# Basic Metric

The original intent of the metric system is to have one name for each characteristic to be measured. For example, the only unit of distance is the meter. That characteristic could then be magnified or diminished with an appropriate prefix. For example, 'deci' means a tenth so that a decimeter is one tenth of a meter. The next important thing is to relate the different characteristics by defining their corresponding units so that 'fudge factors' are not needed. The unit of volute called liter and is defined as a cubic decimeter. The unit of mass called kilogram as defined by the international prototype. The unit of time is called second. The unit of force is called the newton. So if I say 'F=ma', the unit of force was chosen so that it requires a force of one newton to accelerate a mass of one kilogram at a rate of one meter per second every second. The energy needed push an object with a force of one newton for a distance of one meter is one joule. The power needed push an object with a force of one newton at a velocity of one meter per second is one watt. In the design of the metric system, great effort was made to avoid 'fudge factors'.

The metric system was designed to be easy to understand. For each characteristic to be measured [except mass], only one unit is defined. And that unit can be scaled from the very small to the very large. Units for derived characteristics are defined so that fudge factors are not needed. It is for these reasons that we should use the metric system that is the official United States system of weights and measure. Metric units define non-metric units of weights and measure used in the United States.

## Basic Unit Definitions

The standard measurement of length in the United States is the meter (m). What the meter is has been redefined many times. In 1799 the meter was originally defined as one ten millionth of the quadrant of the earth through Paris. Later in 1872, thirty prototype meters were made on metal bars with marks to indicate the meter. In 1960 the meter was redefined as 1650763.73 wavelengths of the orange-red line of krypton-86. And in 1983 the meter was again redefined so that the speed of light in a vacuum was exactly 299729458 meters/second.

In 1964, a liter [L or l] was defined as one cubic decimeter. The kilogram (kg) is the unit of mass and is equal to the mass of the international prototype of the kilogram. One liter of water at maximum density, about 4 degrees Celsius, standard pressure, has a mass very close to 1 kilogram.

## Metric Notation

With English unit, we add a sufix to denote very large or very small quantities. For example, "seven million" In the metric system a prefex is added to the unit, as in "seven megabucks" which would be seven million dollars. Similarly a kilobuck would be \$1000, a decibuck would be a dime and a centibuck would be one cent. Below is a list of some of the prefixes used by the metric system preceded by their abreviation. Also included for completeness are the numbers shown in Scientific Notation and their English name.

```
Metric          Standard            Scientific   United
Notation         Notation             Notation    States

(Y)   yotta 1000000000000000000000000    1024
(Z)   zetta 1000000000000000000000       1021
(E)   exa   1000000000000000000          1018     quintillion
(T)   tera  1000000000000                1012     trillion
(G)   giga  1000000000                   109      billion
(M)   mega  1000000                      106      million
(k)   kilo  1000                         103      thousand
(h)   hecto 100                          102      hundred
(dk)   deca   10                          10       ten
(d)   deci    0.1                        10-1     tenth
(c)   centi   0.01                       10-2     hundredth
(m)   milli   0.001                      10-3     thousandth
(µ)   micro   0.000001                   10-6     millionth
(n)   nano    0.000000001                10-9     billionth
(p)   pico    0.000000000001             10-12    trillionth
(a)   atto    0.000000000000000001       10-18    quintillionth
(z)   zepto   0.000000000000000000001    10-21
(y)   yocto   0.000000000000000000000001 10-24
```

Note that "µ" is the Greek letter mu. According to Paul Trusten, R.Ph., Public Relations Director, U.S. Metric Association (USMA), Inc., the Official U. S. Metric Page:
"... The correct symbol for "microgram" is "µg". However, because the Greek letter "µ" (mu) can be mistaken for other letters when handwritten, the Joint Commission (the accrediting body for U.S. healthcare) has banned the use of the official symbol "µg" in medical records (hospital charts, hospital reports, prescriptions), and has recommended the use of "mcg" instead. ... Once a plan for U.S. metrication is in place, and metric-system education becomes part of the American landscape, I am confident that we will all become fluent in the use of "µ". However, rest assured that, in medicine and pharmacy, both "µg" and "mcg" are understood to be the same shorthand for "microgram." ..."

Please DO NOT confuse "m" and "M". Asprin is measured in mg (milligram) and a Mg (megagram) is called a metric ton. A mm is less than .04 inches but a Mm is more than 621.37 miles, about the distance between San Diego and Salt Lake City. Of course the original definition of 10 Mm was the distance from the equator to the North Pole through Paris.

Note that it is improper to mix English notation (thousand, million, billion, etc.) with metric notation. The total electric power generation capability in California is about 47 gigawatts. It is often incorrectly written as 47 thousand megawatts, or 47 million kilowatts. In a desire to keep the digits significant, writing 47,000 megawatts or 47,000,000 kilowatts should be discouraged. Calling a 1 gram tablet of asprin a 1000 mg tablet is also improper advertising hype.

## Fundamental Metric Units

The unit of length is the meter (m).
The unit of mass is the gram (g).
The unit of time is the second (s).
The unit of temperature is the Kelvin. (K)
The unit of current is the ampre (A).

The meter (m) is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. [i.e. by definition, light travels a distance of 29.9792458 cm in a ns.]

The kilogram (kg) is the unit of mass; it is equal to the mass of the international prototype of the kilogram. [Essentially the kg is the mass of a liter of water at 4°C.]

The second (s) is the duration [or time] of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. [Duration chosen to equal what was a mean solar day divided by 86400.]

The kelvin (K), unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature between the triple point of water and absolute zero. Absolute zero is 0 K.

The ampere (A) is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.

The mole [mol] is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is "mol."

The candela [cd] is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

## Derived Metric Units

The liter (L), a measure of volume, is a cubic decimeter.

The newton (N) is the force needed to accelerate 1 kg at 1 m/sec2. 100000 dynes are equal to 1 Newton. [On the Earth, gravity attracts 1 kg with a force of about 9.8 newtons. This force is the attraction of the 1 kg to all the mass in the universe and is known as the weight of the 1 kg. It varies between the equator and the poles. It also varies with time of day. Standard gravity is defined as 9.80665 m/sec2. ]

The joule (J), a measure of energy, is 1 newton-meter of work. It is also equal to 1 watt-second of energy. There are 3.6 megajoules in a kilowatt-hour. To raise a liter of water one meter is about 9.8 joules of work.

The watt (W), a measure of power, is one joule per second or one newton-meter per second. On Earth, the power needed to raise one liter of water one meter in one second is about 9.8 watts. One ampere through an electrical resistance of one ohm develops an electric force of one volt and dissipates one watt of power.

The coulomb (C) is the measure of electric charge. One amp-second is one coulumb. [About 6.281 x 1018 electrons.]

The volt (V) is the electrical potential (E) across a resister dissapating one watt with a current of one amp. That resister will have a resistance of one ohm.

The ohm (Ω) is electrical resistance (R). It is equal to the voltage across a resistor divided by current through the resister in amps.

The tesla (T) is a measure of magnetic flux density (B). It is webers (Wb) per square meter. Note that 1 Tesla is equal to 10,000 Gauss. [One Gauss = 100 µT]

The weber (Wb) is a measure of magnetic flux. It is volts times time.

The henry (H) is the unit of inductance. It is equal to magnetic flux (Wb) divided by current (A). The voltage across an inductor is equal to the rate of change of current through the inductor times its inductance.

The farad (F) is the unit of capacitance. It is equal to charge (C) divided by voltage (V). The current into a capacitor is equal to the rate of change of voltage across the capacitor times the capacitance.

The degree Celcius (°C) [aka centigrade] is the same difference as the degree Kelvin, but zero on the Celcius scale is at the freezing point of water.

The hertz (Hz), a measure of frequency, is the number of times something happens per second. It replaces the old unit 'Cycles per Second' (CPS). The frequency of a periodic function is the reciprical of the period in seconds. In United States music, 'A' above middle 'C' is 440 Hz.

The diopter, a measure of lens power, is the inverse of the focal length in meters. Thus a lens with a focal length of 1 m has a power of 1 diopter and a lens with a focal length of 0.25 m has a power of 4.0 diopters. When lenses are used together, the total power is the sum of the powers of each lens.

The bel (B) is a logrithmic measure of power ratio. A power ratio of 10 is 1 B, a power ratio of 100 is 2 B, and a power ratio of 1000 is 3 B. A power ratio of 100.1 [about 1.26] is 0.1 B or 1 dB [decibel].

## Relationships between Metric Units

By design, metric units combine to form new units to measure other properties. The following table may prove useful. For all practical purposes:

```        Volume         Volume      Mass of Water
cubic meter     kiloliter       metric ton
cubic decimeter         liter         kilogram
cubic centimeter    milliliter             gram
cubic millimeter    microliter        milligram
```

It shows the relationship between volume expressed in cubic meters, and the equivalent volume in liters, and the mass of water, at maximum density which is about 4 degrees centigrade, that will fill that volume. Note that water therefore has a maximum density of about 1 metric ton per cubic meter [kiloliter], and 1 kilogram per liter, and 1 gram per milliliter, and 1 milligram per microliter.

Similiarly, the unit of pressue called a 'pascal' is a newton per square meter, and is equal to a centinewton per square decimeter, and is equal to 100 micronewtons per square centimeter, and is equal to a micronewton per square millimeter.

## Algebra on Units

A measurement consists of two parts, the number and the unit. Both are important. If I were to say:

9.806650 = 32.17405 = 21.93685

you would say I was wrong. And I would be wrong. But if I were to say:

1 G=9.806650 m/sec2=32.17405 ft/sec2=21.93685 mph/sec

then I would be right. [more or less, these are approximations] Note that here "G" represents the standard [average] acceration of a mass on the surface of the Earth due to the force of gravity.

If I multiply 30 cm. by 40 cm., the answer is 1200 cm2. Note that the units are subject to the same algebraic manipulation as are the numbers. Acceleration is the rate of change of velocity. Rate of change, also known as a time derivative, has the unit 1/sec. Distance is the area under the velocity versus time plot. This area, also known as a time integral, has the unit sec. Thus the rate of change of distance with time is velocity, the rate of change of velocity is acceleration, and the rate of change of acceleration is jerk. If the distance is in meters and the time in seconds, the corresponding units are:

Distance in m
Velocity in m/sec
Acceleration in m/sec2
Jerk in m/sec3

Velocity will always be in the form of unit distance per unit time. Any other unit is wrong. Knowing how to multiply and divide units is essential to understanding science.

## Measurement of Angle

There are four different ways to divide a circle and measure an angle. There are 2 pi radians, 360 degrees, or 400 gradients in a circle. In addition, computers often represent a circle as a fraction between 0 and 1 using Binary Angular Measure or BAMs. The 5 bit Grey coded optical disk shown in Figure 1 is an example of a disk used to encode direction into a computer.

Figure 1

```                code          BAM      deg/min/sec     radians
00     BBBBB  N     0.015625     5° 37' 30"   0.098175
01     BBBBW  NNE   0.046875    16° 52' 30"   0.294524
02     BBBWW  NNE   0.078125    28°  7' 30"   0.490874
03     BBBWB  NE    0.109375    39° 22' 30"   0.687223
04     BBWWB  NE    0.140625    50° 37' 30"   0.883573
05     BBWWW  ENE   0.171875    61° 52' 30"   1.079922
06     BBWBW  ENE   0.203125    73°  7' 30"   1.276272
07     BBWBB  E     0.234375    84° 22' 30"   1.472622
08     BWWBB  E     0.265625    95° 37' 30"   1.668971
09     BWWBW  ESE   0.296875   106° 52' 30"   1.865321
10     BWWWW  ESE   0.328125   118°  7' 30"   2.061670
11     BWWWB  SE    0.359375   129° 22' 30"   2.258020
12     BWBWB  SE    0.390625   140° 37' 30"   2.454369
13     BWBWW  SSE   0.421875   151° 52' 30"   2.650719
14     BWBBW  SSE   0.453125   163°  7' 30"   2.847068
15     BWBBB  S     0.484375   174° 22' 30"   3.043418
16     WWBBB  S     0.515625   185° 37' 30"   3.239767
17     WWBBW  SSW   0.546875   196° 52' 30"   3.436117
18     WWBWW  SSW   0.578125   208°  7' 30"   3.632467
19     WWBWB  SW    0.609375   219° 22' 30"   3.828816
20     WWWWB  SW    0.640625   230° 37' 30"   4.025166
21     WWWWW  WSW   0.671875   241° 52' 30"   4.221515
22     WWWBW  WSW   0.703125   253°  7' 30"   4.417865
23     WWWBB  W     0.734375   264° 22' 30"   4.614214
24     WBWBB  W     0.765625   275° 37' 30"   4.810564
25     WBWBW  WNW   0.796875   286° 52' 30"   5.006913
26     WBWWW  WNW   0.828125   298°  7' 30"   5.203263
27     WBWWB  NW    0.859375   309° 22' 30"   5.399612
28     WBBWB  NW    0.890625   320° 37' 30"   5.595962
29     WBBWW  NNW   0.921875   331° 52' 30"   5.792314
30     WBBBW  NNW   0.953125   343°  7' 30"   5.988661
31     WBBBB  N     0.984375   354° 22' 30"   6.185011
```

The reason for the Grey code is to prevent errors that might occur if two bands were suppose to change at a given angle, but one change was a very small fraction off.

Even though many software programs use BAMs for calculation of angles to describe, for example, direction of travel, computer output is usually converted to degrees and minutes for user display. This is done by multiplying the fractional angle [BAM] by 360 degrees, using the resultant integer as degrees, multiplying the new fraction by 60, then using that integer for minutes. The new fraction could be multiplied by 60 to get the seconds. Unlike this simple example, most encoding disks use 8 or more bits to encode an angle.

## Computer Pseudometric

Computers are binary machines. Everytime a bit is added the memory address space, the amount of addressable membory doubles. So the measurement of computer memory is bases on power of 2. To simplify the description of memory size, the term kilobye is applied to the size of memory assable by 10 bits that is 1024 bytes. This was done because 1024 is close to 1000. So when taking about computer memory or disk size, use the following table:

#### Computer Memory

```
Metric           Standard      Computer     United
Notation          Notation      Notation     States

(TB) terabyte  1099511627776        240        trillion
(GB) gigabyte  1073741824           230         billion
(MB) megabyte  1048576              220         million
(kB) kilobyte  1024                 210        thousand

```

## Metric Review

To summarize, the basic units of the Metric System are the meter [m], kilogram [kg] and the second [s]. 100 inches is the same distance as 2.54 meters. A cubic decimeter [0.1 m or about 3.937" on a side] is a liter [l]. Fill that liter with cold water and the water will have a mass very close to a kilogram [kg]. In Saint Louis, Missouri, that kilogram will have a weight of about 9.8 Newtons [N]. Raise that kilogram up 1 meter and you will have done 9.8 joules [J] of work. Raise that kilogram up 1 meter in 1 second requires a power of 9.8 watts [W]. The pressure at the bottom of that cubic decimeter of water is .98 kilopascal [kP]

### Weight of 100 kg or 220.46223 lbm

as a function of location

```
North Pole              983.217 N      221.036 lbf.
St. Michael, Alaska     982.192 N      220.806 lbf.
Paris, France           980.943 N      220.525 lbf.
Standard Gravity        980.665 N      220.46223 lbf.
New York, New York      980.267 N      220.373 lbf.
Key West, Florida       978.970 N      220.081 lbf.
Equator                 978.039 N      219.872 lbf.
Surface of Mars         369.7 N         83.1 lbf.
Surface of our Moon     162.7 N         36.5 lbf.
Surface of Pluto         65.7 N         14.7 lbf.
```

Mass & Weight are NOT the same thing. The weight of an object varies with location. But its mass remains the same where ever it is. The mass of an object is measured on a balance scale by comparing it with other masses. Weight is a force and must be measured on a spring scale. The chart above shows how the weight of a 100 kg mass changes with location. Weight even changes with time of day. In the United States, things weigh the least at noon, in June, with a new moon! The difference is not much, but it is enough to cause the tides.

Note that when selling sugar, apples, nuts, etc., 1 lb = .45359237 kg. At the standard gravity of 9.80665 m/sec2, a 1 lb [one pound mass] has a weight of 1 lbf [one pound force].

Sometimes the confusion between mass and force results in the use of the meaningless unit of kg/m2 [kilogram per square meter] being used to measure 'pressure'. Presure must be N/m2 which is Pa [pascal].

Below is a list of common metric unit symbols, the name of of the unit and a short description.

```
A      ampere          electric current
B      bell            power ratio
C      coulomb         electric charge
cd     candela         luminous intensity
°      degree          plane angle
°C     degree Celsius  Temperature
Gs     gauss           magnetic flux density [cgs]
g      gram            mass
H      henry           electric inductance
Hz     hertz           frequency
J      joule           energy, work, quantity of heat
kg     kilogram        mass
K      kelvin          absolute temperature
L or l liter           volume
lm     lumen           luminous flux
lx     lux             illuminance
m      meter           length
mol    mole            amount of substance
N      newton          force
Oe     oersted         reluctance
Ω      ohm             electric resistance
Pa     pascal          pressure
s      second          time
S      siemens         electric conductance
t      ton             mass [megagram] [metric ton or tonne]
T      tesla           magnetic flux density
V      volt            electric potential difference, EMF
W      watt            power
Wb     weber           magnetic flux
```

## History

1790 Thomas Jefferson proposed a decimal-based system of measurement for the United States. France's Louis XVI authorized scientific investigations aimed at a reform of French weights and measures. This led to the development of the first "metric" system.

1792 The U.S. Mint was formed to produce decimal currency. (the U.S. dollar consisting of 100 cents). \$25 was then defined as the worth of 1 oz. of gold.

1795 France officially adopted the metric system.

1866 Congress authorized the use of the metric system in the United States and gave each state a set of standard metric weights and measures.

1875 United states became one of the original 17 signatory nations to the Treaty of the Meter.

1893 United States adopted fundamental metric standards for length and mass.

1958 U.S. and imperial yards were adjusted to metric measurement. I.e. the U.S. inch was shortened to exactly 2.54 cm.

1960 The International System of Units, abbreviated SI, was approved by the General Conference of Weights and Measures.

1975 Congress passed the Metric Conversion Act of 1975. The Metric Board was established.

1982 The Metric Board was dissolved.

1988 Congress passed the 'Omnibus Trade and Competitiveness Act of 1988. Designated the metric system as the preferred system of weights and measures for United States trade and commerce.

## Refereces

U.S. Metric Association (USMA), Inc. A must read!

NIST National Institute of Standards and Technology, the Official U. S. Metric Page

## Learning the Metric System

He best way to learn the metric system is to see and feel familiar things that have a metric label. Many of us grew up using 35 mm film with a width of 3.5 cm. The standard CD is 12 cm in diameter. Buy a meter stick and use it. The mass of a 1000 mg tablet of aspirin is 1 g. Buy a liter of bottled water and know that the bottle has a volume of 1 dm3 and the water has a mass of about 1 kg. Two and a half laps around your local high school football field [5 furlongs] is just over 1 km. Tires on a small car often will have a maximum inflation pressure of 300 kPa. Water freezes at 0 °C and boils at 100 °C at sea-level atmospheric pressure. That pressure will support a column of mercury about 76 cm high. Music of a marching band has a 2 Hz beat. In the United States, the 'A' above middle 'C' has a frequency of 440 Hz. If your energy bill says your average energy use is 720 kW-hr/month [30 day month], that is an average power of 1 kW, the power of ten 100 W light bulbs, about the power of bright sunlight on 1 m2 perpendicular to the sun at the Earth's surface. A kW-hr of energy is 3.6 MJ. Light will travel almost 30 cm in a ns.

# Think Metric!

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