding (Photo credit: Wikipedia)

Too Much Sweetness

Sugar is a the name given by chemists to a family of compounds comprised of Carbon, Hydrogen and Oxygen in various configurations. Generally there are the same number of Carbon and Oxygen atoms and twice as many Hydrogen atoms, but this is not always true.

So far as I know all living things contain sugars, which supply the body with energy through a complex series of chemical reactions which function to slow down the release of energy from the sugar, which would otherwise be released in an unusable burst of energy.

The end result of these reactions is the release of water and carbon dioxide. The cycle of reactions needs as many Oxygen molecules as there are sugar molecules, more or less, assuming the the sugar is fully broken down, and that is why all living things need to take in Oxygen and why we breath out carbon dioxide.

We don’t normally notice the sugar that we take into our bodies as it is largely contained in the foods that we eat. With the exception of the sugar in our tea and that we may scatter on our cereals, it is invisible to us. We actually take in a lot of sugar when we eat things.

In fact we are told that we take in far too much sugar in a modern diet. People drink tea and coffee with sugar, or drink copious amounts of sugar filled soft drinks and we know that this is bad for us. It rots our teeth and makes us fat. It predisposes us to diabetes and other metabolic problems.

The reason that it makes us fat is because early humans (and other animals) had a life which saw feasts and famines as the food supply fluctuated. The body evolved to take advantage of the feast phases by storing energy for the famine times by converting the excess sugars that were ingested to fat. Fat is harder to break down to release its stored energy, and the body preferentially uses any sugars circulating in the body (in the form of glucose).

The aldehyde form of glucose (Photo credit: Wikipedia)

As a species we have a taste for sugar. The body wants to take in as much as it can, in order to keep itself running, and to store up energy as fat for the lean times. However, these days, there are rarely any truly lean times for most people, so they take in more sugars than they need and store it as fat.

There is debate sometimes as to whether or not our desire for sugar amounts to an addition. Personally I don’t feel that it is truly an addiction, just a very strong desire to eat as much sugar containing food as possible. People don’t suffer intense reaction to the withdrawal of sugar from their diet as they would if it were an addition.

Generally they don’t suffer hallucinations, pains, or other mental or physical symptoms. Quite often they even feel better. Quite often they lose weight. Diabetes symptoms may abate, they may sleep better, and other physical problems may disappear.

It seems a no brainer that we should reduce the amount of sugar that we consume in our food and drink, yet we continue to consume it to excess. In my opinion there are at least three reasons for this.

Firstly, the body itself makes us crave sugary foods, as it has evolved to make us feel pleasure when eating something that it can use for fuel and additionally store up for the leaner times that never come. We are predisposed to like sweet things, and our experience tells us that a chocolate bar is sweet.

Secondly, we have been given sweet things as a treat from the time that we are small, and we expect our treats to be sweet. In a restaurant the main course is about protein, usually with some meat or other as the star, but desserts are treats and therefore must be loaded with sugar.

Thirdly, it is absurdly easy to obtain sugar these days. The confectionery aisle of the supermarket has the most shelf space. Cakes and desserts take up shelves of their own, as do soft drinks which are laced with sugar.

That’s a big issue. Everything has sugar in it these days. In the past there was sugar in many things, but I believe that the quantity of sugar was lower. These days there is a large slug of sugar in almost everything.



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It’s even crept into the main course. Glazes and marinades have always had sugar in them as the sugar caramelises on the meat as it cooks, and gives it a nice colour, but many pouring sauces and dressings also contain copious amounts of sugar. Even the humble meat pie contains sugar or so I heard. Something to do with giving the right consistency to the gravy, I believe. So what was wrong with the original ways of thickening it?

This has all led to an issue, so far as I am concerned. I can see all the health benefits of reducing one’s intake of sugar, but so far as others are concerned their bodies are their own responsibility. If they wish to make themselves fat and ill, that is up to them.

English: This is a graph showing the rate of o... — English: This is a graph showing the rate of obesity in adults and the rate of being overweight in both children and adults in the United States from 1960 – 2004. (Photo credit: Wikipedia)

What really annoys me about the sugar in everything trend is that almost everything tastes of sugar! I gave up sugar in tea many decades ago, and I didn’t miss it, but now I hate the taste of tea with sugar in it. I can’t drink it. I also gave up sugar on my morning cereal, which was harder, but I now don’t miss it. There’s already some sugar in it of course so why add more?

But everything else? Everything that you touch at the supermarket is laced with sugar, from the white stuff they call bread to the pickles and sauces you put in your sandwich. As a result, you can’t taste much of anything under the overpowering sweet taste.

Someone once commented when I complained about the overbearing sweet taste of everything, that I could just make my own. That’s fine, and I do do it as much as I can, but it’s not always possible, as the ingredients may already contain sugar, and sometimes i just wants the convenience of picking something off the shelf.

English: Organically produced blackstrap molas... — English: Organically produced blackstrap molasses produced in Paraguay. (Photo credit: Wikipedia)

Thus it was written….

1976 The beginning of flexible computing in pu... — 1976 The beginning of flexible computing in public health. Auditorium A at CDC, converted to a war room for the Swine Flu crisis, is filled with epidemiologists and a Digital Equipment PDP 11 minicomputer the size of a refrigerator. A program called SOCRATES, written in FORTRAN by programmer Rick Curtis, allowed an epidemiologist to define questions, enter data, and summarize the results in tabular form without the aid of a programmer. (Photo credit: Wikipedia)

Have you ever noticed that a computer program never does exactly what you want? Oh, it will process your data for you, and produce results that you can use, or it will move those files from here to here, but may move ones that you didn’t want moving or leave ones that you did want to move. But it won’t do exactly what you want, first time with no hassles.

Of course you can make it work for you, make it do what you want more accurately, but you have to work harder to make it do so, and programs are supposed to make things easier, right? Take this very editor that I am using to compose this post. I thought the bold words above as I wrote, but I needed to perform an action to make them come out bold and another to ensure that the text after the bold words didn’t also come out bold.

JWPce is a Japanese Word Processor, this softw... — JWPce is a Japanese Word Processor, this software is under GNU license. (Photo credit: Wikipedia)

Of course this is a very minor consideration and if I were to find a program that changed the font to bold for one word and then stopped, I’d probably find it irksome to bold two words in succession. Also, I’d bet the house that there would be something that the first program did automatically or easily that would be difficult to achieve using the second program.

Those who are not programmers seem to have an ambivalent attitude to programs. On the one hand, if they are using a program and they can’t get it to work, they blame themselves. “I must be doing something wrong!” comes the impassioned plea. So the technophile in the household has to interpret what helpful statement “It doesn’t work!” really means and what how the program expects in the way of input. Problem fixed. Until the next time.

On the other hand, the non-programmer will accept it without question when the hero/heroine of the film briefly types something at the computer and the world is saved! Hooray! The sentient virus destroying humanity is defeated! Hooray again!

However these issues with programs are not bugs. The program is not doing something wrong. The issue is with our expectations of how things ought to work. It may well be that the program is designed to do things differently to those expectations, either because the programmer assumes that the way that he codes it is the way that people will expect things to work.



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If the programmer gets a lot of issues with a particular feature of his program, he may decide to change it to fit more closely with the expectations of the users, and then send out an update. It is likely that he will then get a lot of issues from those people who have been using the program for a while and have come to expect it to behave the old way. The programmer can never win.

I’ve done quite a bit of programming over the years, thankfully mostly for myself. Every program that is written these days usually fits into what might be called an ecosystem. There are programs to gather data, programs to crunch data, and programs to display the results, but there are many programs to perform each of these three tasks.

For example, the results may be printed on a printer, either as a table of data, or as graph or pie chart, or in these days of 3-D printing, a physical object, or it may be displayed on a screen, or in any other way where the recipient can interrelate with it.

Data input may typed in, drawn in on a graphical input device, or it may be collected by some means or other from sensors or other devices. Your heartbeat may be collected from a number of sensors on your skin, read by a machine and be printed off on a roll or paper, while at the same time being displayed on a local screen. It may also be sent off to some remote location where specialists may peruse it.

In the middle, between input and output comes the processing or crunching of the data. This is what most people think of when they think of computer processing. Somewhat unfairly the input and output processes are relegated to only secondary interest. This is, however, what a program or system is written to do. Even so, input and output processes may be complex.

Data may be processed in several different ways in between input and output, and by several different programs. There are layers within layers. For example a program that takes advantage of a database uses many layers. The program which the programmer is writing will no doubt call programs (called APIs or interfaces) within the environment that he is working in which start the process of communicating with the database.

The API puts the programmers request into a form that the database expects and sends it to another process, not usually part of the programmer’s chosen system, which exists only to connect the programmer’s program (and any other similar programs) to the database.

At the database end the receiving part of the database system receives the request and passes it to the main database program, which checks it and executes it. During the execution process the database program makes calls to system programs which perform any necessary retrieval or writing of the data to the system’s file system, via the system’s hardware interface with the storage system.



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I’ve collapsed many layers in the explanation above. The main point is that the programmer’s program is the tip of the iceberg, and there are many layers which are called into action during the execution of the program. To complicate things further, the system that the programmer uses to perform his work is also a program and has its own layers on top of layers.

This explains why programs don’t ever do exactly what you want. The programmer has to use utility programs which, while flexible can’t do everything. The utility programs are also flexible, but are interfacing with other programs, which while flexible also can’t do everything. And so on.

English: TM-database (Photo credit: Wikipedia)

The more flexible a program is, the bigger it is, as it has be programmed to enable the flexibility. So, this forces constraints on it, which impose constraints on the programmer, whose program therefore imposes constraints on the user of the program. And those constraints are why the program can’t do exactly what you want, but usually, it’s close enough that the program is useful to the end user. Most of the time.

Water, the cause of surfing.

I came across one of those pages on the Internet which state something like “At least a few molecules of the water in your body probably passed through the kidneys of Julius Caesar“. They generally use statistics to show that what they are saying is true.

Only those who believe in homeopathy should be disturbed by this. To anyone else, a molecule of water is a molecule of water, and the fact that it had once been contained in a stream of urine is irrelevant. In any case it is too late. 60% of our body is made up of water, so water from Caesar’s urine is already in us.

Water is a fascinating chemical. It carries stuff around as it is the basis for blood and lymph and all the other fluids of our bodies. It carries nutrients up the stems of plants. It wears away mountains and builds rocks, it cools lava to form other rocks. It brings nutrients to our crops and washes them away. It even sinks ships.

“You are water
I’m water
we’re all water in different containers
that’s why it’s so easy to meet
someday we’ll evaporate together.”
― Yoko Ono

It’s difficult to think of any occurrence in our familiar world which is not mediated or affected by water in some way. Shortage of water to a society is a disaster, as food cannot be produced, leading to famine and deaths.

Water is thought by most people to be liquid at usual temperatures, though there are some places where it is to be found in solid form. Actually there is a great deal of it around in the gaseous phase, or vapour. We measure this airborne water in terms of the humidity or wetness of the air.

Water is extraordinarily pervasive and can be found in all the nooks and crannies in the materials that we have around us. It acts as a lubricate, and if the water is driven off, by heating or chemical means things become stiff and fragile. Even so, water cannot be completely removed from things – even a diamond probably has a few entrapped water molecules.

Water molecules attaching to each other by hyd... — Water molecules attaching to each other by hydrogen bonds (Photo credit: Wikipedia)

Water plays a role in rotting things down. A corpse kept in a very dry environment desiccates and turns leathery and fragile. I guess that this is because the organisms that rot a body away cannot function in a water free environment.

A body of liquid water fills things from the bottom up. Gravity pulls the water down to the lowest parts of a container and water continues to layer the container. The surface appears to be flat, but that is an illusion. At a small scale, if a tiny bit of the water happens to be higher than the rest of the water, gravity will pull it down, while the other water molecules resist by being in the way.

Eventually as the water stills, the differences in level even out, and the water surface becomes as level as it can. However, at the molecular level, molecules of water, which are moving relatively fast on these scales, can pop out of the liquid and float away. Other molecules can also pop in to the liquid, so that on average the water is level.

Why doesn’t the surface appear blurry and ill defined then? Well we can’t see at the molecular scale, and also the water molecules form weak electrical bonds with each other. A water droplet is like a large crowd of people all milling about, holding hands much of the time. Those on the outside are not as tightly bound as those further in.



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Imagine now that the crowd is surrounded by a storm of people who are moving faster, and are more spread out so they rarely join hands. One of these gaseous people will now and then bump into the crowd. They may knock loose one of the crowd who will shoot off and become one of the gaseous people, while gaseous person who hit him may now be travelling more slowly and link up with the crowd.

Even in the macro world a water surface is rarely really flat. The dynamic nature of the flatness is apparent when a container is jolted slightly and tiny waves form on the surface as compression waves disturb it. Wind and rain also cause visible disturbances in a lake or pond.

Flowing water often forms a smooth, if not level surface. A submerged rock in a river or a weir or fall in a river can form persistent ripples of flumes as the water flows over them. Kayakers know to aim their craft at a flume to safely descend a rapid or waterfall, although downstream of flumes the river often forms “haystacks” where turbulent water is forced into humps which can prove difficult to navigate.

A little stream may be described as turbulent as it makes its way over and around boulders and small drops, but interestingly it is not random. It is not chaotic. A close look will reveal that the bow waves of stones in the flow may flutter and throw off little whirling vortexes, but the bow wave and the pattern of vortexes persists. The little waterfall over a small stone ledge persists, even though the shape of the waterfall may ripple a little.

In large bodies of water, such as lakes and seas, winds form waves which can travel many thousands of kilometres across oceans and seas. Water waves don’t represent the movement of water over those distances – the only thing that moves is the energy in the wave. Water molecules in a wave move mainly up and down and only a little forwards and backwards.

Circular water current in a wave (Photo credit: Wikipedia)

However when a wave travels over a beach or shoal, the movement in the vertical direction is curtailed, and the energy is transformed into a forward motion – the wave breaks. Water is transported forward, with the water higher up moving faster than the water at the sea bed which may be water draining off the beach from the previous wave and the wave steepens until it collapses. Hence surfing!



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Real or virtual caving? What’s the difference?

A bit less than a year ago I posted about caves virtual and real. I’m going to expand a bit on this today.

I worked for many years as a Linux Systems Administrator, a job which I loved and to a large extent brought home with me. I run a number of Linux systems at home, including the computer that I am writing this on. I have a system that I refer to as my “server” which I use for backups and for things that might get in the way on my desk computer.

Tux, the Linux penguin (Photo credit: Wikipedia)

When Minecraft was a craze some time ago I didn’t get involved at first but when my grandkids got into it I became interested. In Minecraft you can build elaborate structures, but to do so you need to find and extract the necessary materials, and, depending on the server that you are using, fight off automatic monsters (“mobs”) and other players.

I installed the Minecraft client, tried it out and liked what I saw. However, playing on other peoples’ servers soon became less than satisfactory. I didn’t like the unrealistic ability to “fly” (not fall down when unsupported by things) and the combative aspects of the game were found on many servers.



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So, I investigated and tried running my own Minecraft server. This is possible, and not particularly tricky, you still have to pay for the client. There’s nothing wrong with paying for things, of course, and I did pay for it, but being a Linux user I’m always interested in seeing if there is a free version out there. Not of the actual Minecraft naturally, but of a similar program.

Minetest is a very similar program to Minecraft, works in much the same way, but is free, and there were Linux packages available, meaning that the install was simple. So I installed the packages and started playing. (I later downloaded the source code and compiled it, but that is another story. The packages worked fine).

Install debian (Photo credit: Wikipedia)

Pretty much everything that you can do in Minecraft you can also do in Minetest. The differences are minor. So everything I say from here on will probably apply to either program.

The game is all about assembling resources to build things, and you assemble resources mainly by mining. You whack a block (say stone) with a tool (say a steel pickaxe) and it disappears and appears in your inventory. When you have enough blocks you can build a wall of them by taking them from your inventory and placing them appropriately. That is the start of your fortress or palace.

A plain stone building is boring, so you will want to acquire some fancy blocks to smarten it up and that is where the game shines. You can make fancy blocks to place on your building, but to do that you need to acquire resources and fabricate or “craft” them. The resources are found under the ground, which requires you to dig down to get them, which is of course where the “Mine” part of the name comes from.

If you dig downwards (on a slope, so that you can return to the surface eventually) you will sooner or later encounter a void or space where there are no blocks. Hopefully you will avoid falling to the bottom and dying. If you carefully climb down to the bottom of the void you are are the bottom of a virtual cave.

The cave may have blocks which look slightly different to the normal blocks. These blocks contain resources, usually ores, that can be collected and used to make other special blocks. For instance a block with iron ores in it can be used to make steel ingots, which can then be used to make steel pickaxes and other useful tools.

The game’s caves are similar in many ways to real caves. The topography of the caves is varied, with narrow passageways in some places, deep clefts in others, potholes, and sometimes openings to the surface.

Of course it is dark underground. That means that you need to manufacture torches and carry them in your inventory if venturing underground. A torch placed on the wall will illuminate a small area of the cavern leaving dark voids beyond the torchlight which might be interesting to explore!

The game’s caves often have, just like real caves, uneven floors and there will be much jumping over obstacles, and climbing up and down. Some caves do have flatter floors, as do many real caves.

English: Subglacial pothole on Pothole Dome. (Photo credit: Wikipedia)

Some game caves have sloping floor, much like a staircase, and the roof also may slope down, giving the impression of a sloping slit, as is often found in real caves. It is not advisable to skip down the slope into the darkness, of course, as the slope may end in a drop, as may also be found in a real cave.

Game caves are often elongated, just like real caves, and may form large interlinked caverns. Cross passages may start way up on the walls of the caverns, just as happens in real caves. It’s often possible to travel some distance more or less horizontally before one has to climb up or down to continue.



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Game caves do not often contain water, which is unlike most real caves, and when they do the water is either in the form of a waterfall or a lake. Long flowing river of water are not common in game caves. Sometimes a cave will contain a lake of lava, or a lava fall which often looks spectacular. Both lava and water may form short flows but will quickly “seep” into the rocks.

Some of the potholes are spectacular. As they are underground and dark, one cannot see the bottom. Often they can edged or mined around and descended that way. Looking back up one can see the torches that one placed on the way down as tiny lights, often to spectacular effect.

I’ve explored both real caves and game caves, there is a great deal of similarity in the experience. In both cases, one has only limited visibility around one, one has a sense of void, and a sense that one may fall. There’s the thrill of discovery as one explores, and a sense of achievement. Unfortunately in the game caves, there are no stalactites and stalagmites to decorate the cave, but perhaps someone will write some in some time.