Back to index of science pages by Donald Sauter.

Units of Measure -
blessed simplicity at last!

By all rights, this should have been my first web page when I jumped on the web 'round about 1998 (writing in 2006). A ridiculously simple system of units of measure was a lifelong soapbox issue of mine. Maybe I procrastinated because I couldn't see how to make something so sensible glitzy enough for the world to read. It couldn't be because I had doubts about it getting implemented - that's never stopped me before!

You might not have even been aware that there's a problem. We've got more than enough units of measure to keep everybody happy, right? Well, yes. And that's the problem, always painfully converting from this to that, and never developing a feel for more than a few of them.

I found this quote from noted physicist Richard P. Feynman:

I wish to apologize for something that is not my responsibility: Physicists and scientists all over the world have been measuring things in different units, and causing an enormous amount of complexity. As a matter of fact, nearly a third of what you have to learn consists of different ways of measuring the same thing, and I apologize for it.

I believe the quote to be an understatement in the sense that, if we could eliminate all the confusion caused by the profusion of units of measure, and could do away with all the time and effort wasted converting between them, we could all learn far more than three times as much science.

You might be laboring under the misguided notion that the glorious metric system brought order to all this mess. I'd argue that the benefits of metric over the haphazard British system are miniscule; that the metric system is only slightly worse than the British system at keeping everybody stupid. Now there are even more units to juggle. For instance, who can claim they've never been at least momentarily confused when confronted with a measurement given in mega-somethings when you've been thoroughly conditioned to seeing it given in kilo-somethings?

The problem with the metric system is that it completely surrenders to the notion that we will never progress beyond the cave man in our ability to get comfortable with numbers bigger than 1 or a few. With such a huge stockpile of units to choose from (atto-, femto-, pico-, nano-, micro-, milli-, centi-, deci-, deka, hekto-, kilo-, myria-, mega-, giga-, tera- . . . ) we can always find one so that the measure is 1 or, at worst, a few. Or if we're still not communing close enough with ONE-ness, we can make up even more units - witness angstroms, astronomical units, parsecs, light-years... Everything in the universe is ONE something or other - yippee!

My proposed system turns the metric system on its head. I posit that it makes infinitely more sense to get comfortable with big and small numbers, and use a single unit for each property we measure - time, length, and mass being the main ones.

Atomic or galactic dimensions? Same unit! And by using the same unit for any property, besides never ever having to perform another conversion, we might actually start to develop a feel for how things in nature compare.

After choosing new units for time, length, and mass, the new unit for each of the "compound" properties - area, volume, speed, acceleration, momentum, energy, etc. - will fall directly out of the geometric or physical formula for that property.

          Area = length x length
          Volume = area x length
          Speed = length/time
          Acceleration = velocity/time
          Density = mass/volume
          Momentum = mass x velocity 
          Force = mass x acceleration
          Energy = mass x velocity x velocity = force x length
          Power = energy/time
          Pressure = force/area 

Still surprises me after all these years what you can get by mixing up length, mass, and time. And have you cottoned on to something really neat? Because we will only ever use the one unit for each measurable property, we won't even need names for them! To answer the questions, "How long is it?", or "How far is it from here to there?", or "How big is your yard?", or "How much does that jar hold?", or "How fast is he driving?", etc., etc., a simple number will suffice, completely and unambiguously. For the technical stuff, all you need say is, "the force equals .87," or, "the power used was 93," or, "the density of the rock is 2.8", etc.

If all we did was arbitrarily pick a unit of length, time and mass out of thin air this idea would represent a huge step forward. But I'd like to demonstrate that we can do much better than that.

Out of time, length, and mass, time would seem to be the most problematic because we have natural cycles that must be considered and respected - days and years and that sort of jazz. Never fear; I've got it covered.

Bear with me while I recycle a letter to the editor of the Baltimore Sun on the subject. It was in response to an article about calendar reform, in particular, Johns Hopkins scientist Richard Henry Conn's "Calendar-and-Time" plan, and the International World Calendar Association's proposal.

Dear Baltimore Sun,

Your recent article "An idea whose time hasn't come" (Dec 24 2004) presented a couple of lightweight ideas regarding an improved calendar. Before we put them to a vote, let me toss my own calendar proposal into the ring. It is a thing of beauty.

The basis is our own pure, base-ten number system, and the recognition that keeping the calendar in sync with the solar year is not only a total pain in the neck, but completely unnecessary. A year will have 1000 days. Months will be disposed of. The date will be a simple integer from 000 to 999. The date tells you instantly and precisely where you are within the year - something our current month/date system does not. A simple subtraction tells exactly how many days it is to or from another given date.

If that's not exciting enough, we can take this opportunity to drag how we express time from out of the dark ages. The day will be divided into 1000 parts. You may think of it as 10 "hours" of 100 "minutes" each, although that division scheme would no longer be necessary. We would simply give the "minute" within the day as a three-digit number, 000 to 999.

Date/time could then be naturally expressed as a single number, with a decimal point dividing the two. For example, you could schedule your wedding for 721.650, meaning the upcoming day 721 at 650 (half past "hour" 6). Year/date/time could simply have digits for the year tacked on up front.

The Sun's recent cost-cutting measures couldn't possibly allow space for a run-down of all the jaw-dropping advantages of this new calendar and time system, but let me mention one. With a year about 2.74 times longer than the current now-you-see-it-now-you-don't version, birthdays and holidays would become that much more special - we might even start looking forward to them!

Happy New Year!
Donald Sauter

This letter to the Sun was not published. Maybe they had some really world-changing letters to print that day; I suspect the poor kid who opened my email never even knew what hit her.

In the original letter I proposed 100 new "seconds" per new "minute", which would keep it very close to the second we are familiar with. In working up this web page, I discovered that a fundamental unit of time which divides our day into a million parts works much better. It also provides a satisfying balance: three digits apiece allocated to the day within the year, the "minutes" within the day, and the thousand Units of Time within the "minute".

Dividing the day into a million parts yields, exactly:

          New Unit of Time = .0864 seconds.

Just think of it as about a tenth of our familiar second.

Now we need a New Unit of Length that works together well with our New Unit of Time. Hmmm... Let me think. Ok, I've got it!

          New Unit of Length =  7.320628 cm.

Why 7.320628 cm? You'll see why a little further down. I could just say, why not? The ugly number doesn't matter - after the revolution in units it will be, simply, "1". Anyhow, it's a handy little length that's convenient for a lot of common measurements, although, as I hinted above, before we're done you won't care how big or small a number any given measurement works out to. Americans can think of this New Unit of Length as just shy of 3 inches.

Being a "nice" length it instantly yields "nice" units for area and volume. The New Unit of Area is simply the New Unit of Length squared:

          New Unit of Area =  7.320628 cm square.

This works out to about 54 sq cm, but, be honest, isn't it easier just to think of a square about 7 cm on a side? Americans can think of a square about 3 inches on a side.

The New Unit of Volume is simply the New Unit of Length cubed:

          New Unit of Volume =  .3923241 liters.

For this I used 1 liter = 1000 cc, exactly. (I had grown up thinking that 1 ml, defined as the volume occupied by a gram of water, was slightly different from 1 cc, where 1 cm is defined by a number of light wavelengths. Now my references say they are identical.) Americans can think of this as a little less than half a quart, or a little less than a pint - or a cube about 3 inches on a side.

Speed is simply distance divided by time, so:

          New Unit of Speed =  3.050262 km/hr.

See how well our choices for units for time and length are working together so far? (It gets better!) This is a nice and relaxed walking pace, a bit pokey, even. Americans can think of just under 2 miles/hour.

Now watch this. Acceleration is the change in velocity per unit time; or, distance per time per time. Using our new time and length units:

          New Unit of Acceleration =  9.80665 m/s/s.

Look familiar? Yes, it is one standard acceleration due to gravity! What a coincidence! Not really; I loaded the dice with the selection of the New Unit of Length. Why I did so now becomes apparent.

With a unit of acceleration equal to the acceleration due to gravity, the weight of an object will be numerically equal to its mass. Or, a calculated force can be immediately envisioned as the weight of an object having that value as its mass. This is the way we all think, layman and scientist alike, anyhow. Who really thinks in terms of the forces newtons, dynes, and poundals? And who of us who think in pound-forces know its associated mass, the slug? Why do enlightened metric users give their own weights in kilograms and not newtons? Guitar string tensions in kilograms? Stylus weight in grams? Oh mama, what a mess it's been.

The English Engineering System delivers what people want - an object of 1 pound-mass weighs 1 pound-force - but scientists laugh at it because it needs a fudge factor in good old F=ma to make things work out. Or, if not laughing, struggling with it. The loss of the the Mars Climate Orbiter in 1999 resulted from botched calculations in that system. I can sympathize; it had me pulling my hair out trying to reconcile these pound-forces and pound-masses, which go together so well intuitively, with my undergraduate physics while working up this web page. Now we can have what we always wanted - without any fudge factor.

I myself have always been a firm believer in more forceful (haha) and effective education on the difference between mass and the force known as "weight". I claim that teachers who bring the subject up by referring to the "difference between mass and force" have forfeited the battle right there. It's like the situation when somebody starts harping on the difference between "strategy" and "tactics", or the difference between "accuracy" and "precision", or the difference between "rotation" and "revolution", and you know it's safe to take a little mental siesta. If it takes popping blood vessels to explain the difference, how important can it be?

In this case, though, a force, which is a push or pull, has nothing to do with mass, the amount of matter making up something. They are different beasts, entirely.

But, if we insist on carrying this confusion to the end of time, now it won't matter. Or, at least, not so much - as long as we stay here on earth. When we start moving to other worlds, then the pioneers will have to get comfortable with different numbers for the weight of an object and its mass. (Or they could choose new units of time and length to make things work on their world. O will this madness never cease...)

By the way, if you want a good way to think of gravity's 9.8 m/s/s, think of an increase in speed of about 35 km/hr every second. For Americans, it's about 22 mi/hr every second. Pretty impressive, eh?

Continuing our romp through our new units, it's mostly downhill from here, although there is one more nice touch in store. We still have to choose our New Unit of Mass. Any suggestions, class?

          New Unit of Mass = .3923144 kg.

Right! How'd you guess?! (Looks suspiciously close to the New Unit of Volume, hmmmm...) Americans can think "a little less than a pound," or about 14 ounces.

Next up is density, which is mass per volume. Doing the division yields:

          New Unit of Density =  .999975 g/cc

Ding! Ding! Ding! That's the density of water! And that's why you chose the New Unit of Mass above, right? Excellent idea, since earthlings everywhere know water. Sorry about the cranky number - that's what my references say. A gram per cubic centimeter is more precision than most of us will ever need.

Newton's F=ma tells us that force is mass times acceleration, so:

          New Unit of Force =  3.847291 newtons.

Wait a minute, didn't I say that the mass and the weight will be the same? How come the new force isn't .3923144 newtons? Calm down. In the new system, they will be the same. What you see here are mass and force in the metric system, and notice that they are related by a factor of the acceleration of gravity, which is 9.8 (not 1). If we were to convert the 3.847291 newtons into a "kilogram-weight", it would work out to the .3923144 value of our New Unit of Mass. In other words, a 3.8 newton force feels like the weight of a .39 kg block in your hands. Americans, who I said can think of a 14 ounce-mass for the New Unit of Mass, can think of a 14 ounce-weight for the New Unit of Force.

Energy is proportional to mass x speed squared, so:

          New Unit of Energy =  .2816459 joules.

I don't know how many people in the world that has meaning for. Americans can think of it as roughly the work done pushing an object with about 3 ounces of force a distance of one foot. (I've always viewed work and energy, having the same units, as two different names for the same thing. But you should consult your personal physicist first.)

Power is energy per unit time, so:

          New Unit of Power =  3.25979 Watt (Joule/s)

Think of a 3-watt light bulb. If that's not too helpful, think again about the classical mechanics experiment above. One New Unit of Power is used in performing that action about 10 times per second. Equivalently, think of pushing an object with 2 pounds of force a distance of one foot every second. For what it's worth, the New Unit of Power is about 1/200 horsepower.

Momentum is that property of a moving body that stays the same if no force is acting on it. Momentum is mass x speed.

          New Unit of Momentum =  .332406 kg x m/s.

Americans may think of it as the momentum of an object of 1 pound-mass moving at 1.6 miles per hour.

And we'll wrap up with something we have everyday experience with - pressure. Pressure is force per unit area, so:

          New Unit of Pressure =  717.8907 newton/sq m.

As if anybody gives a fig for newtons. This New Unit of Pressure works out to .00708503 atmospheres, but, even though we're subjected to an atmosphere of pressure all the time, I'm guessing most of us still don't have a good feel for it. Call it about 15 pound-force per square inch, so our New Unit of Pressure is about 1.7 ounce-force per square inch. That's about the pressure exerted by a stack of six U.S. quarter dollars.


In summary, our new system of units was derived from dividing the day into a round number of time units, specifically, 1,000,000. Then we chose a unit of length so that the unit of acceleration equals 1 standard acceleration due to gravity. This has the wonderful effect of making the mass of an object and its weight numerically equal (on the surface of the earth.) Then we chose a unit of mass that yields a unit of density equal to the density of water.

A nice side benefit of these choices for time, length, and mass is that all the secondary units derived from them (area, volume, speed, power, etc.) work out to sizes that humans can relate to easily. This isn't actually necessary since, with the adoption of my simplified scientific notation, people will be as perfectly at ease with huge and tiny numbers as they are now with "a handful". But it is a very nice touch nonetheless.

We'll also need to choose new fundamental units for electric current, temperature, and luminous intensity, that work well with these, but I'll leave that to others when this idea gets rolling.

Even before the proposal is universally embraced, I can see a nice, practical use for it. Instead of printing seemingly endless tables of conversion between all of the units we've thought up over the millennia we could give instead just the conversions to and from the New Units of measure. Supposing we have N different units of lengths to deal with, we've just gone from N(N-1) tables or conversion factors to 2N. And this would serve to get us used to the New Units; why not just stop after the first conversion and proceed with the given problem in the new unitless way rather than doing a second conversion and continue to wrestle with the old-fashioned units?

I can also easily imagine streamlined science textbooks being written without a unit shown anywhere in the text. There may be a page of conversion factors to and from the New Units of measure in an appendix, but for the purposes of learning science there would be no reason to refer to it. These students would get a much bigger dose of science, and when they go out into science or engineering world it's hard to imagine they wouldn't prefer the simplicity of the unitless way of doing things.

Notice what a godsend this idea is for the creators of a universal second language. (Read my soapbox thoughts on a universal second language.) As opposed to our national languages, which are not only stinking sick with words for units of measure, but swimming with words for foreign(!!!) units of measure as well, the universal second language will have no words at all for units!

At this point there may be a reader or two still dubious about making all measurements of a given property in the same unit. We can only carry this big and small number thing so far, you think. Please hear me out, and bop on over to my page which tames the scientific notation monster. Thanks!


P.S. I make contributions to society like this all the time with no thought of personal gain or glory, but if you want to call the New Unit of Time the "ald" (short for "donald"), I accept.



APPENDIX 1 - Further discussion of time

Some of these comments presume a familiarity with my proposal for an improved scientific notation.

As admitted at the beginning, time is destined to remain something of a problem child in any system of units because of various natural cycles. The question is: can we achieve the goal of expressing all time periods in terms of the New Unit of Time, and, does it matter if we don't?

Answering the second part first, no, it's not necessary. If we want to view the day as divided into ten "hours" of 100 "minutes" each, and the "minute" divided into 1000 "seconds", that's ok. Likewise, we could view the 1000-day year as divided into ten "months" of ten "weeks", each "week" having 10 days. We could recycle the same terms, or find new ones.

But as to the first part, my answer is, yes, it would be easy as pie. Using nothing but the New Unit of Time, we would express:

     The time within the day = ttt 3 (spoken: the 3-digit number "bip3".)

     The day within the 1000-day "year" = ddd 6 (spoken: the 3-digit number "bip6".)

     A span of 1000-day "years" = yy 9 (spoken: the number "bip9".)  

"Bip" is the syllable chosen to mean "times 10 to the plus". (Did I remember to invite you to my page on a streamlined scientific notation?) About that decimal point in the date/time in the letter to the Sun, its placement depends on the power of ten chosen. We could give the date and time together as ddd.ttt bip6, or as dddttt bip3.

There's no getting rid of the day, of course. That's a "thing" in addition to a unit of time. Moreover, it's the thing on which the New Unit of Time is based. It would take a real meanie to abolish the day. While I propose that science and engineering use only the New Unit of Time in all their work, a regular person may say, "I'm leaving in 4 days", as opposed to "4bip6", without fear of a handslap.

But there's nothing real about minutes, hours, weeks and months. They always bring up the moon when the subject of months come up, but if a month is supposed to be pegged to a lunar cycle, they could hardly have done a shoddier job of it. And except for the werewolves among us, who really pays attention to lunar cycles?

A year is a real thing, too, but it's caused us enough trouble already by being a stupid number of days, and not even an integer, for crying out loud. We'll give it its due by clearly marking all the equinoxes and solstices on our 1000-day calendars. They could all be celebration days, even.

If we want to calculate how many revolutions the earth has made during some period of time, just divide by 365.2422bip6.

If the talk above about 10-day "weeks" sounds like an impossible gear shift from the time-honored 7-day week, don't worry. We can keep our familiar cycle of 7 days going throughout the 1000 day period. I didn't plan to get into any social engineering here, but I'm just noticing that our 7-day week with 5 work days is very similar to a 10-day week with 7 work days. In fact, the latter provides just a touch more free time, adding up to an extra day off every 70 days. Obviously, businesses could adjust to that - if only by being imperceptibly less generous with raises for a few years. This longer "work week" may provide a flexibility not feasible with the shorter one. Some workers might prefer having a 3-day weekend in every 10 days; others might go for a 2-day weekend, and use the 3rd day off to split the remaining 8 days into shorter, more comfortable (and thus more productive?) 3- and 4-day work weeks.

In the letter to the Baltimore Sun, I made the fantastic claim that "keeping the calendar in sync with the solar year is . . . completely unnecessary." What do I mean by that? Speaking mainly to temperate zone dwellers, think about how weakly our daily weather corresponds to the seasonal average. If you were somehow kept ignorant of the date, on a record-breaking hot or cold day you might be able to guess the date to plus or minus 2 months. On a day that's merely hot or cold, about the best you could do is rule out the three months associated with the other extreme. On a day that is not hot or cold, all bets are off (assuming you're blindfolded, too.) Our four neatly labeled seasons are vastly overrated. For non-temperate zone dwellers, the insignificance is even greater. But, if there's a particular holiday that we feel we have to keep in a certain season, we can do that. The 1000-day calendar may show two or three Easters, for example. No problem.

Quick quiz: how many days to Christmas is it? See, you don't know, do you? (Unless this happens to be December.) What's that say about the importance of having a holiday on the exact same date every year? Even when it is, you still don't know how far off it is. With the new decimal calendar, you will always know how far off any given day is. Just the subtract the two day numbers.

For the record, I cast my non-expert vote for the abolishment of the leap second. I say choose a unit of time, and let it rip. Not only would the miniscule divergence of an atomic clock with astronomical time never cause any problems within a person's lifetime, some might find the tiny drift very interesting. Weatherman: "And if trends over the last 6 months continue, we may make up half of the 1.7 seconds lost last year due to California slipping into the ocean..."

He wouldn't say "seconds", of course.



APPENDIX 2 - Summary of the New Units

     New Unit      Equivalent to obsolete...   How to think of the New Unit...
     --------      -------------------------   -------------------------------


     Time          .0864 second (exact)        about a tenth of a second
     Length        7.320628 centimeter         about 3 inches
     Mass          .3923144 kg                 about a 14-ounce mass

   (Derived from above)

     Area          53.5916 sq cm               about a 3-inch square
     Volume        .3923241 liter              about a pint, or a 3-inch cube
     Speed         3.050262 km/hr              slow amble
     Acceleration  980.6653 cm/s/s             the acceleration of gravity, precisely
     Force         3.847291 newtons            about a 14-ounce weight (cf. mass above)
     Energy        .2816459 joules             a 3-ounce force working over 1 foot
     Power         3.25979 Watt (Joule/s)      10x above energy expended per second, 
                                           or, 2-lb force working over 1 ft per sec
     Density       .999975 gram/cc             density of water, precisely
     Pressure      717.8907 newton/sq m        stack of 6 quarters
     Momentum      .332406 kg x m/s            1 pound-mass moving at a crawl

If I haven't goofed up too bad, these numbers should be good to 6 places. I worked in single precision arithmetic in QBasic. I haven't done a careful analysis of where the precision stops; I think the 7th place, where given, may not be precise but good enough for rounding purposes.

If anyone is inclined to check my work, and please do because I wouldn't want the United Nations to mandate it with a blown decimal point, here are the conversion factors I used in working up the numbers on this page.

     1 in = 2.54 cm (exact)
     1 ft = 12 in (exact)
     1 mile = 5280 ft (exact)
     1 pound-mass = 453.59237 gram 
     1 liter = 1000 cu cm (exact)
     1 quart = .946326 liter
     1 mile = 1.609344 km
     1 standard acceleration due to gravity = 9.80665 m/s/s
     1 newton = 100000 dyne
     1 pound-force (lbf) = 4.448218 newton
     1 foot-lbf = 1.35582 joule
     1 horsepower = .7457 kilowatt
     1 atmosphere = 101325 newton/m/m
     density of water = .999975 gram/cc


Contact Donald Sauter: send an email; view guestbook; sign guestbook.
Back to Donald Sauter's main page.
Rather shop than think? Please visit My Little Shop of Rare and Precious Commodities.
Back to the top of this page.

Parents, if you're considering tutoring or supplemental education for your child, you may be interested in my observations on Kumon.