NUMBERS

One of the most endlessly charming parts of the human experience is our capacity to see something we can’t describe and just make something up in order to do so, never mind whether it makes any sense in the long run or not. Countless examples have been demonstrated over the years, but the mother lode of such situations has to be humanity’s invention of counting.

Numbers do not, in and of themselves, exist- they are simply a construct designed by our brains to help us get around the awe-inspiring concept of the relative amounts of things. However, this hasn’t prevented this ‘neat little tool’ spiralling out of control to form the vast field that is mathematics. Once merely a diverting pastime designed to help us get more use out of our counting tools, maths (I’m British, live with the spelling) first tentatively applied itself to shapes and geometry before experimenting with trigonometry, storming onwards to algebra, turning calculus into a total mess about four nanoseconds after its discovery of something useful, before just throwing it all together into a melting point of cross-genre mayhem that eventually ended up as a field that it as close as STEM (science, technology, engineering and mathematics) gets to art, in that it has no discernible purpose other than for the sake of its own existence.

This is not to say that mathematics is not a useful field, far from it. The study of different ways of counting lead to the discovery of binary arithmetic and enabled the birth of modern computing, huge chunks of astronomy and classical scientific experiments were and are reliant on the application of geometric and trigonometric principles, mathematical modelling has allowed us to predict behaviour ranging from economics & statistics to the weather (albeit with varying degrees of accuracy) and just about every aspect of modern science and engineering is grounded in the brute logic that is core mathematics. But… well, perhaps the best way to explain where the modern science of maths has lead over the last century is to study the story of i.

One of the most basic functions we are able to perform to a number is to multiply it by something- a special case, when we multiply it by itself, is ‘squaring’ it (since a number ‘squared’ is equal to the area of a square with side lengths of that number). Naturally, there is a way of reversing this function, known as finding the square root of a number (ie square rooting the square of a number will yield the original number). However, convention dictates that a negative number squared makes a positive one, and hence there is no number squared that makes a negative and there is no such thing as the square root of a negative number, such as -1. So far, all I have done is use a very basic application of logic, something a five-year old could understand, to explain a fact about ‘real’ numbers, but maths decided that it didn’t want to not be able to square root a negative number, so had to find a way round that problem. The solution? Invent an entirely new type of number, based on the quantity i (which equals the square root of -1), with its own totally arbitrary and made up way of fitting  on a number line, and which can in no way exist in real life.

Admittedly, i has turned out to be useful. When considering electromagnetic forces, quantum physicists generally assign the electrical and magnetic components real and imaginary quantities in order to identify said different components, but its main purpose was only ever to satisfy the OCD nature of mathematicians by filling a hole in their theorems. Since then, it has just become another toy in the mathematician’s arsenal, something for them to play with, slip into inappropriate situations to try and solve abstract and largely irrelevant problems, and with which they can push the field of maths in ever more ridiculous directions.

A good example of the way mathematics has started to lose any semblance of its grip on reality concerns the most famous problem in the whole of the mathematical world- Fermat’s last theorem. Pythagoras famously used the fact that, in certain cases, a squared plus b squared equals c squared as a way of solving some basic problems of geometry, but it was never known as to whether a cubed plus b cubed could ever equal c cubed if a, b and c were whole numbers. This was also true for all other powers of a, b and c greater than 2, but in 1637 the brilliant French mathematician Pierre de Fermat claimed, in a scrawled note inside his copy of Diohantus’ Arithmetica, to have a proof for this fact ‘that is too large for this margin to contain’. This statement ensured the immortality of the puzzle, but its eventual solution (not found until 1995, leading most independent observers to conclude that Fermat must have made a mistake somewhere in his ‘marvellous proof’) took one man, Andrew Wiles, around a decade to complete. His proof involved showing that the terms involved in the theorem could be expressed in the form of an incredibly weird equation that doesn’t exist in the real world, and that all equations of this type had a counterpart equation of an equally irrelevant type. However, since the ‘Fermat equation’ was too weird to exist in the other format, it could not logically be true.

To a mathematician, this was the holy grail; not only did it finally lay to rest an ages-old riddle, but it linked two hitherto unrelated branches of algebraic mathematics by way of proving what is (now it’s been solved) known as the Taniyama-Shimura theorem. To anyone interested in the real world, this exercise made no contribution to it whatsoever- apart from satisfying a few nerds, nobody’s life was made easier by the solution, it didn’t solve any real-world problem, and it did not make the world a tangibly better place. In this respect then, it was a total waste of time.

However, despite everything I’ve just said, I’m not going to decide that all modern day mathematics is a waste of time; very few human activities ever are. Mathematics is many things; among them ridiculous, confusing, full of contradictions and potential slip-ups and, in a field whose age of winning a major prize is younger than in any other STEM field, apparently full of those likely to belittle you out of future success should you enter the world of serious academia. But, for some people, maths is just what makes the world makes sense, and at its heart that was all it was ever created to do. And if some people want their life to be all about the little symbols that make the world make sense, then well done to the world for making a place for them.

Oh, and there’s a theory doing the rounds of cosmology nowadays that reality is nothing more than a mathematical construct. Who knows in what obscure branch of reverse logarithmic integrals we’ll find answers about that one…

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In my last post (well, last excepting Wednesday’s little topical deviation), I talked about the real nuts and bolts of a computer, detailing the function of the transistors that are so vital to the workings of a computer. Today, I’m going to take one step up and study a slightly broader picture, this time concerned with the integrated circuits that utilise such components to do the real grunt work of computing.

An integrated circuit is simply a circuit that is not comprised of multiple, separate, electronic components- in effect, whilst a standard circuit might consist of a few bits of metal and plastic connected to one another by wires, in an IC they are all stuck in the same place and all assembled as one. The main advantage of this is that since all the components don’t have to be manually stuck to one another, but are built in circuit form from the start, there is no worrying about the fiddliness of assembly and they can be mass-produced quickly and cheaply with components on a truly microscopic scale. They generally consist of several layers on top of the silicon itself, simply to allow space for all of the metal connecting tracks and insulating materials to run over one another (this pattern is usually, perhaps ironically, worked out on a computer), and the sheer detail required of their manufacture surely makes it one of the marvels of the engineering world.

But… how do they make a computer work? Well, let’s start by looking at a computer’s memory, which in all modern computers takes the form of semiconductor memory. Memory takes the form of millions upon millions of microscopically small circuits known as memory circuits, each of which consists of one or more transistors. Computers are electronic, meaning to only thing they understand is electricity- for the sake of simplicity and reliability, this takes the form of whether the current flowing in a given memory circuit is ‘on’ or ‘off’. If the switch is on, then the circuit is represented as a 1, or a 0 if it is switched off. These memory circuits are generally grouped together, and so each group will consist of an ordered pattern of ones and zeroes, of which there are many different permutations. This method of counting in ones and zeroes is known as binary arithmetic, and is sometimes thought of as the simplest form of counting. On a hard disk, patches of magnetically charged material represent binary information rather than memory circuits.

Each little memory circuit, with its simple on/off value, represents one bit of information. 8 bits grouped together forms a byte, and there may be billions of bytes in a computer’s memory. The key task of a computer programmer is, therefore, to ensure that all the data that a computer needs to process is written in binary form- but this pattern of 1s and 0s might be needed to represent any information from the content of an email to the colour of one pixel of a video. Clearly, memory on its own is not enough, and the computer needs some way of translating the information stored into the appropriate form.

A computer’s tool for doing this is known as a logic gate, a simple electronic device consisting of (you guessed it) yet more transistor switches. This takes one or two inputs, either ‘on’ or ‘off’ binary ones, and translates them into another value. There are three basic types:  AND gates (if both inputs equal 1, output equals 1- otherwise, output equals 0), OR gates (if either input equals 1, output equals 1- if both inputs equal 0, output equals 0), and NOT gates (if input equals 1, output equals 0, if input equals 0, output equals 1). The NOT gate is the only one of these with a single input, and combinations of these gates can perform other functions too, such as NAND (not-and) or XOR (exclusive OR; if either input equals 1, output equals 1, but if both inputs equal 1 or 0, output equals 0) gates. A computer’s CPU (central processing unit) will contain hundreds of these, connected up in such a way as to link various parts of the computer together appropriately, translate the instructions of the memory into what function a given program should be performing, and thus cause the relevant bit (if you’ll pardon the pun) of information to translate into the correct process for the computer to perform.

For example, if you click on an icon on your desktop, your computer will put the position of your mouse and the input of the clicking action through an AND gate to determine that it should first highlight that icon. To do this, it orders the three different parts of each of the many pixels of that symbol to change their shade by a certain degree, and the the part of the computer responsible for the monitor’s colour sends a message to the Arithmetic Logic Unit (ALU), the computer’s counting department, to ask what the numerical values of the old shades plus the highlighting is, to give it the new shades of colour for the various pictures. Oh, and the CPU should also open the program. To do this, its connections send a signal off to the memory to say that program X should open now. Another bit of the computer then searches through the memory to find program X, giving it the master ‘1’ signal that causes it to open. Now that it is open, this program routes a huge amount of data back through the CPU to tell it to change the pattern of pretty colours on the screen again, requiring another slue of data to go through the ALU, and that areas of the screen A, B and C are now all buttons, so if you click there then we’re going to have to go through this business all over again. Basically the CPU’s logical function consists of ‘IF this AND/OR this happens, which signal do I send off to ask the right part of the memory what to do next?’. And it will do all this in a miniscule fraction of a second. Computers are amazing.

Obviously, nobody in their right mind is going to go through the whole business of telling the computer exactly what to do with each individual piece of binary data manually, because if they did nothing would ever get done. For this purpose, therefore, programmers have invented programming languages to translate their wishes into binary, and for a little more detail about them, tune in to my final post on the subject…

What we know and what we understand are two very different things…

If the whole Y2K debacle over a decade ago taught us anything, it was that the vast majority of the population did not understand the little plastic boxes known as computers that were rapidly filling up their homes. Nothing especially wrong or unusual about this- there’s a lot of things that only a few nerds understand properly, an awful lot of other stuff in our life to understand, and in any case the personal computer had only just started to become commonplace. However, over 12 and a half years later, the general understanding of a lot of us does not appear to have increased to any significant degree, and we still remain largely ignorant of these little feats of electronic witchcraft. Oh sure, we can work and operate them (most of us anyway), and we know roughly what they do, but as to exactly how they operate, precisely how they carry out their tasks? Sorry, not a clue.

This is largely understandable, particularly given the value of ‘understand’ that is applicable in computer-based situations. Computers are a rare example of a complex system that an expert is genuinely capable of understanding, in minute detail, every single aspect of the system’s working, both what it does, why it is there, and why it is (or, in some cases, shouldn’t be) constructed to that particular specification. To understand a computer in its entirety, therefore, is an equally complex job, and this is one very good reason why computer nerds tend to be a quite solitary bunch, with quite few links to the rest of us and, indeed, the outside world at large.

One person who does not understand computers very well is me, despite the fact that I have been using them, in one form or another, for as long as I can comfortably remember. Over this summer, however, I had quite a lot of free time on my hands, and part of that time was spent finally relenting to the badgering of a friend and having a go with Linux (Ubuntu if you really want to know) for the first time. Since I like to do my background research before getting stuck into any project, this necessitated quite some research into the hows and whys of its installation, along with which came quite a lot of info as to the hows and practicalities of my computer generally. I thought, then, that I might spend the next couple of posts or so detailing some of what I learned, building up a picture of a computer’s functioning from the ground up, and starting with a bit of a history lesson…

‘Computer’ was originally a job title, the job itself being akin to accountancy without the imagination. A computer was a number-cruncher, a supposedly infallible data processing machine employed to perform a range of jobs ranging from astronomical prediction to calculating interest. The job was a fairly good one, anyone clever enough to land it probably doing well by the standards of his age, but the output wasn’t. The human brain is not built for infallibility and, not infrequently, would make mistakes. Most of these undoubtedly went unnoticed or at least rarely caused significant harm, but the system was nonetheless inefficient. Abacuses, log tables and slide rules all aided arithmetic manipulation to a great degree in their respective fields, but true infallibility was unachievable whilst still reliant on the human mind.

Enter Blaise Pascal, 17th century mathematician and pioneer of probability theory (among other things), who invented the mechanical calculator aged just 19, in 1642. His original design wasn’t much more than a counting machine, a sequence of cogs and wheels so constructed as to able to count and convert between units, tens, hundreds and so on (ie a turn of 4 spaces on the ‘units’ cog whilst a seven was already counted would bring up eleven), as well as being able to work with currency denominations and distances as well. However, it could also subtract, multiply and divide (with some difficulty), and moreover proved an important point- that a mechanical machine could cut out the human error factor and reduce any inaccuracy to one of simply entering the wrong number.

Pascal’s machine was both expensive and complicated, meaning only twenty were ever made, but his was the only working mechanical calculator of the 17th century. Several, of a range of designs, were built during the 18th century as show pieces, but by the 19th the release of Thomas de Colmar’s Arithmometer, after 30 years of development, signified the birth of an industry. It wasn’t a large one, since the machines were still expensive and only of limited use, but de Colmar’s machine was the simplest and most reliable model yet. Around 3,000 mechanical calculators, of various designs and manufacturers, were sold by 1890, but by then the field had been given an unexpected shuffling.

Just two years after de Colmar had first patented his pre-development Arithmometer, an Englishmen by the name of Charles Babbage showed an interesting-looking pile of brass to a few friends and associates- a small assembly of cogs and wheels that he said was merely a precursor to the design of a far larger machine: his difference engine. The mathematical workings of his design were based on Newton polynomials, a fiddly bit of maths that I won’t even pretend to understand, but that could be used to closely approximate logarithmic and trigonometric functions. However, what made the difference engine special was that the original setup of the device, the positions of the various columns and so forth, determined what function the machine performed. This was more than just a simple device for adding up, this was beginning to look like a programmable computer.

Babbage’s machine was not the all-conquering revolutionary design the hype about it might have you believe. Babbage was commissioned to build one by the British government for military purposes, but since Babbage was often brash, once claiming that he could not fathom the idiocy of the mind that would think up a question an MP had just asked him, and prized academia above fiscal matters & practicality, the idea fell through. After investing £17,000 in his machine before realising that he had switched to working on a new and improved design known as the analytical engine, they pulled the plug and the machine never got made. Neither did the analytical engine, which is a crying shame; this was the first true computer design, with two separate inputs for both data and the required program, which could be a lot more complicated than just adding or subtracting, and an integrated memory system. It could even print results on one of three printers, in what could be considered the first human interfacing system (akin to a modern-day monitor), and had ‘control flow systems’ incorporated to ensure the performing of programs occurred in the correct order. We may never know, since it has never been built, whether Babbage’s analytical engine would have worked, but a later model of his difference engine was built for the London Science Museum in 1991, yielding accurate results to 31 decimal places.

…and I appear to have run on a bit further than intended. No matter- my next post will continue this journey down the history of the computer, and we’ll see if I can get onto any actual explanation of how the things work.