my name is JD, and I am new to the forum.

I would like to know why the numbers 16, 8, and 4 are used so often in Imperial units and why they are so popular in general.

Just for example, There are:
-16 oz in a lb,
-16 dr in an oz,
-many multiples of 16 in capacity measurements,
-8 pts in a gall,
-8 furlongs in a mile,
-an 1/8 of a pound is (was...) a half crown,
-4 roods in an acre
-4 palms in a foot,
and I even found all the multiples of 16 highlighted on a measurement tape (an Imperial one - every 16 inches)

thank you

Welcome JD...

I guess the easy answer to the question of the use of binary divisions is that these are simple to understand and were effective for the task in question. The binary divisions were practical.

Though dozenal doesn't directly support binary, or at least not to the extent that hexadecimal does, the power of base 12 to handle sixteenths in two digits is convenient.

1/*14 = *0.09 Nice!

An ideal base would have all integers "round numbers". Example, 25 and 58 would "come out", end in a zero. This would make all numbers "transparent", easily factorable so you could readily work with it. With an ideal base you could immediately factor any number. There is no way to create an ideal base, so you have to select a set of primes to "detect" in a quantity. The most useful primes are the most frequent: 2, 3, 5, and 7, etc., progressively less important as the prime increases in size.

Twelve accounts for 2 twice and 3 once, so it simultaneously tests for the first two primes, with more power to resolve binary content. It's eggs aren't in one basket, testing only for 2, as does hexadecimal and octal.

Sixty accounts for 2, 3, and 5. It is like twelve but is burdened with more relatively prime digits and a large multiplication table. Because it accounts for 5 in addition to twelve's prime factors, its ability to resolve a quantity is superior than 12, but it is significantly more difficult to wield. (the field carpenter will be hard pressed to multiply base-60 digits 41 and 43 in his or her head on the second story of a new wood frame house; this task is pretty simple in dozenal or decimal).

Decimal, as with its cousin base 14, splits the primes it tests for, introducing some inefficiency, rendering relatively simple and very common factors "strange" (3 in decimal's case; 3 and 5 in base 14's case). Decimal accentuates the importance of 5, base 14 does the same for 7. For a base composed of two primes, 6 (2 x 3) is the optimum choice, because 2 and 3 are the most common prime factors.

So the binary measures are neatly accommodated by dozenal, because dozenal can resolve for content of twos and fours directly, and by extension to powers of twelve, the further binary divisions. Twelve is superior to decimal but not as specialized to the task as hexadecimal; the superiority perhaps determined by the degree of representation of the prime factor 2 in the base, and whatever else is included in the base (powers of 3, 5, 7, etc) that make the base larger than it needs to be to handle the task. So hexadecimal is superior to dozenal and further decimal to handle the specific task of geometrical binary measurements. Dozenal is a more general base than hexadecimal, because dozenal can account for threes and still has a manageable base size.

The answer lies in the market place.
Halving is the easiest action, so it was logical to divide units by two and two again to get exact pricing (in the case of sterling this would appear in the farthing and half-farthing*).
Think of a pound of butter, which you can halve and halve again for quarter-lb blocks, or again and again for 1-oz pats (for your toast). In pre-metric India the currency-unit was also divided in the same way (16 annas to the rupee).

And for the sixteenths, thirty-seconds and so on - these were units used by people, not machines, and they chose divisions which they found convenient for themselves and their work.

Oh, and for us who hark back a bit, there are still 8 half-crowns to the pound sterling (which itself began as a pound of 240 sterlings - hence the name); I'll have it on my gravestone: "Give me back my half-crown!" ... (Bit more polite than "you know where you can stick your decimal point" ...)


* the half-farthing was issued in 1844 and did circulate in Britain though intended for the Colonies; there were also (for Colonial use) a third-farthing and a quarter-farthing (matching local currency-units and also suggesting the natives weren't paid all that much in those days).

Also, the New York Stock Exchanges used binary fractions of a dollar until 2001.

JD,
First Hi there and welcome.

I think you will find that the markings at intervals of 16 inches on tape rules is an addition to the Imperial rules introduced by the US building industry. It is used to mark the locations of wall studs, floor and ceiling joist, rafters, and trusses.

As for the 16 ounce pound, it should be noted that previous to the avoirdupois pound standardized to 16 ounces, all pounds were originally 12 ounces and the apothecary and Troy pound were never changed and that gold and some other precious items are still quoted in troy ounces even in metric countries.

The furlong was in use by the Saxons long before the mile was standardized to 5280 feet in 1592 by the English Parliament, defining the statute mile to be exactly 8 furlongs. I suppose this was just to bring the two measures into one common set of measures and the number 8 just happened to be the closest in terms of furlongs to the previous existing mile.

Why 8 pints to the gallon. One suggestion is that the the gallon was once set at 8 Troy pounds, and so the pint would have been 1 Troy pound.

OK, the half crown = 1/8 of a pound sterling is just a consequence of the original definition of the pound in such a way as to maximize the possible subdivisions possible. The pound sterling and the British system of pounds, shillings and pence originates with Charlemagne and King Offa in Britain who adopted Chalemagne's system, and was itself a revival of the system used by Greeks and Romans (i.e. 1 libra = 20 solidi = 240 denarii) and was introduced by Charlemagne's father Pepin the Short. Charlemagne's equivalents were the 'livre' of 20 'sou', and the 'sou' of 12 deniers. The livre was both a monetary value and a weight in the same way as the Roman 'As' which was also had a twin usage as money and weight. This livre and thus the pound sterling allowed for subdivisions into 1/2 (10 shillings) , 1/4 (5 shillings or a crown), 1/8 (2 shillings and sixpence or a half crown), 1/16 (1 shilling and three pennies {thruppence}), and with halfpennies and farthings even 1/32 ( 7 pence h'appeny) and 1/64 ( thruppence 3 farthings). Then of course there are the subdivisions of 2/3 (13s 4d), 1/3 (6s 8d), 1/6 (3s 4d), 1/12 (1s 8d), 1/24 (10d), 1/48 (5d), 1/96 (2[DOHTML]<SUP>1</SUP>/<SUB>2</SUB>[/DOHTML]d and even 1/192 giving the penny farthing. Just think how much easier it would have been with a dozenal number base!!!

Now, 4 roods in an acre. I don't know, but hazzard a guess that it was another result of tying different measures into one system. The rood was 40 rods by 1 rod wide. The rod was 16.5 feet or 5.5 yards. Why? Again it was an ancient measure used by the Anglo-Saxons before the Norman conquest in 1066 and in the 12th century was set at 16.5 of the newly introduced 'feet'. This was done since so much land was already recorded in terms of rods and by this means these measures would still be of the same length.

So, 4 palms in a foot. Well the Romans used their foot or 'pes' with 12 uncia to the pes. They also used the digit and there were 16 digits to the pes. Thus, 4 digits equalled 3 uncia or in our terms 3 inches and there are 4 x 3 inches in a foot!

As a final note to Dan, how do you think the NYSE would have viewed a dozenal system of fractions before they replaced their binary fractions with decimals?
[DOHTML]
<TABLE>
<TR><TD> 1/2 </TD><TD> = </TD><TD> 0.5 </TD><TD> = </TD><TD> 1/2* </TD><TD> = </TD><TD> 0.6* </TD></TR>
<TR><TD> 1/4 </TD><TD> = </TD><TD> 0.25 </TD><TD> = </TD><TD> 1/4* </TD><TD> = </TD><TD> 0.3* </TD></TR>
<TR><TD> 1/8 </TD><TD> = </TD><TD> 0.125 </TD><TD> = </TD><TD> 1/8* </TD><TD> = </TD><TD> 0.16* </TD></TR>
<TR><TD> 1/16 </TD><TD> = </TD><TD> 0.0625 </TD><TD> = </TD><TD> 1/14* </TD><TD> = </TD><TD> 0.09* </TD></TR>
</TABLE>
[/DOHTML]

I'd say that the influence of the entire world using decimal currency was too great for them to consider dozenalization.

For a more realistic question, if the U.S. had chosen a dozenal monetary system (or more likely, a hybrid system of 8 bits to a dollar and 12 ____s to a bit) in 1792, would we still have it today?

also 5 "Northern" yards, each of 3 "Northern" feet (or 1.1 Imperial foot). See DSGB site for "A British metre".
The Northern foot was the same as the Sumerian foot; Sumerian cubit = 1/10 pole.
(See Berriman's "Historical Metrology")

Sorry folks - but I disagree about the fundemental reason for the use of binary in measurement.

Surely it largely comes from the use of 2-pan scales as the primary tool of weight measurement?

This is an extract from somethig I wrote elsewhere on the matter:

4) The two pan balance was the earliest, and most reliable, weighing instrument available to pre-modern man. Use of this device has interesting consequences, since given a standard weight (a stone say), there are only a few operations that can be carried out with a two pan scale:

a) One can use the scale, plus the standard weight, to make duplicate copies of the standard weight.

b ) One can put a number of these duplicate weights into one pan, so as to weigh integer multiples of the standard weight

c) Using say fine sand one can create a weight of sand exactly matching the standard weight, and then tip this into the (empty) second pan, until balance is again reached and one has a (in fact two) heap of sand weighing exactly half the standard weight. One then can use this to calibrate a new stone weight to a half the original standard weight.

d) The process with the sand can be repeated, to make the series of lesser binary divisions of the standard, the quarter, eighth, sixteenth etc.

e) Other fractional weights can be constructed by added or subtracting the binary weights, and their own binary fractions can also be constructed.

f) One can make approximate versions of ‘non binary’ weights. For instance one can through a series of steps construct (1/2 + 1/4) x 15/16 - which at about 0.703 is quite a good approximation of 7/10ths of the standard. But this procedure would be quite time consuming, nor can it ever be theoretically exactly accurate.

best

rob

PS More fundementally still - that we can only bisect weight with a 2-pan scale is a corrolary of the fact that we can only bisect an angle or a length wth a pair of compasses - a result finally proved by Galois - after others had tried for 2,000 years to prove it and failed.......

Thing is, it is inherently easier to divide things in half than it is into thirds, fifths, sevenths, and so on. Anyone who's ever had to fold a piece of paper into three equal parts, with no ruler, can testify to this.

But no dout what you're saying has played a part, too.

Uh? :huh: I've never had much difficulty folding a piece of paper into three parts. Take the piece and bend it into an S shape, visually adjusting the lengths of the sections as you make each end of the piece coincide with one of the folds. It only requires a bit of attention. The result may not be as accurate as folding it into halves, but is a fairly decent approximation. It is far more difficult to get a decent result by trying to fold it into fifths, let alone sevenths. A rope or cable can be "thirded" by a similar method. Dividing a line segment into thirds is also much easier than into fifths or sevenths, although of course not as straightforward and accurate as into halves.

A round cake can be divided into thirds (and sixths and twelfths) with about as much accuracy as into halves and quarters, using the fact that sin 30 = cos 60 = [DOHTML]<SUP>1</SUP>/<SUB>2</SUB>[/DOHTML]. Start by determining the center of the cake and make the first cut. Then calculate the middle point of the other half of the diameter defined by that cut. Now visually draw a perpendicular from that middle point until it crosses the circumference, and that will be the second point. So make the second cut, and there you are the first third (corresponding to 4 o'clock in a clock face). Then repeat the method for the third point or simply calculate the bisectrix visually which is fairly easy.

Dividing a cake into fifths and sevenths can be approximated using their dozenal or sexagesimal representations. For fifths and using sexagesimal, just remember that one fifth of an hour is 12 minutes, and then visually imagine where approximately the 12, 24, 36, 48 and 60 minute marks would go on a clock face; although it will be a little difficult since it actually requires visually dividing 30 degree arcs into fifths. Using dozenals, start by visually dividing the circle into twelfths as in a clock face (this is fairly easy to do with decent accuracy, using the method described above for thirds), then set the first point at 12 o'clock and use the points corresponding to a bit more than 2 1/3 and 4 2/3 for the fifth points in one half (mnemonic: even digits, ascending thirds, a bit more), and then get the remaining fifth points using symmetry. For sevenths, the method works the same as for fifths, but this time using the points corresponding to a bit less than 1 3/4, 3 1/2 and 5 1/4 (mnemonic: odd digits, descending quarters, a bit less). What we're doing with this is calculating dozenal 0.24+, 0.48+, 0.74− and 0.98− for fifths, and dozenal 0.19−, 0.36−, 0.53−, 0.69+, 0.86+ and 0.A3+ for sevenths, which are easy-to-remember rough approximations to the dozenal fifths (0.2497..., 0.4972..., 0.7249... and 0.9724...) and to the dozenal sevenths (0.186A35..., 0.35186A..., 0.5186A35..., 0.6A3518..., 0.86A351... and 0.A35186...).

Me neither. Might not be perfect, but it's good enough to fit in an envelope.

I always just construct equilateral triangles.

Me: "it is inherently easier to divide things in half than it is into thirds, fifths, sevenths, and so on."

uax: "The result may not be as accurate as folding it into halves"


You have just made my point for me, uax, with this clause. My point being, as I said, that it is much easier to divide stuff binarily than any other way. So I'm not sure what your "uh" comment was about :P

That is precisely what I was describing. How do you inscribe an equilateral triangle within a quadrant visually? Calculating the middle point of the base (which is the cosine of 60 degrees) and tracing the perpendicular at that point (the height of the triangle) in order to find its intersection with the circumference. Using compasses you can calculate the perpendicular at the middle point of the base and its intersection with the circumference in fewer steps, but you're basically doing the same. However, one doesn't usually have a compass at hand when trying to divide a birthday cake, and anyway they would probably make funny faces at you if you took out a ruler and compass and started drawing lines and arcs on the surface of a delicious chocolate cake. :P