America’s Fractional Mind

By The Metric Maven

“This is dialectics, it’s very simple dialectics, one through nine, no maybes, no supposes, no fractions. You can’t travel in space, you can’t go out into space, you know, without like you know, with fractions. What are you going to land on?—one-quarter?, three-eights? What are you gonna do when you go to from here to Venus or something?”

Dennis Hopper — Apocalypse Now!

While Dennis Hopper’s character in Apocalypse Now! might have been traveling into space without a spacecraft, those who went to the moon in the 1960s did follow his advise about fractions—thankfully! The navigation computer on Apollo 11 was programmed with decimal numbers—in metric. The data internal to the computer was entirely metric. Only when the desired navigational data had been computed was it converted to the “inch-pound” non-system. The computer had such minimal memory that the conversion was a burden. Those who recall the flights might remember that the distances were all given in nautical miles. In America, a sea going romantic metaphor triumphs over measurement clarity.

Every common US measuring tape or ruler with inches on it has fractional divisions. In the US we are all familiar with 1, 1/2, 1/4, 1/8, 1/16 and 1/32 of an inch.

The very first thing one is taught in school, is that in order to add fractions they must have a common denominator. If one measures 1 3/4″ and wants to subtract that from 2 7/32″ well we have to change 1 3/4″ to 1 24/32″ and then there is more work…..I won’t take you through it, I think I’ve made my point. In decimal inches it would be about 2.22″-1.75″ or 0.47″, but good luck finding an inch ruler with decimal divisions in the US. The other choice is to use an all metric, mm only tape measure, like the Australian construction industry does. As I’ve pointed out in a previous blog, finding millimeter only tape measures in the US is only marginally easier than locating Amelia Earhart. In this case we would have 56 mm – 45 mm = 11 mm. That was easier, now wasn’t it? Ok, for those who want sub millimeter accuracy in everyday life it’s  56.39 mm – 44.45 mm  = 11.94 mm, yes it’s off by almost a millimeter or 1/25 of an inch.

Tape measures in mm only — Good luck finding one in the US. Click on image to enlarge

The proverbial video which illustrates how arithmetic with fractions can cause confusion features the crew of American Choppers in an epic numerical struggle. American rulers have several sets of fractional divisions. Most of the time, two measurements on an American ruler do not have the same denominator. One can almost never directly add two fractions, but with millimeters and decimals, you always can. If you are willing to live with 1/25 of an inch or so of tolerance, then just use integers.

My friend Thern has asserted many times that even Americans who have worked in construction for many years seldom learn how to read fractional tape measures. Not because they are intellectually challenged, but it’s the user unfriendly nature of fractional divisions. This is why Australians have saved about 10-15% on construction costs compared with the United States since the 1970s. Less material gets wasted and time is more efficiently used. Apparently confusing the fractional divisions on a scale is so easy that some manufacturers have four different fractional scales separated out on the same scale:

A scale with separate scales on each of its edges. Click on image to enlarge

It has been asserted that not having the metric system costs each of us $16.00 per day. When one adds up all the small efficiencies encountered every day in the US, it’s not hard to believe. My friend Lapin showed me a situation that illustrates this point. Lapin is an amateur radio enthusiast, and like most of his cohort has an interest in antenna design. He directed me to a website which has a computer program which will compute the design values for basic types of antennas. The RFID frequency in the US is 915 MHz. I put that number into the program as an example; the output is below:

Click on image to enlarge

Notice all the 1/32 fractions it spits out. My mind recoiled at the output, but I quickly realized that in the US, tape measures are all fractional, so the programmer did what he needed to, to output fractions so one could use an American ruler. The computer program itself is available to view, and uses decimal arithmetic as you might expect, to compute the lengths of A, B, C, D and E. Similar to the computer navigation program on the Apollo 11 mission, it has to compute the fractional values for output at the end of the program, so they will correspond with our 17th century rulers. Don’t be afraid of what is shown next. I only put it in for illustration, you don’t need to understand it. The computer code below is used by the antenna design program to sort out what fractions correspond to what decimal values for display:

if (temp3 <= .03125) fract = “1/32”
else if ((temp3 > .03125) && (temp3 <=.0625)) fract= “1/16”
else if ((temp3 > .0625) && (temp3 <=.09375)) fract = “3/32”
else if ((temp3 > .09375) && (temp3 <=.125)) fract= “1/8”
else if ((temp3 > .125) && (temp3 <=.15625)) fract= “5/32”
else if ((temp3 > .15625) && (temp3 <=.1875)) fract= “3/16”
else if ((temp3 > .1875) && (temp3 <=.21875)) fract= “7/32″
else if ((temp3 > .21875) && (temp3 <=.25)) fract=”1/4″
else if ((temp3 > .25) && (temp3 <=.28125)) fract=”9/32″
else if ((temp3 > .28125) && (temp3 <=.3125)) fract=”5/16″
else if ((temp3 > .3125) && (temp3 <=.34375)) fract=”11/32″
else if ((temp3 > .34375) && (temp3 <=.375)) fract=”3/8″
else if ((temp3 > .375) && (temp3 <=.40625)) fract=”13/32″
else if ((temp3 > .40625) && (temp3 <=.4375)) fract=”7/16″
else if ((temp3 > .4375) && (temp3 <=.46875)) fract=”15/32″
else if ((temp3 > .46875) && (temp3 <=.5)) fract=”1/2″
else if ((temp3 > .5) && (temp3 <=.53125)) fract=”17/32″
else if ((temp3 > .53125) && (temp3 <=.5625)) fract=”9/16″
else if ((temp3 > .5625) && (temp3 <=.59375)) fract=”19/32″
else if ((temp3 > .59375) && (temp3 <=.625)) fract=”5/8″
else if ((temp3 > .625) && (temp3 <=.65625)) fract=”21/32″
else if ((temp3 > .65625) && (temp3 <=.6875)) fract=”11/16″
else if ((temp3 > .6875) && (temp3 <=.71875)) fract=”23/32″
else if ((temp3 > .71875) && (temp3 <=.75)) fract=”3/4″
else if ((temp3 > .75) && (temp3 <=.78125)) fract=”25/32″
else if ((temp3 > .78125) && (temp3 <=.8125)) fract=”13/16″
else if ((temp3 > .8125) && (temp3 <=.84375)) fract=”27/32″
else if ((temp3 > .84375) && (temp3 <=.875)) fract=”7/8″
else if ((temp3 > .875) && (temp3 <=.90625)) fract=”29/32″
else if ((temp3 > .90625) && (temp3 <=.9375)) fract=”15/16″
else if (temp3 > .9375) fract=”31/32”;

If we had the metric system in this country, writing code like this would be eliminated. Clearly creating and writing this sorting section of computer code took a long time to accomplish.  Had we become metric in the 1970s, like Australia did,  this computer code would never have been written, there would be no need.

The word fraction comes from the latin word fractio which means to break. I cannot think of a more appropriate rubric. It’s long past time to break with the past, and embrace the decimal metric system, so we can live in the future, rather than living in the past.

Related essays:

Stickin’ it to Yardsticks

The American “Metric Ruler”

The Design of Everyday Rulers

Postscript:

My friend Pierre was asked to fill out a form which provides information for a medical report. The form asked for his height, and defaulted to centimeters (tsk…tsk millimeters please!) Below is the first image:

We won’t go into the fact that it is in centimeters, but requests a person’s height to tenths of millimeters. Straight millimeters would have been just fine, which is a single unit, and may be expressed without a decimal point. Pierre is a proud user of Ye Olde English units and attempted to set the software to feet and inches. You can see in the screenshot below his what his choices are:

Indeed the software designer did not use USC, Ye Olde English, Imperial, WOMBAT, “royal measurements” or whatever, to describe the choice, but something perhaps even more appropriate: archaic. The final screen after the selection is shown below:

Now the programmer needs to also embrace the international date format. Thanks to Pierre for letting me share this.

18 thoughts on “America’s Fractional Mind

  1. Regarding that last form where it’s asking for cm, it looks like it’s really asking for metres, so if you enter a value of “1.85cm”, that would really mean “1.85m” or 185cm. It’s completely irrational to ask for height to a precision of 0.1mm, and it looks like the developer has not learned how to use metric properly. You can test this by entering a value, and then seeing what coversion is applied when you switch to archaic.

    Ordinarily, I agree with you about the use of mm over cm, but I disagree about using mm for height. It’s too precise for a persons height which fluctuates by more than that throughout the day. Using intregral centimeter values is adequate. There’s still no need for any decimal point.

    • True. Metric is so versatile that one can use millimetres where they work best and centimetres elsewhere. To go between them is very simple. Some countries express height in centimetres and others in metres, but this is not a problem. The numbers are the same and someone use to hearing centimetres but is told in metres can mentally convert in milliseconds. One can not do the same if they are expecting a height in feet and inches and are given inches only.

      It should always be remembered that the centimetre, millimetre, etc are not units. The unit is the metre. The prefixes are just a form of scaling the numbers to eliminate a lot of zeros on either side of the decimal point.

      The problem and confusion result when the form of the metric system is treated like USC.

  2. This actually brings up a point that I would call the one good thing about these legacy units. As we are forced to work with fractions to do much measuring, it could be that Americans do end up better at fractions than our counter parts around the world. (This hardly justifies forcing us to wear these wooden shoes). But for some reason, US students are NOT better at math. (I was surprised to learn that students from common schools in the Philippines were about 2 years ahead of US students in math.)

    US medicine not only uses a weird mix of legacy units, the lab results lack error banding. The lack of error bands matters – if some result (say TSH is +/- .7) and the cut off for treatment is 2.0 what happens if someone comes in with 2.2? Next time you get a lab back ask your doctor how accurate the results are (he won’t know).

    (An aside – the tests run for Lp(a) should be run in nmol/ but they don’t – they run mg/l – one company converts this to nmol/l based on an average – totally misrepresenting the results of the test. ( people’s molar levels vary greatly via particle size due to genetic differences – in this case, it greatly matters).)

    A true story about legacy rulers fro the 1980’s: There was a professor at KU that taught occupational therapy. One of the requirements was his students had to ‘pass’ his ruler test. They had to answer questions such as, “what is larger 3/4″ or 3/8”? If they couldn’t, they would fail his course (75% was passing back then). The University got complaints and changed his grades, passing students that couldn’t read a ruler. (So after that he told everyone they would get an A as the grades and credits were meaningless. )

    Metric would have saved these students, but I have a hard time respecting any degree universities hand out that don’t require at least one calculus course. Not that calculus is required for most jobs – it is the discipline of thought that it teaches that matters. Do mathematically illiterate students belong in universities?

    • I’ll never understand the very bad practice of incorrectly writing a decimal number less than one without the leading zero. That has to be the epitome of innumeracy.

  3. … and nowadays young people are amazed if you can add whole numbers with 3 or 4 digits in your head.

    Good God, I don’t understand why decimal-inch tape measures are all but impossible to find.

    • http://www.grainger.com/Grainger/WESTWARD-Measuring-Tape-4YP86

      Sorry couldn’t find it, could only find this dual 1/8″/1/10″ but I recall, think it was aircraft supply company that made a standard 25′ tape made to 10/1000″ increments.

      We agree that tenths of an inch are useful in certain instances. If I had a tape marked in hundredths, though, I have enough neuron power that I could at least resolve to 16ths and the first and last four or five 32nds. . .

      There’s also a tape measure I saw once (dangerously present at a stage concert involving hanging a huge metal truss) that is designed in decimal feet primarily for surveyors.

  4. “Do mathematically illiterate students belong in universities?”

    In a word: no. Unless it’s either in a remedial math course, or in a literature or arts type course in which math skills are pretty much irrelevant.
    Even to follow a recipe for a cake you still need to be comfortable with measurements.

    • In the US anyone qualifies to be in a “university” if they have the money to pay for it. It’s all about the money.

      The term university is misapplied. In a real university, a person graduates as a PhD. What most Americans go to a “university” for would in other countries be a technical college or a business college or an engineering high school, but not a university.

      In countries where university education is free, you have to be smart enough to qualify to go there.

      Even what Americans call High School, isn’t really High School but Secondary School. High School is where you go to learn your trade.

  5. “The navigation computer on Apollo 11 was programmed with decimal numbers—in metric. The data internal to the computer was entirely metric. Only when the desired navigational data had been computed was it converted to the “inch-pound” non-system.”

    Do you have proof of this? I know the Werner von Braun loathe USC and did all of his calculations in metric, but I was under the impression he had to convert and all of the design, manufacturing and operations were entirely USC. What proof is there that their computer worked in metric? Possibly decimal inches or decimal feet, but metric units?

    NASA along with the long existing aerospace companies in the US are strong metric opponents. NASA couldn’t even design the Constellation in metric claiming millions in cost. When it was pressured to go metric, its partner company refused and the result was the crash of a probe on mars.

    It would be nice if they did use metric internally, but where is the proof?

    • This is well known. Here is the quote from Wikipedia:

      Although data were stored internally in metric units, they were displayed as United States customary units. This calculator-style interface[nb 1].

      You could have found this with a minimal amount of research.

      Centimeters are a vestigial hold-out for a pseudo-inch. They are an echo of Ye Olde English which waste time and energy. The desire to use them appears based on tradition rather than consideration in my view. They just get in the way and add no utility.

      • Not as well known as you assume. Metric opponents insist the US went to the moon using USC only and metric supporters just assume they are right. There are few proofs to indicate otherwise.

        It is understandable that it internally calculated in metric as that is the system MIT engineers are taught in. So, it makes sense they would design it to work in metric and only convert at the end.

        There is nothing wrong with using the centi prefix when dealing with height and clothing sizes. It makes no sense to tell someone your height is 1800 mm, when 180 cm works just fine and uses a properly ranged value with no decimal numbers.

        In engineering applications and construction/manufacturing only the millimetre should be used. It is not a one size fits all scenario.

      • >Centimeters are a vestigial hold-out for a pseudo-inch. They are an echo of Ye Olde English which waste time and energy. The desire to use them appears based on tradition rather than consideration in my view. They just get in the way and add no utility.

        This war on the centimetre from some metric supporters got me to thinking about the resistance to metric in some product marketing. It made me wonder why TV screens and computer monitors are universally marketed in inches.

        Is it possible that marketers have been trapped into the false belief that the centimetre is a bad unit to use and the only real choice is either metres or millimetres? It seems that not only TVs but clothing where the centimetre would fit in perfectly are also still sold in inches.

        When marketing or selling products to consumers large 3-digit sizes might be a turn off and seen to have a negative sales effect and two digit numbers are preferred. Which seems simple to a consumer to deal with a 3 digit number like 800 mm or a two digit number like 80 cm or 32 inches? But if centimetres are to be avoided, then the only choice is to use inches.

        The problem worsens when screens exceed 1m. Now you have a 4 digit number if expressed in millimetres, where as the inch descriptions are still two digits.

        I’ve seen a number of consumer products that use rounded inches in their descriptions and the conversion is to single decimal centimetres. The number looks clumsy but is preferred by marketers to a 3 digit millimetre even if it would be a whole number.

        Marketers most likely loathe decimal parts and never express inches in parts unless it is a simple fraction.

        If we stopped putting down the centimetre for descriptions of consumer products, we may see preference for the centimetre over the inch as something in centimetres sounds bigger, provided the centimetres are whole numbers.

  6. That code fragment is simply horrid. Rewritten (in Java):

    long denom = 32;
    long numer = Math.round(temp3 * 32.0);
    while(numer % 2 == 0 && denom % 2 == 0) {
    numer /= 2;
    denom /= 2;
    }
    frac = numer + “/” + denom;

    That’s much more elegant. ^_^ I assume temp3 is a double-precision floating-point number and frac is a String (in Java). Some adjustments would need to be made for other languages, but the basic logic works.

  7. I have to take some issue with the claim that “Every common US measuring tape or ruler with inches on it has fractional divisions. In the US we are all familiar with 1, 1/2, 1/4, 1/8, 1/16 and 1/32 of an inch.” I would agree if you said “Most” instead of “Every.”

    Measuring tapes with decimal inches are offered by Stanley, Lufkin, Starrett, and probably others. An example is Stanley 33-272, which is fractional inches on one side, decimal inches on the other. Resolutions are 1/16″ and 0.1″ for most of the tape, 1/32 and 0.02″ (1/50) for the first 6″.

    Also, in long tapes (open reel, usually >50 ft), decimal feet are very common because surveyors use decimal feet. Usually numbered graduations to 0.1 ft, minor graduations to 0.01 ft. Most leveling rods are also graduated in hundredths of a foot. Note that 0.01 ft = 0.12 in, nearly the same resolution as 1/8″. Many drafting scales, calipers, micrometers, etc use decimal inches. Most machinists think fractions are for carpenters.

    You may need to order online or go to a store that caters to these trades, not the big-box hardware stores. I agree with your major premise that fractions are a major PITA. I also agree everybody should go metric, but there are other solutions to fractions and not everyone who uses Customary uses fractions.

    • I believe the word common mitigates the word every. I have written about chains in a previous essay. I have purchased mm only metric rulers from Australia, yes they exist, but one can hardly call them common. I have never encountered a decimal tape measure in all my years as an Engineer or elsewhere. They are hardly common. I will check into a Stanley 33-272, but I’m not sure I’m excited about owning a Ye Olde English tape measure of any sort these days. The units would still be too large to produce non-decimal integers. Perhaps a barelycorn tape measure would be better? What also struck me was that these decimal inch tapes do exist, but the notion of market darwinism has not produced any pressure for the even the small change to decimals in the US.

      MM

    • Sorry John, missed your comment replying above.

      Couldn’t find the one I was looking for, thought it was a 25′ model, but I did find this 12′ tape in both binary and 10/1000″ increments. Pretty sure the longer one I saw was also advertised for or made available through a store catering to aviation and/or aircraft maintenance:

      http://www.aircraftspruce.com/catalog/topages/tapemeas1.php

      Decimal feet seem too rough, unless they are carried to hundreds, and even then aren’t compatible with more common inch measure, so I prefer decimal inches because they can be converted easily into feet, links, even chains with a tape. So they are suited for all types of engineering, surveying, and are backwards compatible with binary fractions that are so often encountered.

      They are versatile, of course, not as much if only the first few inches are graduated in 10/1000″ increments. Would like to get one that has the fine graduations all the way up, and this is the only reason I’ve held off getting one. . .

  8. Take at look at the Wikipedia article on leveling rods:
    http://en.wikipedia.org/wiki/Level_staff
    I can’t say as I care for the English/metric rod in the illustration vs one-sided, but note the scale description. The English side is readable to 0.01 ft. I don’t much like the invisible decimal on the metric side. Easier to just say numbers are in decimeters.

    I am an electrical engineer, not civil, but it certainly seems that all crews doing grading, drainage, pipe slopes etc, for building sites continue use of surveyor’s units (decimal feet) in their work. If you look at the ads in Google, all the English leveling rods are decimal feet.

    These are only common in certain trades and you have to shop at a store that caters to the trade, but they are not as hard to find as a metric only tape measure in the US. Big box stores will probably only have fractional inch tapes. There are stealth decimal users just as there are stealth metric users, and suppliers who cater to them. The “average American” may have never seen (or know how to use) decimal foot or decimal inch measures.

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