## Hexadecimalised, Metric-aligned Imperial Measures

Despite enduring optimism from the USA and defiance from quaint, antiquated Britain, it seems inevitable that American standard and British imperial measurements will eventually die. As I see it, imperial and standard measures have two problems; firstly that they are bizarre; the following diagram of relationships between imperial units demonstrates this with stark, bewildering clarity:

Secondly, it is not easy to convert between imperial and metric units; conversion factors need memorising, and tables consulting when going between one and the other. Nobody likes this.

However, there is an awesome, cosmic coincidence which exists at a pivotal point between the two systems which, if exploited through a slight tweaking and reform of imperial measures, could not only save these units from fading into insignificance, but could even turn the tables with this neo-imperial system becoming the preferred system of scientists, engineers, merchants and laymen alike, with the comparatively inelegant metric system being consigned to the history books.

This idea has two components: Firstly, a slight adjustment of the inch and the mile. Secondly, the hexadecimalisation of Imperial measures and with it the creation of succinct new word prefixes meaning 16^{1}, 16^{2}, 16^{3} etc., just as 10^{1}, 10^{2} and 10^{3} are defined as deka, hecto and kilo respectively in the metric system.

Now this is certainly not the first proposal for an adjustment of imperial measurements; some have previously advocated, for example, redefining the foot as 1/3 of a metre, others have suggested 0.3m, in order to shoe-horn them neatly into the metric system. However, I propose redefining the foot as 0.29296875m; a quantity with mathematical properties far preferable to 0.33̅ or 0.3. Allow me to explain why.

### The Problem with Decimal: Our Binary Intuition

Although now ubiquitous, the metric system and a decimal system of arithmetic have no shortage of detractors. Even in 1859 John Nystrom submitted a scathing critique of the inelegance of the decimal metric system in his book advocating that base 16 be adopted as the universal radix. At the time his recommendations faded into obscurity as the metric system gained international acceptance. However, since the dawn of computing and with it the notion of binary numbers, generations of scientists and engineers have re-discovered the elegance of number radices which are a power of two such as hexadecimal and base 256. Not that this was a secret; many traditional systems of weights and measures feature halves, quarters, eighths, sixteenths and so on, not least the imperial inch and avoirdupois pound.

Succinctly, the reason powers-of-two have such utility is that halving and doubling are about the simplest and most intuitive operations that humans and computers can perform on a given quantity. Practically, using a simple weighing scale and no weights, it is possible to split a pound into halves, then each of those into halves again, and so on until you are left with sixteenths – or in other words, ounces. Despite the apparent elegance of having ten hectograms to the kilogram, it is impossible to split a kilo into hectos using just a scale; you can achieve 125g (1/8th), 62.5g (1/16th), but not 100g. The same problem arises when dividing lengths; in secondary school my geography teacher Mr Pelton assured me that estimating tenths of a grid square on a map was intuitive and that humans were very good at this; however it is clear that humans are far more adept at splitting lengths into halves, then halves again, and again, than they are at estimating fifths or tenths, and there is a neat online Java applet (see links) which demonstrates this by asking you to click to divide an object into pieces: it is demonstrably easier to divide into binary quantities than tenths.

### Metric-Binary Abominations

The most surprising place you can observe humankind’s preference for power-of-two quantities is in products sold in metric quantities in metric countries. Looking through my (European) kitchen cupboard, there are a surprising number of products which are sold in 500, 250 or 125 gram or mililitre quantities – these all being binary divisions of one kilogram or litre.

One would expect, if decimal really did possess the utility that its proponents claim, that you would rarely see 250g and 125g quantites, but instead 100g, 200g or 300g; that is, similar amounts in rounded hectograms. This observation is testament to the fact that people prefer to divide things in a binary way.

What is alarming about my European pantry however, is that you will **never** see products which are 62.5g – that is, half of 125g; 1/16th of a kilogram. As soon as the amount cannot be expressed neatly in whole grams or ml, people abandon this intuitive behaviour and instead plump for 100g – and then start using binary divisions of 100g; there are no shortage of products in 50g, 25g, etc. amounts! This highlights the cognitive dissonance present in the mind of anyone trying to achieve neat binary divisions **and** neat rounded numbers at the same time, but failing due to the nature of the decimal base.

### Hexadecimalisation – Metres or Hectometres?

One approach for a hexadecimal system of measures would be to simply invent words for base-16 multipliers similar to kilo- and mili-. For example, a “dexametre” might be 16 metres and a “heximetre” might be 0.1_{16} (in hexadecimal, i.e. 1/16th) metres. This has the advantage of the metre remaining the standard universal unit, whatever the radix. This would mean a complete abandoning of the Imperial systems, as these quantities (16m and 62.5mm) have no similar Imperial equivalents.

However, it may be desirable to instead have 100 metres to be the common point between the two systems, due to a handful of delightful coincidences which exist.

### Imperial and Metric – Separated at Birth?

Notice that 1/16th of a mile is 100.584 metres – almost exactly a hectometre.

Notice that there are 63360 inches in a mile.

Notice that 16^{4} is 65536 (a number that will be familiar to any programmer as the number of values expressible by a 16-bit byte), and that this is close to 63360.

By exploiting the similarities between these figures; we could redefine a mile as being 65536 inches, thus hexadecimalising Imperial measures with exactly 10000_{16} (hexadecimal 10000) inches to the mile.

With a 100m imperial city block, a mile could therefore become 1600m; or 16 city blocks.

With these adjustments, 100m becomes a handy point at which conversions can take place between the metric and imperial systems, without the need for look-up tables of awkward conversion factors.

With the revised, “neo-imperial” system, the inch changes from 25.4mm to 24.4140625mm – an ugly number *in decimal*, but you will never have to memorise it because it is simply one block (100m) ÷ 16 ÷ 16 ÷ 16. In hexadecimal it is not an ugly number; it is exactly 0.001_{base 16} blocks. Twelve of these neo-inches would equal 0.29296875m – hence my earlier comment that one foot would be this length. The foot itself, as 12 inches, would cease to be a useful measure, and would be superseded by the new cubit (16”).

So, there would be 16 inches in a cubit, 16 cubits in a lug, 16 lugs in a block and 16 blocks in a mile. Or, using new binary prefixes, the inch, cubit, lug and block could become synonymous with, say, germi-mile, sulfi-mile, cromi-mile and oxy-mile respectively, supposing new hexadecimal prefixes might be named after elements in the periodic table: E.g., oxygen has atomic number 8. 2^{8} = 16^{2} = 256. 1 mile / 256 = one oxy-mile.

### Do Not Open Until 2050

A final note about the strategy behind implementing of such a reform: if I were Mayor of Measurement in the USA and had the power to execute this change tomorrow, I *would not*. Moving from one system to another very similar system overnight would be disastrous. Instead, I would look to the present day UK to see what opportunities the future might hold. A move like this **would** be practically possible in the UK today, because imperial measures are never used in situations requiring great accuracy in Britain; you hear people casually talking about x inches of snow or y feet of rope, but in technical contexts metric is always used. Therefore, if the British governing bodies were to declare a new inch as being 24.4140625mm tomorrow, there would be no great impact; 20 inches of snow would still be 20 inches of snow to ordinary people; yet 132 micrometres would still be exactly 132 micrometres to the engineer. Similarly, roughly 50 miles would still be roughly 50 miles to a car driver; yet 80km would still be exactly 80km to the surveyor. The only people affected would be those who do not require great accuracy.

If the SI metric system is adopted in the USA, as it inevitably will be, especially if metric China becomes the world’s dominant economy thus denying the USA its ability to impose avoirdupois on the rest of the world, American standard measures could conceivably go the way of British imperial system in mere decades, relegated to imprecise, informal use only. If this happens, it could be the perfect time for a reform and revival of a re-invented hexadecimal standard system of weights, measures and arithmetic; one which can co-exist happily with the decimal metric system and which could gain great favour with engineers, merchants and laymen the world over.

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