## Round numbers

It seems that we have a fascination for numbers that end in zeros. One thousand (1,000) and one million (1,000,000) and so on and to a lesser extent numbers like one hundred (100) and ten thousand (10,000). Fractions of round numbers also appeal to people. Reaching the age of 50 (half of 100) or 75 (3/4 of 100) is considered an interesting milestone while reaching 74 or 51 for some reason is not so interesting.

However some fractions are not even noticed. At the age of 33 years and 4 months you will be 1/3 of 100 years old and at 66 years and 8 months you will be 2/3 of 100 years old. When I passed the second milestone, I mentioned it to people and they didn’t seem to care. No prezzies were forthcoming.

There is one special number that could be considered a round number, and the wellspring of all round numbers and that is the number zero. The first number which is usually considered a round number is ten (10), where the zero indicates the absence of any digit (except zero itself, which is normally considered a numeric digit), and the 1 is positional and represents the number ten or ten units. Stick another zero on the end has the effect of multiplying all the digits in the number by ten, so 10 becomes 100 which represents one hundred.

The number 100 could be considered to be, reading from the right, zero ‘units’, zero ‘tens’ and one ‘hundred’. 110 is zero ‘units’, one ‘ten’ and one ‘hundred’ giving the number one hundred and ten. When I was learning arithmetic as a small boy, while I grasped the principles quickly enough, I continued to ponder this mapping process from numbers, like seven hundred and thirty one to the mathematical representation of 731. Maybe I was a strange child. I still ponder it, even today. I look on “seven hundred and thirty one” as a name of the number, ‘731’ as a representation of the number, and the number itself as some ineffable thing. Maybe I grew up to be a strange adult!

Notice that above I said that “seven hundred and thirty one” is a name of the number and ‘731’ is a representation of the number. This is because there can be other representations of the same number, mainly in different bases. For all the numbers above I’ve used the number ten (10) as the base, but I could have used another base, such as sixteen (16) which is frequently used in computers. The number “seven hundred and thirty one” in base 10 representation is represented as ‘2D8’ in base 16 or hexadecimal representation. The ‘D’ is in there to represent the number thirteen (decimal 13).

Any positive integer can be used as a base. The bigger the base the more ‘digits’ are required to represent numbers, making them hard to read and hard to calculate with, so a base of ten (10) is a reasonable choice for general use. Negative integers can also be used as bases, but then things get very difficult! I’ve occasionally wondered if rational numbers or real numbers could be used as bases, but I can’t see how that would work.

Computers are interesting, since they, at the lowest level, appear to use a base of two (2), which is the smallest possible positive integer base. The numbers are conceptually simple strings of ones (1) and zeros (0) called ‘bits’. It’s not as simple as that however as in the computer’s central processor the data and programs are shunted around like little trains of bits, switching from track to track and in many cases circulating round small loops, merging with other trains of bits, eventually arriving in stations called buffers.

These buffers can be 8 bits long (one byte) or 16 bits long (2 bytes) or even longer. The length is related to the architecture of the processor, and a 64-bit processor can handle addresses, integers and data path widths up to 64 bits, so effectively they naturally use numbers up (but not including) decimal 18,446,744,073,709,551,616! Computer people can’t read such long strings of bits of course so they convert the numbers to base sixteen (16) otherwise known as hexadecimal. It’s still very long, but can be handled and is less error prone than long strings of zeros and ones.

Round numbers are very useful as abbreviations. Saying nine thousand, eight hundred and seventy three is a lot more verbose than “about ten thousand” and is sufficiently accurate for many purposes.

One interesting use of round numbers is found in the nominal sizing of disk drives. To a computer person one byte is the smallest unit of storage. Bytes are usually grouped into ‘kilobytes’ where in this sense the prefix ‘kilo’ stands for one thousand and twenty four, and kilobytes are grouped into ‘megabytes’ where in this sense the prefix ‘mega’ stands for one thousand and twenty four again, and megabytes are grouped into ‘gigabytes’. This means that to a computer person a gigabyte contains 1,073,741,824 bytes. So this number (and numbers with the smaller prefixes of kilo and mega) are round numbers to computer people, because, if expressed in hexadecimal or binary bases these numbers end with long strings of zeros!)

There is a source of confusion here, because outside of the computer world, the prefixes of kilo, mega and giga are defined in terms of thousands. A kilogram is one thousand grams. Technically a megagram would be a thousand kilograms or a million of grams. This confusion impacts the computer world when computer disks size are given. To a computer disk manufacturer a gigabyte is one thousand million bytes, not a bit over one thousand and seventy three million bytes as mentioned above.

This leads to disappointment when purchasing disks. A nominally one hundred gigabyte disk will contain one hundred thousand thousand thousand bytes (100,000,000,000) but when when formatted will be able to contain less than ninety three gigabytes as the computer world counts bytes and that doesn’t take into account the overhead of the method of storing data on the disk. This overhead is necessitated by the need to hold the file names and locations on the disk itself so that the files can be retrieved.

There is no right or wrong way to consider bytes on disks and so computer people are in general now aware that if they buy a disk it will not seem (to them) to be quite as big as advertised. The moral is to ask what people mean when they use round numbers.

I was going to go into topics like giving change and Swedish rounding, but this post is already long enough. I will just mention that the topic of round numbers came to me because this is my fiftieth post! Fifty is sort of a round number, I suppose. It is halfway to a proper round number.

## Sickness

Today I am going to reflect on sickness. As an aside, my aim was to write something every Friday and post it here, but lately the deadlines have been slipping past and I didn’t complete the previous post until Tuesday. This Friday I was still suffering from the bug that I caught, and motivation and energy levels were low, so I didn’t start this until Sunday. The effects hang on, but if I don’t start now, I may not get a post done at all! So here goes.

Last Monday I was feeling like I was coming down with something but struggled into work anyway. A couple of hours into the day it was obvious to me that I was rapidly getting worse so I headed home and put my feet up. I fully expected to be over the worst by Wednesday but on Wednesday morning it was obvious that I wasn’t recovered enough to return to work, so I visited the doctor who confirmed a flu-type illness.

The doctor didn’t prescribe anything apart from rest, which suited me. I must write a post about over prescribing of medicines by doctors as I see it sometime. So the rest of the week was taken up by lying around, drinking copious tea, coughing and aching. I believe that one of the symptoms of the sickness I am still suffering from is to make everything ache. Of course, the constant coughing results in aching of the chest muscles, but my arms and legs and head also ached. Not nice.

Add on on shivering fits and sweats and it all makes for a fun week. Did I mention a sore throat?

I think that I am suffering the attack of a flu virus, but obviously not a strain that was targeted by the flu jab that I had. Since it was presumably a virus there is no treatment possible, apart from alleviating the symptoms.

Speaking anthropomorphically, it is in the virus’ interest to not reduce the functional level of the organism that it attacks to the level where it quickly dies and so cannot pass on the infection, so viruses tend to merely make you sick. So infected organisms remain more or less functional. They still eat, drink, and interrelate with others of their type, which allows the virus to spread by coughs and sneezes which fill the air with the virus which can then be inhaled by well individuals.

It is good strategy for the virus to irritate the nose and the the chest, increasing the possibility of the virus being passed on. I say “strategy”, though of course it is pure evolution in action – viruses which don’t cause you to cough and sneeze don’t get spread around so easily and so tend to die out. Of course there are other ways to spread a virus or other disease, STDs and diseases transferred by physical contact spring to mind.

When you think about it, sneezes and coughs are a pretty damn efficient way of spreading a virus. A cough or sneezes creates a mist of tiny virus-laden particles that can be inhaled or picked up from surfaces where they settle. It follows that viruses at least of this type would spread most efficiently in enclosed spaces such as homes and workplaces. A farmer could sneeze in the fields and not affect anyone, but a sneeze in a packed classroom could result in several pupils being missing in the next day or two, not to mention the teacher.

One of the silly things about employment laws is that a person who takes leave from work because of sickness can be asked to provide a medical certificate, even if the employer doesn’t believe that the worker is faking the sickness. Usually there is a day or two’s grace to allow the sick person to obtain a certificate from the doctor. This usually means that the sick person has to go out into the community, sit in a waiting room which is probably a miasma of viruses, and talk to a doctor who is then subjected to the airborne virus! It’s possible that evolution will favour viruses which reach maximum infectiveness in 2 – 3 days!

The reason for the laws is to prevent people from claiming to be sick when they aren’t (known colloquially as “taking a sickie”). While this is obviously a problem it does mean that people may struggle into work when sick in order to avoid the expense of a doctor’s visit, and they may spread the virus around the workplace, resulting in more absences and more costs to the employer.

Viruses are amazing things, on the borders of death and life. They are simply little packets of genetic code for self-replication which utilise the organism’s own machinery against it. Of course all living organisms are packets of genetic code for self-replication, but viruses are the smallest possible, with the possible exception of things like prions. (Which, I’ve just read, don’t contain any genetic code).

The immune system of the body is triggered by viruses (which results in all the coughing and sneezing) and so the body is not defenceless. However viruses mutate quite quickly, so we have many ‘strains’ of common viruses. The common cold is, I believe, a particularly mutable virus which is probably why research into it has not gone far in combating it. The flu virus that attacked me is likely to have been a mutation of a strain of the flu virus that was targeted by the flu jab that I had.

And so the war goes on.

## Gambling

Gambling has probably been a human activity since two cavemen had a bet over their respective hunting prowess. Or maybe it was over which of them could stay upright longest after sampling the newly invented alcoholic grog. Gambling games generally have probabilistic component, though the contestants generally try to remove or circumvent it, usually by such techniques as remembering the order that cards come out or ‘card counting’. This latter technique involves keeping track of the high cards that come into play.

For some people gambling can become a problem. Sometimes susceptible individuals can become ‘addicted’ to gambling to the extent that they embezzle and steal so that they can continue to gamble. They may rationalise this by claiming that they are only trying to regain what they lost, or repay the people who they have stolen from, and indeed, because of the probabilistic nature  there is a chance that they might be able to do that. However the chance is very very small.

When a gambler starts gambling the reason that they gamble is the thrill of the possibility of winning big. Once the gambler has used up all his or her resources and has borrowed or stolen to keep gambling then the fear of losing and the fear of people finding out would be the predominant emotions, especially the fear of being found out.

They probably think to themselves that their luck must turn sooner or later and they must start winning, however this is just not true. Suppose the gambler is \$100,000 in debt and chooses odds of evens. Then to win \$100,000 he or she must wager \$100,000 and to do that he or she would have to steal another \$100,000. Such a theft is more likely to be noticed than a smaller amount and there is an even chance of losing and being \$200,000 in the hole.

As a result, it is likely that a ‘problem gambler’ would choose to go for longer odds and therefore smaller amounts of stake money, but with less chance of winning.

Statistically, over a large number of gamblers and a large number of wagers on something like a horse race, if all the money taken on wagers is paid back to punters then the average return, over all punters taking into account the stakes and the winnings paid out, is exactly zero! Of course, not all the money taken in in bets is paid to the punters. If the bets are on a totalizator system then the organisation running the ‘tote’ takes out taxes and administration costs so the payout will be less than the amount taken in.

If the system is a ‘bookmaker’ run system then the bookie needs to cover his costs so he (or she) arranges his books so that not all the money taken in bets is returned to the punters. There is a very small chance that under some circumstances he cannot cover all the bets made, but it is rare for this to happen. A bookie will sometimes ‘lay off’ a bet somewhere else if he feels exposed as a result of a large bet.

What this means to the gambler is that, on average, he is going to get back from the system less than he puts in. What a gambler hopes is that his personal return is positive, and he will in fact beat the odds. It is highly likely that he won’t though. A ‘problem gambler’ is unlikely to be a clever gambler and is likely to continue to lose.

Lotteries are different. Although Lotto is referred to as gambling, it is different from card games like poker or betting on horses. Again the average return is going to be zero or negative. However, with a lottery, the only way to increase your chance of winning is to buy more tickets. There is no real or illusory skill involved. As a result the lottery is unlikely to attract the ‘problem gambler’.

Of those who do take part in the lottery there are some who fall foul of the “Gambler’s Fallacy”. Some people use the same numbers draw after draw after draw in the belief that their particular set of numbers must come up sometime. This is not so at all. It doesn’t do any harm, though, as their particular numbers are just as likely to come up in one draw as any other. In fact, if their numbers do come up, they are equally as likely as any set of numbers to come up the next week too.

I confess that I don’t see much sense in betting on horses or dogs or whatever. I don’t have the skills necessary to increase the odds in my favour, though the so-called ‘professional gamblers’ appear to have those skills, and ‘problem gamblers’ definitely don’t.  I do buy lotto tickets, though, but I’m not too upset when I lose and the rare small win is fun.

## The number of the universe.

Anything that can be measured can be encoded in a single number. Take for instance the trajectory of a stone thrown into the air. Its position in relation to the point of launch and the time it has taken to reach that point can be encoded into a set of numbers, three for the spacial dimensions and one for the time dimension. This can be done for all the points that it passes through. These individual numbers can then be encoded into a single number that uniquely identifies the trajectory of the stone.

Or, a physicist can describe the motion of the thrown stone by using generic equations and plug in the starting position and starting velocity of the stone, which can then be encoded, probably in a simpler fashion than the above point by point encoding.

If we can imagine a set of equations that describe all the possible physical processes (the “laws of nature”?) and we can imagine that we can measure the positions of all the particles (including photons,’dark matter’ and any more esoteric things that might be out these), then we could encode all this in a huge number which we could call the ‘number of the universe’. Such a number would be literally astronomical and I do mean ‘literally’ here.

The most concise expression of the state of the universe over all time is probably the universe itself and the laws that govern it. Each individual particle has its own attribute, like charge, mass, position and so on as well as things like spin, charm and color. Some of these change over time and some are fundamental to the particle itself – if they change so does the nature of the particle. The rest of the universe consists of other particles which have a lesser or greater effect on the particle, all of which sum together to describe the forces which affect the particle.

There are a couple of things which might derail the concept of the number of the universe. Firstly there is Heisenberg’s Uncertainty Principle and secondly there is the apparent probabilistic nature of some physical processes.

What follows is my take on these two issues. It may make a physicist laugh, or maybe grimace, but, hey, I’m trying to make sense of the universe to the best on my abilities.

People may have heard of the Uncertainty Principle, which states that there are pairs of physical properties which cannot both be accurately known at the same time. You may be able to know the position of a particle accurately, but you would not then be able to tell its momentum, for example.

It is usually explained in terms of how one measures the position of something, which boils down to hitting it with something else, such as a photon or other particle. The trouble here is that if you hit the particle with something else, you change its momentum. This is, at best, only a metaphor, as the uncertainty principle is more fundamental to quantum physics than this.

Wikipedia talks about waveforms and Fourier analysis and an aspect of waves that I’ve noticed myself over the years. If you send a sound wave to a frequency analyser you will see a number of peaks at various frequencies but you cannot tell how the amplitude of the wave changes with time. However, if you display the signal on an oscilloscope you can get a picture of the shape of the wave, that is the amplitude at any point in time, but not the frequencies of the wave and its side bands. Err. I know what I mean, but I don’t know if I can communicate what I mean!

The picture above shows a spectrum analysis of a waveform. I don’t have the oscilloscope version of the above, but below is a time-based view of a waveform.

In any case, the uncertainty doesn’t imply any indeterminacy. A particle doesn’t know its position and momentum, and these values are the result of its properties and the state of the rest of the universe and the history of both. This means that the uncertainty principle doesn’t introduce any possible indeterminacy into the number of the universe.

On the second point, some physical processes are probabilistic, such as the decay of a radioactive atom. I don’t believe that this has any effect on the number of the universe. The number incorporates the probabilistic nature of the decay, including all the possibilities.

There is an interpretation of quantum physics called the “Many Worlds Interpretation“, where each possible outcome of a probabilistic process splits off into a separate world, resulting in an infinity of separate worlds. I don’t believe that this tree of probabilistic worlds is a useful view of the situation.

No, I think that there is a probabilistic dimension, just like time or space. All the things that can happen, ‘happen’ in some sense. The probability of you throwing 100 tails in a row with a fair coin is very small, but it is possible. As I see it the main objection to this view is the fact that we only see one view of the universe and we don’t appear to experience any other possible views of the universe, but this is exactly the same with the dimensions of space and time. We only experience one view of space at a time as we can’t be in two places at the same time. While we could be in the same place at two times they are two distinct views of the universe.

In any case the number of the universe encompasses all probabilities so if you still adhere to the single probability model of the universe, our universe and all possible universes are encoded by it. The question then becomes how you can extract the smaller number that encoded the single universe that we experience. I believe that that is not a question that needs to be answered.

The question that does remain open is – why is that number the number of our universe? Why not some other number?

## America’s Cup revisited

I think that it is fair to say that New Zealanders expected their team to win from 8-1 in the lead as they only needed to win one race to win the regatta and the cup. It is also fair to say that almost everyone in the country both knew about and supported Emirates Team New Zealand, to the extent that the whole country almost came to a standstill at 8am which is when the racing started, New Zealand time. I can definitely state that the traffic on the motorway as I drove to work at 8am was much lighter than usual. Many people in Auckland decided to make their way to Shed 10 on the waterfront where the matches were shown on giant TV screens.

When the score was 8-1 the supporters were ecstatic, expecting a quick finish to the regatta, but as the Oracle Team USA yacht continued to win, the feelings changed first to worry then to despair, however I don’t think that many if any failed to hang on until the bitter end.

Contrast this to the situation in the US where most people didn’t even know that there was a yacht race on!

There is no doubt that Oracle Team New Zealand appreciated the support from the supporters in New Zealand and for that matter in the US, and acknowledged the continuing support in defeat. They would have also felt that they were representing New Zealand and would have felt a huge responsibility as a result. The looks on the faces of the team members after the final race would have reflected their own personal disappointment and the disappointment for having let their supporters down.

Explanations for the extraordinary comeback by Oracle Team USA are naturally speculative. It is probably down to a number of factors, but was likely to stem firstly from the alterations that Oracle Team USA made to their yacht during the regatta, making it faster and more manoeuverable and secondly from better boat handling learnt during the regatta, not least of all from Emirates Team New Zealand.

Reactions in the US to the win are interesting. The Slate expounds at length on the fact that Larry Ellison, chief executive of Oracle has expended multi-millions of dollars on the America’s Cup, that holder of the cup can pretty much determine the rules of the regatta and that the yachts are so expensive. They touched on the fact that the Oracle Team USA yacht had only one American on board, the team being predominately from New Zealand and Australia, with the only other American starting the regatta on the boat being kicked off in favour of Ben Ainslie, who is British. It almost seems that the Slate was on the side of Emirates Team New Zealand!

The New York Times is more restrained, merely pointing out the huge input of cash that Larry Ellison has injected into Oracle Team USA, and commenting on the fact that yachting is a niche sport in the US but that 1 million out of 4 and a half million in New Zealand were watching the cup. (I actually think that it was much higher than a mere one million!). So, no denigration of Oracle Team USA, but no real congratulations either.

An interesting thing about all the reporting of the 2013 America’s Cup is that the American yacht has been almost universally been referred to as “Oracle” while the other yacht has generally been referred to as “Team New Zealand”, which says a lot about the general perception of who was actually racing for the America’s Cup. I believe that the Oracle Team USA yacht does not have a name beyond “17” but the Emirates Team New Zealand yacht is named “Aotearoa”, which caused American commentators some problems.

The challenger of record for the next America’s Cup regatta is likely to be an Australian yacht club, which brings back memories of the other remarkable comeback in the America’s Cup history, when the Australian team came back from 3-1 down to win 4-3 and end the longest winning run in sport, but nothing is likely to top the sheer spectacle of the wonderful AC72 yachts flying on foils in the 2013 America’s Cup regatta using techniques developed and perfected by Emirates Team New Zealand and adopted with such success by Oracle Team USA.

There has been no criticism or vilification of Oracle Team USA by New Zealanders or the New Zealand press so far as I know, although the Oracle Team USA team boss, Russell Coutts, has come in for a little criticism, simply because Coutts is a New Zealander. There is talk, though, that Coutts may return to Team New Zealand, if Team New Zealand in fact survives. This might be an issue since Team New Zealand do not have a billionaire backer and mounting an America’s Cup campaign takes a mountain of money.

Ben Ainslie, the British tactician on board Oracle Team USA in the later races has mentioned that he would see a British challenge for the America’s Cup. It would certainly be fitting if a British challenge were to prevail and finally take home the America’s Cup. Although it officially belongs to the New York Yacht Club I believe, it was originally awarded by the Royal Yacht Squadron (Based on the Isle of Wight in England) to the yacht America in the very first race for the cup.