AN INTRODUCTION TO THE METRIC SYSTEM

Prologue

Today, with the metrication of the United States quite imminent, many commercial concerns are alleging to offer "easy" and "simple" methods and materials to acquaint the masses with the metric system.  This propaganda campaign is aimed directly at the educational community.  It has become quite clear that all of us should become acquainted with a minimum of certain fundamentals in order to intelligently evaluate the materials being thrust upon us.  It is to this end that I have prepared this report.

November 1974
Axxxxx J. Dubrexxx
Mathematics Dept. Chairman
xxxxx (CT) High School
06xxx

Prologue II

President Ford signed the metrication law on 23 December 1975.  This provides ten years for full conversion to the metric system in the United States.  As educators responsible for Implementation of this law in the school systems, you should find this report quite useful.  A final reminder, this paper is intended to be a cursory outline, not an extensive study.

October 1976
A. J. D.
AJD/dlh

7/’98  for Historical Perspective   (Section II – designed for overhead projector.)
          Much effort spent to preserve ‘original’ type-written pages.

**i**
 
 

 OBJECTIVES Part I

To provide an appreciation of:

1. The differentiation of measurable physical quantities.

2. The meaning and nature of Derived Quantities.

3. The preference of Mass over Force as a fundamental quantity.

4. The Uniformity of the Metric Prefixes.

5. The exactness of definition for the fundamental MKS units.

6. The relation of basic MKS units to their corresponding English units.

7. The overall simplicity of the MKS system.
 

OBJECTIVES Part II

Further discussion (another report) would include:

1. The definitions for the remaining fundamental quantities.

2. Definitions for derived quantities such as Force, Work, Power, and Energy.

3. Common translations and everyday uses of the above

**ii**
 
 

 TABLE OF CONTENTS

Prologue                                          p.  i

Prologue II                                      p.  i

Objectives I                                    p. ii

Objectives II                                   p. ii

Divisions of Physical Quantities      p. 1

Systems of Measurement FPS         p. 1

Systems of Measurement MKS       p. 2

Metric Prefixes                               p. 3

Standard Fundamental Units Plus    p. 3

Metric-English Conversions
 Length                                p. 4
Mass                                   p. 5
Time                                   p. 5
Liquid                                 p. 5
Area                                    p. 6
Temperature                        p. 6

Conversion Summary I                     p. 7

Bibliography and Footnotes              p. 8

Appendix
 Supplementary Lecture Notes         p. 9

Pages are numbed in tradition format: ( ** # ** ).

**iii**
 
 

 Divisions of Physical Quantities

     There are two distinct divisions of physical quantities in our universe. The first division we refer to as the fundamental quantities upon which all others can be defined. The six fundamental quantities are:
           length,   mass (amount of matter),   time,    temperature,
           luminous intensity    and    electric current.

     The second division we refer to as the derived quantities. These physical quantities provide rules through their definitions for determining their values in terms of the measurable fundamental quantities.  Examples of derived quantities are:
          velocity  :         length/time (kilometers/ hour)
          acceleration  :  length/time/time (meter/sec/sec)
          force :             (via Newton's 2nd Law: F= m * a) - mass * length/time/time (1 Newton = 1-   ……….                     kilogram* meter/sec/sec)
           work  :           force * length (1 Joule = Newton meter)
           power  :          work/time (1 Watt = Joule/sec)

     Once having recognized the six fundamental quantities, it remains to choose a standard for each with which unknowns may be quantitatively compared (measured). In choosing a standard we must consider its reproducibility and invariance, and in selecting a measuring device we must consider its accuracy and dependability.
 

Systems of Measurement: FPS

     Presently, we are confronted by two systems, which provide standards to quantitatively compare the fundamental quantities. The more familiar is the English Gravitational System (FPS). This system's mechanics is based on the foot (length), pound (force) and second (time). These units have evolved from common usage rather than rational thought.
                                                                                                              --1--, --2--
---------------------------------------------------------------

NOTE:   ‘ * ‘   means "times", operation of multiplication

**1**

   = = = = =

 **2**

     The FPS system is quite cumbersome for converting from sub-units to units, to multiple-units since the multiples are not consistent with-in the same unit and are very different for each fundamental unit:
          Length: 12 in = ft,           3 ft = 1 yd,             1760 yd = 1 mi.
          Liquid:   8 oz = 1 cup,     2 cups = 1 pt,         2 pts = 1 qt,          4 qts = 1 gal.

     The FPS system employs the pound (force) as a fundamental unit. However, force is not invariant. The weight (gravitational attractive force) of an object is quite dependent on its location. An object has significantly different weight on the Moon than it does on the Earth or in an orbiting space laboratory. It is the mass of an object, the amount of matter it contains, which is invariant.
 

Systems of measurement:  MKS

          The second "new" and unfamiliar system confronting us is commonly referred to as the Metric System. This system was developed through the collective rational thought of enlightened men during the latter part of the 18th century to describe the fundamental quantities of the universe in a systematic and consistent manner.

          There are two basic subdivisions of the Metric System. The CGS system employs the centimeter (length), gram (mass, amount of matter), and second (time) to describe the phenomena of mechanics.  It is essentially a laboratory-oriented system, particularly in used thermodynamics.

           The MKS system employs the meter (length), kilogram (mass, amount of matter), and second (time) to describe the fundamental units of mechanics. The MKS system is best applicable to the macroscopic "real" world. This system's mechanics will concern us most.

--------------------------------------------

NOTE:    FPS short for North American United States Engineering System
 
 

 **3**

Metric Prefixes


     The subdivisions and multiples of the MKS units necessary for everyday use are relatively easy to learn.  We further note that all are powers of ten.
     The most often used subdivisions are identified by the Latin prefixes: milli (1/1,000), and centi (1/100).
     The Greek prefixes required for multiple-units are: kilo (1,000), mega (1,000,000) and giga
(1,000,000,000).
     Deci  (1/10), deka (10) and hecto (100) are not commonly employed.

     The milli-unit is familiar to us in the form of the mill in our property tax rate. In financial circles, the centi-unit is commonly referred to as a penny. Thus, we need only learn to replace thousand by kilo, million by mega and billion by giga to complete our working knowledge of the most useful metric sub-unit and multiple-unit prefixes. These same prefixes apply to all the metric units.
 

Standard Fundamental Units plus

                                               --3--,--4--

     The MKS unit of length Is the meter, 1 meter = 196,50,763-73 wavelengths of the orange-red light of krypton-86 as measured with a Michaelson interferometer.  This definition is invariant, reproducible and sufficiently accurate.

     The unit of mass (amount of matter things contain) in the MKS system is the kilogram (1,000 grams).  Mass measurements are made with an equal arm balance.  The standard kilogram is a platinum-iridium cylinder kept at the International Bureau of Weights and Measures at Serves, France.

     The MKS unit of time is the second.  1 second = 9,192,631, 770 oscillations of cesium-133 atom.  An atomic clock device is employed to record time.
 

 **4**

     The unit of liquid measure is a derived unit, the liter ( l ). it has a volume of exactly 1,000 cubic centimeters.
 

Metric-English Conversions

     The meter (m) = 39.37 inches, i.e., is approximately 9.36% larger than a yard. Distances, which are appropriately approximated in yards, may be roughly converted to meters directly on a one-to-one basis.  For example, the length of a football field (100 yd) Is fairly well approx-imated by 100 meters (to 4 significant figures it is 91 .44.)

     1 kilometer (km) = 1,000 meters (1,000+  yd) while 1 mile (mi)  = 1760 yards.  More to the point, and to two significant figures, 1 km  = .62 mi and 1 mi = 1.6 km.  Thus, distances that would be appropriately expressed in miles may be converted to kilometers by multiplying the number of units by 8/5ths.  Therefore, a distance of 5 miles becomes 8 kilometers (5*8/5 = 8), 25 miles becomes 40 kilometers (25 * 8/5 = 40) and 100 mi becomes 160 km (100 * 8/5 = 160 or 100 * 1.60 = 160).

     1 meter = 100 centimeters (cm) and 1 meter = 39..37 inches combine to yield the relation 1 in = 2.54 cm.  Thus distances that would be appropriately expressed in inches may be roughly converted to centimeters by doubling the number of units and adding half the number to this result.  Therefore:
6 in .ap=. 15 cm (2*6 + .5*6 = 15 or 6*2.54 = 15.24) ,
12 in  .ap=.  30 cm (2*12 + .5*12 = 30 or 12*2.54 = 30.48).
A 15-cm shoe size would be appropriate for a woman whereas 25 cm would usually be for a man.

--------------------------------------------------------

NOTE      .ap=.   means "approximately equals"
                  A football playing field plus one end zone .ap=. 100m  (100.58)
                  ‘ l ‘  is the symbol for liter
                   1 in = 2.54 cm by U. S. definition
 

 **5**

 MASS (AMOUNT OF MATTER)

     The conversion from kilograms (a mass unit) to pounds (a force unit) has caused some confusion for the unenlightened and been a source of much aggravation and controversy for the "experts". The controversy arises from attempts, by some, to directly equate kilogram (mass) and pound (force). This is a quite incompatible situation, to say the least, however, the problem may be simply resolved by learning the correct wording immediately.
     "A kilogram mass has the weight (gravitational attract-ive force) of 2.2 pounds" at the   Earth's surface.
     This wording is proper and exact, and with the exception of extreme cases such as in the deep recesses of an African mine or at the top of Mount Everest, is a fair (2 sig-figs.) approximation of the weight of a kilogram mass.

     Thus the weight of a kilogram mass is approximately 2.2 pounds and a 1 pound weight  has the mass of about .4536 kilograms.
     For a rough approximation of mass given the weight, divide the number of units in half.
For example, 1 lb .ap=.  ½ kgm,  2 lbs .ap=.  1 kgm.
     And. reversing direction 1 kgm .ap=.  2.2 lb, 1.5 kgm .ap=.  3.3 lb, 2 kgm .ap=.  4.4 lb, 2.5 kgm .ap=.  5.5 lb.

   TIME

  The unit of time, the second, is universal to the three systems we have introduced.  Thus the minute and hour retain their usual defini-tions: 60 seconds = 1 minute and 60 minutes = 1hour.
 

  LIQUID

     The liter ( l ) the MKS unit of liquid measure, equals approximately 1.057 quarts, i.e., is 5.7% larger than a quart.  Thus, as with the yard-meter conversion, a one-to-one direct conversion yields good approximation.
      Therefore, 1 qt .ap=.  1 l (.9463),   ½  gal  .ap=.   2  l (1.893), 1 gal .ap=. 4 l (3.785), and reversing direction),
            1 l  .ap=.  1 qt (1.057),  2 .ap=.  2 qt (2.114),  4 l .ap=. 4 qt (4.228) .ap=.  1 gal (1.057).

---------------------- -----------------   ---------   ---------
NOTE      “.ap=.”  still means approximately equal.
                 ‘ in^2 ‘ means ‘in * in’ or square inches
 

 **6**

 AREA

     In the English system, we commonly express area in square inches (in^2), square feet (1-ft^2 = 144 in^2), square yards (1 yd^2 = 9 ft^2 = 1296 in^2) and acres (43,560 ft^2).
     In the metric system, we most often express area in square centimeters (cm^2), square meters (1 m^2 = 10,000 cm^2) and hectares (10,000 m^2).
     From this and previous definitions we my derive the following relations: 1 in^2  = 6.452 cm^2,  1 cm^2  = .1550 in^2, 1 ft^2  = 929.0 cm^2,  1 m^2 = 1550 in^2,  1 m^2 = 1.196 yd^2, and 1 yd^2  = .8361 m^2.

     The hectare is a square hectometer (1 hm = 100 m = 328.1 ft).   Thus, 1 hectare = 107, 650 square feet. Therefore 1 hectare = 2.471 (2.5) acres and 1 acre = .4047 (2/5) hectare.
 

TEMPERATURE

     There are two temperature scales employed in the metric system. The more commonly known is the centigrade (Celsius) scale. For most practical purposes, the centigrade scale assigns the arbitrary number zero (0) to the freezing point of water and the arbitrary number one hundred (100) to the boiling point of the same. Thus, each centigrade degree (deg-C) is exactly 1/100 of the temperature difference between the boiling and freezing points of water.

     We will remember that our "usual" temperature scale, the Fahrenheit scale, assigns the arbitrary number 32 to waters freezing point and the arbitrary number 212 to the boiling point of the same. Thus, each Fahrenheit degree (deg-F) is exactly 1/180 the temperature difference between water's boiling  and freezing points.

     Conversion from one arbitrary scale to another is "simple" once we have established the  relationship between them.  The first thing to be done is establish the relationship of their "zero"  points.  The second thing to be done is establish the relationship the relative size of each unit.
 

 **7**

     As we have seen, the Fahrenheit scale starts off  with a +32 at water's freezing point (O deg-C).  Thus, when converting from Fahrenheit to centigrade, we must make the zero adjustment first by subtracting 32 from the Fahrenheit reading, then we may complete the unit conversion.  When converting  from centigrade to Fahrenheit, we must make the zero adjustment after the unit conversion by adding 32 to the result.

     Since there are fewer centigrade degrees between the two fixed points than there are Fahrenheit degrees, the centigrade degree must be larger than the Fahrenheit degree.  We note that this ratio is 180/100 or 9/5.  This means it requires only 5/9ths as many centigrade degrees as Fahrenheit degrees to express the same temperature difference. In a like manner, it requires 9/5ths times more Fahrenheit degrees as centigrade degrees to express equivalent temperature differences.

     The above discussion leads naturally to the following conversion formulas:
C = 5/9*(F - 32) and F = 9/5*C + 32.
Some solved examples:
  -10  deg-F    by   C = 5/9*(-10 - 32)  = 5/9*(-42)     = -23-33    yields      -23.3    deg-C
 +10  deg-F    by   C = 5/9*(+10 - 32) = 5/9*(-22)     = -12.22     yields      -12.2    deg-C
    30 deg-F    by   C = 5/9*( 30 - 32) = 519*(- 2)      = - 1.11      yields       - 1.1    deg-C
    50 deg-F    by   C = 5/9*( 50 - 32) = 5/9*(+18)     = +10.00    yields      +10.0    deg-C
    70 deg-F    by   C = 5/9*( 70 - 32) = 5/9*( 38)       = 21.11     yields         21.1   deg-C
    90 deg-F    by   C = 5/9*( 90 - 32) = 5/9*( 58)       = 32.22     yields         32.2    deg-C
 

Conversion Summary I

     It is the conversion between the English and metric units which is the source of confusion and exasperation for many. The sub-unit and multiple-unit prefixes are quite simple and lucid. They are the same for all the metric units, except the familiar time relations. In the words of a friend, a history major and high teacher: "It seems too simple to believe. "

- - - - - - - - - - - - - - - --

NOTE:   centigrade is used throughout to emphasize the metric prefix centi
 

 **8**

     The key to the whole thing is to attain a conceptual comprehension of the basic metric units in terms of the more familiar English units. This simply means to relate the meter to 1+ yards the kilogram mass to increments of 2+ pounds, the second to itself, the liter to 1+ quarts, the hectare to 2.5 acres, and the double relation of  0 deg-C to 32 deg-F and 100 deg-C to 212 deg-F. Possibly, a brief summary table, as listed above, for the more "common" temperatures (or any other unit needed) may aid in learning the "new” metric units.

     Once we have achieved this qualitative relationship for the basic metric units and learned the meanings of the metric prefixes, we will be better able to understand the specific metric sub-units and multiple units for themselves.  We will NOT attempt to memorize the conversion decimals for each English unit (1 in = 2..54 cm, 1 ft = 30.48 cm, etc.) as these are useless units in the metric system (and may be found in tables if needed).  Such an approach would require more memorization and effort than our present elementary and secondary students are willing to put forth.  And, would be a momentous task for our adult population.
 

-1-   Black, N. H. (Harvard), Little, E. P. (Wayne); An Introductory Course In COLLEGE
           PHYSICS,  4th ed., The MacMillan Co., N. Y., 1956.

-2-   Dull, C. E., Metcalfe, H. C., Williams J E., MODERN PHYSICS, Holt, Rinehart &
           Wilson, Inc., N. Y., 1960.

-3-    Sears, F. W. (Dartmouth), Zemansky M. W. (City College, N. Y.), UNIVERSITY
            PHYSICS, 4th ed., Addison-Wesley Pub. Co., Inc., 1970

-4-   Williams J E, Metcalfe H. C., Trinklein, F. E., Lefler, R. W., MODERN PHYSICS, Holt,
           Rinehart & Wilson, Inc., New York, 1968
 

 **9**

 INTRODUCTION

      RESPECTING THE VARIED BACKGROUNDS AND EXPERIENCES OF THOSE
HERE ASSEMBLED, I SHALL ATTEMPT TO RESTRICT NY TALK TO AS UNIVERSAL A LEVEL AS IS POSSIBLE.
     APPROPRIATELY ENOUGH, LET US BEGIN WITH OUR A, B, and C’s.

A.        IN LANGUAGE, A STATEMENT (DECLARATIVE SENTENCE) CONTAINS TWO
             ESSENTIAL PARTS: 1) a SUBJECT and 2) a VERB.

B.        OFTEN, STATEMENTS ARE SO CONSTRUCTED AS TO HAVE TWO, OR MORE,
             INTERPRETATIONS.  THAT IS, THEY ARE JUST SO MUCH RHETORIC.

C.        IN MATHEMATICS AND SCIENCE, A STATEMENT MUST MEET THE  SAME
             CRITERIA AS IN "A" ABOVE,
             HOWEVER,
             IN CONTRAST TO "B" ABOVE,
             A STATEMENT CAN HAVE ONE  AND  ONLY ONE  INTERPRETATION !!!

     TO UNDERSTAND AND APPRECIATE THE BEAUTY OF THE METRIC SYSTEM, AND INDEED THE SCIENCE FROM WHICH IT SPRUNG.  WE MUST KEEP THIS RELATION CONSTANTLY IN MIND.

Note:
          r^2  =>  r * r     \\    ‘ * ‘  =>  multiplication     \\    ‘ => ‘   means ‘ implies ‘
          Newton’s 2nd Law   \\   N’s L of Universal Gravitation    \\   N’s 1st Law
 

 **10**

N’s 2nd L :          F = m*a  or  F*dt = d(m*v)  =>  F = d(m*v)/dt

N’s L of UG :     F = G*(m*M)/r^2
 

KEY

F = FORCE

m = MASS ( m or M )

a = ACCELERATION

v = VELOCITY

r = RADIAL DISTANCE

t = time

G = GRAVITATIONAL CONSTANT (a number)

IGNORE "d"    \\     Use ‘s’ for displacement and ‘r’ for radius
 

NOTE :

1.   FORCE (F) STANDS ALONE ON THE LEFT SIDE OF EACH
      EQUATION DIRECTLY ABOVE.

2.   MASS ( m or M ) IS NEVER  ALONE ON THE
       RIGHT SIDE OF THESE EQUATIONS.

THIS TELLS US :

1. THERE IS A RELATION BETWEEN THE FORCE F AND THE MASS ( m or M ).

2. BUT!!!  THEY ARE NOT EQUAL!  THEY ARE DIFFERENT QUANTITIES!!!
 

 **11**

A.     N's lst L :   MOVING OBJECTS (MASSES) MUST MAINTAIN A STRAIGHT
                          LINE PATH UNLESS ACTED UPON BY AN EXTERNAL FORCE.

B.     K's lst L :  THE PLANETS MOVE IN ELLIPTICAL (CURVED) ORBITS.

C.     THUS, THE PLANETS REVOLVING ABOUT THE SUN, THE MOONS
         RVOLVING ABOUT THEIR RESPECTIVE PLANETS, OUR ARTIFICIAL
         SATELLITES REVOLVING ABOUT THE EARTH, MOON AND MARS
         MUST EXPERIENCE AN EXTERNAL FORCE.
         N’s L OF U G   DESCRIBES THIS FORCE:

                                               F = G * m * M/ r^2

           Me  …   O < -------------------------- ‘r’  ------------------------ >  o … ms

EARTH  (Me)                              Mass earth => ‘M’, in equation above.
SPACE LABORATORY  (ms)     mass space-lab => ‘m’, in equation above

          mv  …   o < -------------------------- ‘r’  ------------------------ >  O … Mm

VIKING ORBITER  (mv)            mass viking-orbiter => ‘m’, in equation above.
MARS  (Mm)                               Mass mars => ‘M’, in equation above.
 

       ‘r’  =  THE RADIAL DISTANCE OF SEPARATION BETWEEN EACH SATELLITE (ms) AND ITS REFERENCE MASS (Me or Mm )  DIVIDES THE NUMERATOR (GmM) BY ITS SQUARE.
 

 **12**

     FOR A GIVEN SYSTEMS, THE PRODUCT GmM REMAINS CONSTANT. THUS DIVIDING BY r^2, AS "ri" BECOMES LARGER  RESULTS IN A SMALLER VALUE FOR F.  IF WE LET G * m * M = 1000, THEN  (for study purposes  only)

                          F  =    1000/ r ^2

r = 1                  F   =    1000/ 1^2           =       1000/1                      =       1000

r = 1                   F  =   1000/10^2          =       1000/100                  =           10

r = 100               F  =    1000/100^2       =       1000/10000              =              .1

r = 1000              F  =   1000/ 1000^2     =       1000/(1000* 1000)   =              .001
 

     THIS CLEARLY INDICATES THAT FORCE (F) IS NOT INVARIANT, IT CHANGES FROM PLACE TO PLACE.   FORCE (OR WEIGHT) DEPENDS ON WHERE WE REFERENCE IT FROM.

     THE QUANTITY OF MATTER AN OBJECT CONTAINS, ITS MASS, IS INVARIANT.
 

 **13**

M    =     AMOUNT OF MATTER AN OBJECT CONTAINS. (( MASS ))

F     =     THE  PRODUCT OF AN OBJECTS MASS AND THE ACCELERATI0N IT
               EXPERIENCES :           (( FORCE ))  ( via N's 2nd L: F = m * a )
 

WEIGHT  =  THE GRAVITATIONAL ATTRACTIVE FORCE BETWEEN AN OBJECT
             AND SOME CELESTIAL REFERENCE BODY SUCH AS THE EARTH, MOON,
             MARS,  VENUS,  SUN, ETC.
 

     THIS NOW LEADS US TO CONSIDER THE FUNDAMENTAL QUANTITIES OF OUR UNIVERSE. THE AFORE MENTIONED EQUATIONS (OF MECHANICS) CAN ALL BE EXPRESSED IN TERM OF ONLY THREE ELEMENTARY CONCEPTS.

        FUNDAMENTAL
        QUANTITY                    M K S UNIT                               F P S UNIT  ---1---

LENGTH (s)                             METER (m)                                FOOT (ft)

MASS (m or M)                        KILOGRAM (kgm)                     ((SLUG))

TIME (t)                                    SECOND (sec)                             SECOND (sec)
 

---1---
         North American U. S. Engineering System or English Gravitational System.
 - - - - - - - - - - - - - - -
         ‘ s ‘  is used for displacement or distance.
 

 **14**

     ALL THE REMAING QUANTITIES (OF MECHANICS), VIA THEIR DEFINITIONS, MAY BE EXPRESSED IN TERMS OF JUST THREE FUNDAMETAL METRIC UNITS ( LENGTH, MASS & TIME ).

QUANTITY        DEFINITION                     EQUATION      MKS UNIT     FPS UNIT

VELOCITY (v)    TIME (t)  RATE   OF                 v = s/t                m/sec               ft/sec
                           CHANGE OF LENGTH (s)

ACCELERATION (a)   TIME (t) RATE OF          a = v/t                 m/sec^2         ft/sec^2
                          CHANGE OF VELOCITY (v)

MOMENTUM (p)   MASS TIMES VELOCITY   p = m*v             kgm*m/sec    slug*ft/sec
                                                                                                                   = N              = lb

WORK (W)         FORCE TIMES LENGTH (s)      W = F*s            N*m  =  Joule    ft*lb

POWER (P)        WORK DIVIDED BY TIME        P = W/t              J/sec = Watt     ft*lb/sec

NEWTON (N)     JOULE (J)    SLUG = Weight/32ft/sec^2   (the FPS unit of mass)
 

 **I**

METRIC PREFIXES

TERA      T      10^12       1,000,000,000,000            TRILLION
GIGA      G      10^9         1,000,000,000                    BILLION
MEGA    M      10^6         1,000,000                          MILLION
KILO      k       10^3          1,000                                THOUSAND
HECTO   h       10^2          100                                   HUNDRED
DEKA     D      10^1           10                                    TEN

UNIT       *       10 ^0 1                                               ONE

DECI       d        10^-1         1/10                                 TENTH
CENTI     c        10^-2         1/100                               HUNDREDTH
MILLI      m       10^-3          1/1,000                           THOUSANDTH
MICRO   'mu'    10^-6          1/1,000,000                    MILLIONTH
NANO     n        10^-9          1/1,000,000,000             BILLIONTH
PICA        p        10^-1 2       1/1,000,000,000,000      TRILLIONTH

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