How Many Threes Are There In The Universe?

The obvious answer is “quite a lot”. There’s the three on the front of my house, there are a couple more in my phone number and there’s one on the number plate of the car across the street from me right now. There are also several copies of my house number three, stored in the databases of various shopping sites who have been kind enough to deliver goodies to my home address over the years. Ditto for my phone number threes; and the DVLA, police ANPR, and several insurance companies presumably have copies of the three on my neighbour’s car number plate. Now in some ways these threes are very different from each other – my neighbour’s car has little to do with my phone. And neither of these have very much at all to do with that group of three wildebeest over yonder on the African savannah. And yet, in some ways, some of them are very similar – the pair of threes in my phone number, for example, share quite a lot of context and, other than their position in the string of digits, you’d be hard pressed to distinguish one from the other. You’ll notice though that I’ve not used the digit ‘3’ anywhere in the discussion so far. And yet that’s what’s encoded in Amazon’s database, even though my house is adorned with the word ‘three’ on a little piece of green Lakeland slate. If I happened to live in certain parts of Canada or in France, I might have the word ‘trois’ in place of the word ‘three’. So, not only are there many different threes, there are also many different representations of the same three. What they all have in common though is some notion of ‘threeness’. All threes are clearly distinct from all fours, fives and 3.14159s. That’s what it means to be three. We can take this platonic idea of three and call it the master three. If we only have one master three then it makes sense that all the other threes we have, the real-world threes rather than the platonic, idealised, ‘perfect’ three, can really just refer to the master three to get their threeness characteristic. In fact they probably should do that so that we’re not duplicating information needlessly, and potentially having conflicting ideas of threeness. Threeness is threeness and it only needs to be defined in one place. All the other bits and pieces of information related to the real-world threes are specific to to those threes rather than having anything to do with the general idea of threeness. The encoding format of the bit pattern on my phone’s SIM card, the font and colour of the numberplate, the origin of the stone used to make my house number. None of these things help us understand the threeness of these threes. This idea of a separation of concerns is at the heart of referential semantics. But, apart from the very real benefit of letting me express clearly what I mean, what does it allow me to do in practice? Well, as we’ve seen, I can refer to the threeness attribute of the master three from my own threes, maintaining a strong and secure threeness without risk of corrupting it and crucially, from a practical standpoint, without incurring any extra storage or processing costs.

Hexadecimal

This is just a stub for now but in order to avoid your inevitable disappointment have a lovely link.

Memory addresses are often written in hexadecimal and will look something like 0x1234abcd. Hexadecimal is a base 16 number system in the same way binary is base 2 and decimal is base ten. Instead of numbers being made up of the binary digits 0 and 1 or the decimal digits 0,1,2,3,4,5,6,7,8 and 9 they are instead made up of the hexadecimal digits 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e and f.

Grouping of Bits

At the most fundamental level computers are dealing with voltage levels. In reality these voltages vary quite a bit but one of the jobs of the electronics is to ‘condition’ or ‘tidy up’ these voltages to ensure they can be considered either a low voltage or a high voltage. Now, what exactly these voltages are and how low and high are defined varies quite widely between machines but for our purposes just assume that voltages in a computer are low or high.

So, we’ve got two voltage states – low and high. These correspond to what are called bit values, short for binary digit, which are digital 0s and 1s. A low voltage represents a 0 and a high voltage represents a 1. Computer memory stores 0s and 1s. And that’s all it stores.

But it can group these bits together using the conventions of the binary number system into nibbles (4 bits), bytes (generally 8 bits) and words (often 16, 32 or 64 bits – so-called 16, 32 or 64 bit architectures). Why would you want to do that? For two reasons.

Firstly for semantic reasons, in a similar manner to the way we group decimal digits together into numbers. For example the digits 3, 7 and 9 have meaning on their own as individual digits but to express the notion of the number 379 you need to combine the three digits together and treat them as a group.

Secondly, for efficiency reasons. If you only have one wire you can only pass one bit (either a high or low voltage) at any one time (unless you do clever multiplexing things that are not relevant here). It’s quicker to be able to transfer information in parallel which you can do if you have, say, eight individual wires bundled together in a cable (often referred to as a bus in computing) all running in parallel and each carrying its own bit or voltage.

When we’re addressing computer memory we’re usually addressing (or accessing) words rather than individual bits or bytes.