## Apr 10, 2007

### Metric vs. Imperial

In many math/science classes across the US, teachers teach how wonderful the metric system is, and get students to agree that it is unfortunate that we are still using the imperial system in the US. The argument is something along the lines that it is trivial to convert between meters and kilometers - just move the decimal place. This "knowledge" that the metric system is superior to the imperial system seems pervasive(beyond school age), even in the US where we use the imperial system for distances, weights, and often volumes.

What is often assumed from the statement that "metric is better than imperial" is that metric is the ideal system. Indeed, at first glance, it seems very clean - what could be better? This is unfortunately far from the truth.

Metric (otherwise known as Base 10)
The basic premise that makes the metric system generally better than imperial is that you have a single "base" multiplier between different units:
1cm x 10 = 1dm
1dm x 10 = 1m
1m x 10 = 1dam
1dam x 10 = 1hm
1hm x 10 = 1km

I've bolded the units most people are familiar with (centimeter, meter, kilometer). Anyway, the problem with metric is the number 10. It makes the math "easy" because our number system also uses 10, but that is also broken, I claim. Most computer-related people would guess that I'm about to suggestion using a base that is a power of 2, like 8 or 16. Instead, 12.

Duodecimal

If both our number system and metric system were base 12, life would be even simpler. The problem with 10 (or powers of 2) is that it can't be cut in thirds. 1/3 of a mile is 5,280 ft / 3 = 1,760 feet - an integer. 1/3 of a kilometer is 333.3333333.... meters, neither an integer nor a trivially representable decimal. 1/3 of a unit in base 12 is 4 of the next smaller units. 4 is cleaner than 1,760 or 3.3.... Base 12 units are the smallest units cleanly divisible by 2,3, and 4. We could go up to 2,3,4 and 5, but that would require base 60 which for a 1st grader is a little unwieldy when learning all of the digits. 12 is the sweet spot. I'd rather be able to divide things by 3 (more often needed) than 5 if I had to pick. You could even go down to base 6 to simplify further, but that prevents you from dividing cleanly by 4, which seems useful.

What already uses a power of base 12 as it's base?
1. Geometry: 360 degrees in a circle: 12*30 = 360
2. Time: 60 seconds in a minute, 60 minutes in a hour, 24 hours in a day: 12*2 = 24, 12*5 = 60
3. Imperial: 5,280 ft in a mile (12 * 440 ft), 12 inches in a foot
4. Eggs: 12 eggs in a dozen: dozen = 12 * 1 egg

Apparently there is already a term for a system like this: Duodecimal
We should all switch immediately.

Interested in other things misunderstood in school? Check out:

Thomas said...

The metric system is the easiest to use and calculate. Imperial units are easier to visualize, at least in the USA. The imperial system was created at a time when most trade was local and it was common practice to use certain units with specific commodities. This has survived as Troy weight for gold, silver, and platinum, and avoirdupois for cheaper materials. Metric does not make this distinction, but does distinguish between mass and force, more significant for space missions. Imperial was adequate when we were earthbound.
Imperial has many special-purpose units, one set of units for cloth, others for various foods, another for land, still another for watches, yet another for shoes. Metric has very few special units, and they are of such extreme sizes that they would not be useful for ordinary use.

Frank said...

I totally disagree with using base 12 as a starting point for a new system.

Here is why.
You state that it is often easy to divide by 2, 3 or even 4. As in street-racing, where 1/4 of a mile is raced.
If we change the system to base 12... a mile will suddenly yield a different number of base-12 meter/yard like equivalents. Suddenly, we will need to go to 0.27 times a base-12 km/mile.
(This numbers are not correct, as you state that a mile is already 'compatible' with base-12.)

Now, the problem is, that for every situation where you come-up with a nice clean solution, anybody could come up with 2 situations where it does not give any benefit. (If we took more time than to simply put our disagreements on paper/on-line, not taking the time to actually convert base-12 into anything useful. Which would require me to use pen/paper/calculator, as base-12 is NOT easy to calculate with.)

Also, dividing and multiplying by 12 are not easy (*3 *2 *2... is quite ok, /2 /2 /3 is NOT ok). Whereas some people (me amongst them) are totally comfortable with multiplying by 2, everybody with even the most basic knowledge of calculation is comfortable with multiplying/dividing by 10 (I wouldn't even call it mathematics, as the word is already more complex).

In short, there is no single advantage to base-12 that is not negated by the fact that there are more disadvantages.

To make sure I also kick Thomas against his ankles (it's a saying here in Holland)...
The Imperial system is indeed easier to visualize, for those living in the US. In an international environment, image what a foot or inch (width of thumb) could be, we Dutch are the tallest people on earth, in China people are generally less tall than anywhere else... suddenly imagining the size of a foot or width of a thumb becomes less easy, and definitely less constant.

By using the (also standardized) notations on milli, micro, centi, kilo, mega, etc, all of those 'extreme units' can be used for anything. Making it useful and comparable to many other fiels or applications.
Example: Comparing inches to feet to yards to miles. Comparing centimeters to decimeters to meters to kilometers. See the simplicity.

Greg said...

To Frank, I posit the following. Lets say we could start from scratch. Clearly we can't, and in this world of momentum it would be nice if the US would switch over to metric just for the sake of everyone being on the same page.

If we could start from scratch though, we could just use a base 12 number system as well. Instead of 0-9, we would have 2 new symbols, say A and B:

Multiplying by 12 would then be quite easy in our new system:

9*10 = 90 <- I just multiplied by twelve.

B*10 = A0 <- multiplied by twelve again.

Multiplying by decimal ten (in digits: A) would be slightly harder:

9*A = 76

Dividing by 3 or 4 would be easy as cake:

112 / 3 = 44
112 / 4 = 33

Thomas said...

There have been numerous attempts for the USA to convert to metric, but were unsuccessful. We do have 2-liter bottles of pop, water is sometimes sold in 500-ml and 1-l bottles, tape measures are often marked in centimeters in addition to inches, many scales read in kilograms. For a short period in 1975, gasoline was sold by the liter, which caused great confusion. This was done when gasoline cost more than \$1/gallon for the first time, and the dials cannot accommodate this. Switching to liters was the first attempt to deal with a gas price beyond the dials' range, but later switched to half-gallons until the new dials, which can handle up to \$10/gallon, were introduced.

palpatine92 said...

imperial all the way!

paul said...

Smart people need metric. its the best to do calculations. simple minded individuals need imperial, as its more of a visual thing. USA is one of the few countries that still use imperial.. what does that say about them? its logical thinkers vs visual people. use your brains guys, not just your eyes >_<

Harley said...

The Imperial twelve base is still used because of the divisibility of 2,3,and 4. If you have enough experience with any type of craftsmanship which uses fractions a lot, you CANNOT do accurate work with ten base. Dividing work pieces by halves, thirds, and quarters is way too common to use ten base. I wish I knew what it was like to do carpentry with the metric system. Accuracy must be difficult.

If you study history, we used two, four, ten, and twenty base first (different cultures). Thats because some only needed up to four for the same reason as above (1/3, 1/2, 1/4). The other cultures used ten and twenty base because they had ten fingers and toes to count with. This seems too obvious to ignore.

The metric system is superior because of its universal nature. That is different units are relative like one cubic centimeter of water at standard room temp. weighs exactly one gram. That is volume, mass, and temperature for the most common substance as a standard.

The best thing for me is to use metric units but in twelve base.

If anyone wants to help me earn a Nobel we could try to recalculate Pi with twelve base. I think that will yield a repeating number which would be an extremely important contribution to science.

Ask me why, I welcome emails.
Harley Borgais
harleyborgais@gmail.com
"Genesis of Relativity Unifying Fractal Model of Physics" (GRUF model)tm.

Sebastian said...

There is no reason to post, since this thread is so damn old, but you are retarded.

Imperial is not for visual people. Imperial is for visual people who are ALREADY familiar with imperial.

Oh boo-hoo, you cant divide by 2,3, or 4? Who cares, decimals work fine. What about dividing by 5? Oops, your fucked.

Greg said...

This particular post gets all sorts of interesting comments.

I agree with your first point Sebastian, but not the second.

Obviously base 12 isn't divisible by 5. I address that in the post saying that you could use base 60 to deal with that issue, but it's a bit unwieldy to have that many numerical characters to learn. However, It is more useful in practice to be able to divide by both 3 and 4 than it is to be able to divide by 5, if one had to pick between the two.

Harley said...

Sebastian, Have you ever worked construction, with a tape measure?

Carpenters practically never need to divide by 5. Only 2s, 3s, and factors of those (2,3,4,6,8,9,12,16...).

If its so damn old, and there is no reason to post, why did you?

If I am retarded, why are you wasting your time posting? I am trying to achieve good things.

So, are you a metric user?

Harley said...

I have been communicating with many people (including two from Dozenal Society) about Pi (as well as Unified Field Theory Topics).
The point is to see if Base 12 can give a rational number for Pi. So far, I have not found one instance where Pi was really calculated in Base 12, ever! Nor has Base 12 been used to calculate any other important numbers in math, or physics, even though astrology IS based on 12s.
Can anyone show me a real calculation of Pi in Base 12, or has that really NEVER been done?
harleyborgais@gmail.com

Harley said...

If Pi can be recalculated using Base 12 math (and it creates a rational number), then the human species may understand how to make great new technology like free energy, force fields, faster than light space travel, time travel, and an amazing plethora of new things that were impossible without an accurate value for Pi.

An inaccurate Pi is problematic for many fields like physics, astronomy, rocket science, electronics, and nuclear engineering.
(hopefully these related Key words will attract more visitors- mystery, particle, higgs, quantum, light, atom, electron, Phi, secrets of the universe, ultimate answer to everything)

Sebastian said...

Let me clarify, I was not calling you retarded Harley, but I was referring to others who have replied to the OP. I'm sorry if it seemed like I was replying to you, and admit that you have a valid point, worth noting (that the conversion to base 12 would benefit carpenters). I would mention who I was directly responding to, but I feel that would just make people angry, so I'll let that go.

To answer your question, I am a metric user. I have no idea why the US hasn't switch to metric, as it costs the country millions of dollars in calculation errors each year, but that is not the topic of this discussion.

Pi is an irrational number. What that means is that it cannot be expressed in the form x/y, where x and y are both integers. A number cannot be rational in a particular base, if it is irrational in one base, it is irrational in them all (since different bases are different methods of communicating the same number).

Furthermore, the accuracy of pi isn't very important to science in general. There are no scientists saying "I could build a time machine, but only if pi were a rational number :(" that never happens. The expression of pi to 5 digits is more than necessary for almost any scientific calculation, and if more digits are needed, thats fine, since it has been calculated in the millions of digits.

You may say that infinite accuracy for pi (as in a rational expression) would be helpful to science, but I say that it is not. At a certain point the usefulness of pi would break down. At the molecular level perfect circles do not exist (hexagons and tetrahedral structures are more common). Hence, pi to a million digits cant even be used in any meaningful application.

Now, to address the OP. Wherever you can say: "changing to base 12 would benefit a carpenter" or "a young student" the fact is that it would have a negative effect on just as many, or more disciplines. In chemistry many solutions are standardized at 0.1mol, which is 0.124972497249725 in base 12. Not a happy number... You could tell the chemistry industry to switch over to 0.1 (base 12) standard solutions, and they might, but in all cases this isn't necessarily possible. There are hundreds of other examples like this, it is difficult to consider all the viewpoints.

I feel I've written too much, so I'm gonna end it there.

Tarjei said...

You COMPLETLY overlook the single most important benefit with the metric system.

The metric system isn't just for length. It's for length, weight, volume etc. And THE CONVERSION and crosspoints between these.

The decimal argument is one of the lamest ones ive heard.
What's the problem with x.x ???
You really think it's only the US who have carpenters?

(BTW: Computers have a binary system because of the logic nature of a single switch - it can be either ON or OFF).

Valentin said...

Hmmm.

Well, the metric system is flat out superior to the imperial in practical terms.

Jesper said...

Hi! I realize that this is a really old post, but I'll comment anyway.

I'm from Sweden and I'm used to the metric system, so of course I'm pro-metric for all reasons (and more) stated in previous posts.

And since Harley said something about the Nobel Prize (which is Swedish, by the way!) I helped him calculate PI in base 12: 3.23577346643391A7271857241177987A20AB6839502BA326B398964A984A52925542293831B21876812A56410B135293B152

So, can I be mentioned when you receive your Nobel Prize? At least let me shake our King's hand :)

I haven't included more digits, since, as Sebastian said - it's an irrational number in ANY base (unless that base is a multiplier of PI itself, which is not really practical at all).

Harley said...

Jesper, did you use a calculator?

It makes all the difference!

Jesper said...

Changing the base of any number is fairly straight forward - even using real numbers.

I know this has nothing to do with the original post, but since you want the Nobel Prize (btw, there is no Nobel Prize for math), here's how you do it. And I suggest writing your own software to do the conversion, since a normal calculator won't be able to calculate with more than about 16 digits:

As you might remember from first grade math a normal base 10 number has the following values for every position (counting from 1000 downwards):

10^3 (1000)
10^2 (100)
10^1 (10)
10^0 (1)
10^-1 (0.1)
10^-2 (0.01)
10^-3 (0.001)
and so on

Changing the number base to, say 2 (binary) gives you the following numbers:

2^3 (8)
2^2 (4)
2^1 (2)
2^0 (1)
2^-1 (0.5)
2^-2 (0.25)
2^-3 (0.125)

... and to 12 gives you:
12^3 (1728)
12^2 (144)
12^1 (12)
12^0 (1)
12^-1 (1/12)
12^-2 (1/144)
12^-3 (1/1728)

And the way you convert a number from our normal base 10 to another base is by first finding the largest number in the series of values that is not larger than the number we're trying to convert.

Then divide the number you wish to convert by the value number. The number on the left side of the decimal point is the first number in your base conversion.

Now, multiply the result you got with the value number and then subtract that value from the number you wanted to convert. This gives you the next number to convert. Go one step down in the value list and repeat the steps above until you have reached zero.

Example: convert 6.75 to base 2

Values:
2^3 (8)
2^2 (4)
2^1 (2)
2^0 (1)
2^-1 (0.5)
2^-2 (0.25)

First find the largest value that is not larger than 6.75 => 4

Divide 6.75 by 4 => 1.6875
The integer part (1) is our first binary number
That number multiplied by the value is 4 (1 * 4). Subtract that number from 6.75 and you get 2.75 (6.75 - 4). That's the number we're continuing with.

The next number in the value series is 2. Repeat the previous step:

Divide 2.75 by 2 => 1.375
The integer part (1) is our second binary number
That number multiplied by the value is 2 (1 * 2). Subtract that number from 2.75 and you get 0.75 (2.75 - 2). That's the number we're continuing with.

... and continue (value 1):
0.75 / 1 = 0.75
Integer number is 0
0.75 - 0 = 0.75

Value 0.5:
0.75 / 0.5 = 1.5
Integer number is 1
0.75 - 0.5 = 0.25

Value 0.25:
0.25 / 0.25 = 1
Integer number is 1
0.25 - 0.25 = 0

We reached 0, so we're done!
Hence 6.75 (base 10) equals 110.11 (base 2).

Now you know how to do the calculations - now you do the calculation using π (pi).

As anyone with some knowledge in mathematics know - an irrational number is a number that cannot be written as a fraction between two integers. Pi is an irrational number. An irrational number will not be rational in any base unless that base is a multiplier of the irrational number itself.

Phil said...

In chronology it would be nice to have a base of 60. Geometry it would be nice to somehow have the base of pi. Calculations are made easier as we use symbols, and as humans (especially in gradeschool and high school) a convenient base is 10 because we can wrap our brains around it. In the US convenient standards are feet, inches, cups, etc, because our culter facilitates us to get our brains around it.

jocelyn said...

I would totally agree with a base twelve system; but you would have to make twelve the first double digit number, twelve would have to called 'ten'. That means altering the entire numbering system for values above nine. Ten x ten would still equal 100; but not as we know it. Let's face it, we only use ten because we have five digits on each hand. If we had six?

Interestingly; old money, Pounds, Shillings, and Pence combined base ten, twelve and the power of two. I challenge anyone to find a more versatile system in the history on man.

The problem with base ten is its' puritanical and compulsory imposition.

John said...

We can learn the metric sytem, we can understand the metric system, but we dont see the advantages of the metric system until we start to use the metric system. Reading through the comments I suspect that most commentators have never used the metric system.
Its often stated, that because the metric system is based on the decimal system. and uses powers of 10, that because 10 is only divisable by 2 and 5 then the metric system measurements are only divisable by 2 and 5. This is incorrect. The metric system has flexibility ..A meter can not only be described as 1 meter but also as 100 cm or 1000 mm. Take 1000 mm for example..1000 mm can be divided evenly into 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, and 500.
Also its easier to divide metric measures by three, than Imperial measures..For example one kilometer divided by 3 is 333 and 1/3 meters. (ie 333.333 meters) Of course you can have more reoccuring threes if you require futher precision.
Imperial measures are not consistant. There are 64 parts to an inch, 12 inches to a foot, 3 foot to a yard, 22 yards to a chain, 10 chains to a furlong, 8 furlong to a mile etc and also 16 ounces to a pound, 100 pound to a hundredweight etc and many more.
Many units are not divisable by 3. For example how would you measure 1/3 of an inch or 1/3 of a pound.
Whether 12 is divisable by 3 and 10 is not divisable by 3 is mainly related to theory and mathematics. It doesnt matter very much in the real world of using Imperial measures or using the metric measures. That is why I suspect that many commentators have not used the metric system. Measurements in the real world are more than just the basic unit. (ie; more than the inch or foot, more than the meter or milliliter) For example Can 21 inches be divided by 3.. Can 33 feet be divided by 3.. Can 39 meters be divided by 3..Can 330 milliliters be divided by 3...Yes they all can..
Can 22 inches be divided by 3.. Can 40 meters be divided by 3 .. No they cant ..not exactly... but in most situations they can be divided by 3 with enough accuracy for practical use.
There is a perception that metric measures must be difficult for carpenters and builders. This is incorrect. Carpenters/builders use millimeters ( mm ). Many measurements used in the building industry are designed around the length 100 mm or multiples of it.. For example measurements of 300 mm 600 mm 1800 mm and 2400 mm are common in metric designed buildings. All the measurements divisable by 3.
Also Renard numbers are easily conpatable with metric measures, and very difficult to use with Imperial measures. Renard numbers are very important, not just in designing buildings but in designing generally. Renard numbers would not be compatable with a base 12 or base 60 system.
As you have stated there may have been some merit in basing a number system on 12 or 60 if we were starting from scratch. But the metric system is well established worldwide, used by all countries and the primary measurement system in over 170 countries and 95% of the worlds population. It would very difficult to change it to base 12 or 60.

harleyborgais said...

It is really this simple:
In the real world, we need to divide things by the three simplest units, 1,2,&3.
With base 12, there is no need for decimals or rounding. thirds are accurate and solid numbers.

The universality of the metric as you describe, how you can convert units, and how for example:
One cubic centimeter of water is one gram at room temp.
These are part of a system that is priceless, and the way that these things evolve, is by hybridization.

We need only to use base 12 during tasks of simple construction, and we need to use base 10 with tasks of complexity, but we should really use the metric units the entire time.

Fake Furby said...

You linked to the British Imperial system which is different to the US. US uses United States customary units, many of which are identical to the British system but there are differences, which is why it's important to make the distinction.