## 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.

## Nothing

Nothing is an interesting concept with many different aspects. Maths, science, philosophy and many other fields of endeavour have their own overlapping concepts of nothing, zero, null or just the absence of anything.

Some computer languages have a concept of ‘null’. This is not the same as the concept of ‘zero’. To use the usual analogy of pigeonholes, numbers and other things in computers are conceptually stored like objects stored in pigeonholes. Each pigeonhole must have a location, sort of like ‘third row down, fourth hole in the row’. A pigeonhole could be empty or it could contain a number or a string of characters or more complicated objects that the computer recognizes. It could optionally have a label so that it can be found quickly.

A computer moves things around and in the process it manipulates them. Given this analogy, what is ‘nothing’ to a computer?  It could mean several things. It could mean the number zero, stored in a pigeonhole or it could refer to an ’empty string’ stored in a pigeonhole. (An ’empty string’ is like the object ‘where’ when the individual letters ‘w’, ‘h’, ‘r’, and the two ‘e’s have been removed. It is represented by two ). It can be a more complicated object that hasn’t been completely set up. Alternatively it could refer to an empty pigeonhole. It could even refer to a label which has not yet been allocated to a pigeonhole. Pity the poor programmer who has to keep all these ‘nothings’ separate in his or her mind (and a few others that I’ve not mentioned!).

In mathematics we have the concept of zero, but this is a fairly newly introduced concept. Some number systems, such the Roman Numeral system do not have a zero, and it was a big conceptual jump to add zero to the mathematical number systems. After all, what do you hold when you have two oranges and you give them away? Nothing! You can’t see zero oranges in your hands, unless you are a modern mathematician of course.

So mathematically ‘nothing’ is zero then? It could be, though ‘nothing’ could be integer zero, ‘0’, rational zero, ‘0/any number’, real number zero, ‘0.0’, complex zero, ‘0 + 0i’, or many many other versions of zero. Maths also has a concept of a set, which is just a collection of objects, which can be pretty much anything. An analogy often used is to liken a set to a bag which contains any sort of object. Statisticians are fond of sets which comprise a set of balls which can be of more than one colour but are usually otherwise identical. If all the balls are removed from the bag, what do you have? A bag with nothing in it! It is usually referred to as an ’empty set’. Note the similarity with the ’empty string’ mentioned above. There’s nothing coincidental there.

There are other sorts of ‘nothing’ in mathematics. A mathematical ‘function’ is a way of relating ‘variables’. The details don’t matter, just the fact that functions have ‘zeros’. They may have one or more zeros or they may have none. Having no zeroes could be considered a sort of ‘nothing’, in a way, though the functions in question are no less proper functions than any other. I’m sure that there are other more esoteric ‘nothings’ in maths.

In physics things should be clearer, right? In physics a vacuum is created is all matter is removed, leaving … nothing. Except that it appears to be impossible to actually remove everything from a container leaving nothing. Even the best pumps will leave a considerable numbers of atoms floating around inside the container. Other methods of emptying the container may reduce slightly the number of atoms in it, but we can’t even reach the very low densities found in the gas clouds visible to astronomers. Even in the depths of space between the galaxies we still find the occasional atom, usually of hydrogen.

Maybe we should look between the atoms for nothing? Most people have an image of an atom as a sort of miniature solar system with the nucleus standing in for the sun and the electrons standing in for the planets. Unfortunately the analogy breaks down if you look closely. Electrons are only found in certain orbits around an atom and even that is an over-simplification. Their location depends on a probability function and in some views this means that the electron is sort of smeared out in space and doesn’t have a strict location and you can’t say specifically that it is ‘there’ at a particular location, only that it has a particular possibility of being there.

One consequence of this is that you can’t say that is isn’t at a particular location, so it is impossible to declare that there is nothing at a particular point in space at any one time. If you consider all the particles in the universe, they all have a probability of being there, so you might be surprised not to find a particle there at a particular moment in time.

In addition to this, I have read article which describe ’empty space’ as a seething mass of pseudo particles or virtual particles. These come in pairs of particle and anti-particle which are continually coming into existence, mutually annihilating each other out of existence again. Viewed in this way it is difficult to describe ’empty space’ as containing nothing, so we still haven’t found ‘nothing’. Although physics has the concept it is hard to find a physical instance of it.

Cosmologists talk about the “Big Bang” when everything came into existence. Before the Big Bang, they say, there was nothing. Nothing! But what does this mean. I like to think of it by analogy. If you take a piece of paper and draw a circle on it, you can consider this circle to contain all space and time and everything that exists in space and time. If you draw a line horizontally through it you can label the big inside the circle as ‘time’. Note that the line should not extend beyond the circle.

The point where the line reaches the left hand side of the circle is the Big Bang. The point where the line reaches the right hand side of the circle is the point where everything collapses on itself and space and time cease to exist.

Some cosmologists think that there will not be a collapse, so the curve is not a circle but a curve open to the right. This doesn’t affect my argument – everything and every time is included inside the curve.

If you now draw a line vertically, not extending beyond the curve, and label it ‘space’. If you move the line to the left, the graphical distance between the top point and the bottom shrinks. Moving the line to the left moves it back along the time axis and represents an earlier state of everything. When the line just touches the curve the point of intersection of the two lines represents the Big Bang.

What about the points outside of the curve? This is where the analogy breaks down. Since we have included all space and time inside the curve the points outside the curve do not represent real points in space and time at all. In short, they do not exist. We could loosely say that nothing exists outside the curve of space and time, but that is not true. ‘Nothing’ is a concept based on space and time, being the opposite of ‘something’ or the potentiality for ‘something’ and as such needs a space-time framework to mean anything. If there is no space and time, there can be no ‘something’ and therefore ‘nothing’ is meaningless. Beginners in science and astronomy might ask what is beyond the boundary of the universe, but the question doesn’t mean anything. The universe contains everything.

If there were other universes, with their own space and time, they would have to be right alongside our universe (that is an analogy of course – language fails us in this situation) as there is nothing to be between the two universes. If you were able to travel from one universe to the other, a concept which I don’t believe stands up to examination, you probably wouldn’t notice the difference. Maybe nothing is a sort of inability to be. But that language implies an intent, which implies a lot of other things and maybe leads to pantheism and I don’t wish to go there.

Well, I’ve used over 1300 words to talk about ‘nothing’, so I will stop here. What comes after the end of this post? Why, nothing, of course!

## Predicting the future

The farmer fed the chicken every morning at the same. The chicken realised this and ran up to the farmer every morning to be fed. One morning the chicken ran up to the farmer who grabbed it and chopped off its head. This demonstrates the dangers of inductive reasoning. The old turkey was a little more sophisticated however. When asked by a younger turkey when Thanksgiving was, he replied that it was on the fourth Friday in November. The younger turkey was incensed to find out that it was the fourth Thursday in November. The older turkey said to him “Boy, the humans celebrate it on the Thursday, but if I wake up on Friday morning, then I give thanks”.

Induction is looking at the past in a particular way to predict the future. Specifically, induction looks at a series of events in the past to predict the future. The sun has risen like clockwork every day, whether or not you can see it, for as long as anyone can remember and for as long as we can determine from reports from the past. Will it rise tomorrow morning?  I would put money on it because either it will, and I win, or it won’t and it won’t matter because we will almost certainly be dead. The argument comes down to “It has always happened in the past, so it will (or it is extremely like to) happen in the future.

The alternative method of reasoning is deductive reasoning. The deductive argument is that the rising of the sun is a consequence of the rotation of the earth. As the earth rotates, the sun appears to us on the earth’s surface to appear from beneath the horizon and travel across the sky. Actually, it is us who move, a good demonstration of relativity (but maybe I’ll go there another day). The argument goes stepwise from fact to fact and leads inevitably or logically to a conclusion.

The trouble with this approach is that, for all its logical stepwise approach it is built on two things, a theory and a set of past observations. A scientist has a theory or decides to check a theory, so he does an experiment, and the results of his experiment support or do not support the experiment. The scientist assumes that the theory is true and bases his predictions on this. Unfortunately there is an inductive element to this – if the theory is true for the experiment, there is no guarantee that it will be true for subsequent experiments, even given that ‘ceteris paribus’ (all things remain the same). Some other unconsidered cause could affect the result. The argument is deductive, proceeding in logical steps from the theory, but the practise is inductive – the data has always supported the theory in the past, so it will continue to support the theory in the future.

To be fair to the inductivists, todays’ inductivists tend to specify the results of their arguments in terms of probabilities: the probability of the sun rising tomorrow is close to 100%, given that it has always risen in the morning for as far back as we can see, but there is a minute but finite possibility that it won’t for known or unknown reasons.

Let’s consider the case of the sun rising each day and suppose that the fact that the earth rotates is not known. To make the argument more deductive we can postulate causes and so long as the cause fits the facts, we can tentatively label the cause as a hypothesis. Suppose we conjecture that some deity causes the sun to rise each morning. This hypothesis certainly fits the facts and predicts with accuracy that the sun will continue to rise each morning. Such a hypothesis would not be accepted today, of course, except by some individuals.

Is there any great difference between the theist and the scientist? The theist says “all things happen because of God”. The scientist says “all things happen because of the laws of nature”. They both explain things on the basis of their fundamental beliefs.

It is possible that people in the future may look at our theories of the sun rising and other things and consider them naive and consider our view of everything happening according to the laws of nature to be a quaint misunderstanding, in much the same way as many people would consider the “deity hypothesis” to be today.

In mathematics the situation is different. Induction is a much more formal process and is applied on top of an axiomatic system. Proved theorems are the results of the applying the axioms repeatedly to another proved theorem or the axioms themselves. Unproven assertions can be proved and turned into theorems or disproved and discarded (or possibly modified so that they can be proved). If something is proved in an axiomatic system, it is true for all time, and cannot be disproved in that system.

Specifically an inductive proof would go something like this: firstly the theorem would be proved for a generic case (eg if statement N is true, then statement N + 1 is true) and secondly it is proved for a specific case (eg statement 1 is true). Then all applicable statements are true because, if statement 1 is true, the generic case means that statement 2 is true, and so on for all cases. Because of the rigor of the argument and the undeniable conclusion of the argument, mathematical inductive proofs are of the same order of reliability as deductive proofs, that is, they are only wrong if there is an error in the logic.

Why the difference between scientific induction and mathematical induction? Well, I think that it is related to the fact that mathematics is axiomatic and therefore certain, whereas scientific induction is based on the laws of nature which are not and never will be, in my opinion, completely defined. If the basis of your argument is not certain, how can your conclusion be certain?