The Metric System refers to the system
of measurement based on powers of 10. Because of this, moving
between various "sizes" of a measurement only involves
moving the decimal point to the right or to the left in the number
itself.
In this system
the unit of length is the meter, abbreviated with 'm'. As
we'll see later, 1 meter is slightly larger than a yard. Well we
already know that a yard contains 3 feet, where each foot has 12
inches, we get one yard has 36 inches. To find smaller units, we
start dividing this inch by powers of 2, and we get 72 half inches,
144 quarter inches, 288 eighth inches, 676 16th inches, etc.
Let's go back
to the meter. Remember a meter is a bit larger than a yard, so it
doesn't hurt to think of a meter as yard for now. We start dividing
by 10 to get smaller units of measurements. One meter has 10
decimeters. Each decimeter has 10 centimeters. Each centimeter has
10 millimeters, and we'll stop here.
So 1 meter =
10 decimeters, each decimeter has 10 centimeters, so
1
meter = 10 * 10 = 100 centimeters.
1
meter = 10 * 10 * 10 = 1000 millimeters
Now looking at
this the other way:
1
decimeter is 1/10 meter = 0.1 meters
1
centimeter is 1/100 meter = 0.01 meters
1
millimeter is 1/1000 meter = 0.001 meters
Compared to the
English system this is very simple, we just move the decimal
point by one in
this case to get to the next smallest unit of measure. Moving to the
next larger unit works the same.
The first
division of a yard is the foot. There are 3 of them, in other words
the yard is divided into 3 equal parts.
Remember a
meter is about the length of a yard. The first division of a meter
is the decimeter, that is, the meter is divided into 10 equal parts.
Next the foot
is divided into 12 equal parts, 12 inches.
Likewise the
decimeter is divided into 10 equal parts, 10 centimeters.
We'll stop
here. The point of this is to develop an analogy between the two
systems with a common point of reference, the yard. In particular,
a meter is to a yard as a decimeter is to a foot as a centimeter is
to an inch.
Here is how a
cm compares to an inch: follow the dashed red line from the bottom
(1 inch) to just beyond 2.5 cm, in actuality we use 2.54 cm. So a cm
is about 1/2 inch.
Suppose we
want to speak of larger lengths. For instance, say we have 1 mile.
1 mile is 5280 feet. If we wanted to know how many yards this is,
we'd divide by 3 to get 1760 yards. Here we had to think of miles
to feet then to yards to get the result.
Now one mile
is about 1.600 kilometers which is 1600 meters. All we had to know
here is 1 kilometer is 1000 meters. Note the decimal point moved 3
places to the right. (1.600 > 1600)
In the English
system we talk about inches, feet, yards, miles, etc. The relative
magnitudes are implied by the unit itself, which changes from
magnitude to magnitude, and the corresponding factors to move between
these magnitudes follow no pattern. Also, different quantities
measured in length, volume, mass, force, etc., have their own names
and their own multipliers. In the metric system the quantity
being measured may change, meter to gram to liter, etc, but the
multipliers are the same within each quantity.
... 1/1000,
1/100, 1/10, 10, 100, 1000, 10000 ...
We have special
names for these multipliers
milli
centi deci deka hecto kilo mega
There are more,
but these are the most commonly used for day to day purposes.
More
importantly, no matter what physical quantity we're measuring, these
multipliers are used the same way and mean the same thing.
What must be
learned here (memorized if need be) is how these multipliers are
related. Powers of 10. Powers of 10.
These
multipliers are further abbreviated when used with the quantity being
measured. For a meter we have
nm "nanometer"
µ
"micrometer"
mm
"millimeter"
cm "centimeter"
dm "decimeter"
1 "meter"
dkm
"dekameter
hm
"hectometer"
km "kilometer"
Mm "megameter"
Gm "gigameter"
Tm "terameter"
These units are
commonly spoke one of two ways (consult a dictionary for pronounciation.)
1)
no accent on any syllable; the o in km is long as is the first e,
Mm: second e long rest are short same in Gm and Tm
2)
accent on the second syllable: all vowels short
Interesting
note: The multiplier G has the same Greek root as does "gigantic."
Therefore it should be (and was originally) spoken with a soft g
followed by a hard g. Common practice is to ignore this root and
instead use a hard g sound.
It sounds more
impressive I suppose and is just one example of our ever changing
dynamic English language. I, on the other hand, purposely use both
in the same conversation, even the same sentence, to keep everyone on
their toes. :)
The unit of
mass is a gram, 'g'. So we would have mg, cg, dg, ..., kg, etc.
The same power of tens apply.
So a mg is how
many grams? 1/1000, that is 0.001 g.
A kg is how
many cg? write it out... 1kg = 1000g, now move to cgs
1000 00 cg
(100 cg in a g) writing this normally, 100,000 cg
A cm is how
many dkm? again write it out. 0.01m
In one dkm there are 10
meters so dividing we get 0.01 / 10 = 0.001 dkm
The unit of
volume is liter, 'l', (lower case L). Se we would have ml, cl,
dl, l, dkl, hl, kl, Ml, etc.
A dl is 100
ml, 10 cl, and 1/10 l.
In review: (this
needs to be memorized, no way around that)

multiplier

name

power of ten

magnitude

T

tera

12

1,000,000,000,000

G

giga

9

1,000,000,000

M

mega

6

1,000,000

k

kilo

3

1,000

h

hecto

2

100

dk

deka

1

10

1

unit

0

1

d

deci

1

0.1

c

centi

2

0.01

m

milli

3

0.001

µ

micro

6

0.000001

n

nano

9

0.000000001

p

pico

12

0.00000000001

The Greek
letter 'µ', micro, is used
instead of 'm' to name the micro multiplier.
Notice that moving from milli
to kilo requires a multiplication or division by 10 at each step. These
multipliers are sufficient for every day quantities. Beyond these multipliers we
increase or decrease by a factor of 1000. These relatively large and small
multipliers are heavily used in Science.
Many
times in Physics, extreme numbers have units renamed for what is
being measured. For example, the very large distances between
objects in space especially when in different galaxies, are measured
in lightyears. A lightyear is the distance light travels in one
year. The speed of light is about 3 Х 10^{8} m/s. In one year there are 31, 557, 600 seconds, which means in one
year light travels
3
Х 10^{8} X 31, 557, 600 = 9.46728 X 10^{15} m
= 9.47 Pm (petameters, next higher from Tm on the chart, but not
shown, exponent 15).
So,
1 lightyear is approximately 9.5 Pm.
In
practical terms, the closest galaxy to our solar system is reported
to be the Canis
Major galaxy and is about 25,000 lightyears away. Ee
don't say 25,000 Pm away, nor 25 Em (Exa meters, exponent 18), but
25,000 lightyears away.
Back
to the chart:
Using
this chart you can readily move from one multiplier to another. For
example,
convert
245.671 hm to mm. Subtract the power of tens, 2  (3) = 5.
move
the decimal point 5 places to the right to get 24567100.0 mm,
written more clearly: 24,567,100.0.
So
which way do we move the decimal? Ask yourself this, is the
resulting multiplier larger or smaller than the starting multiplier?
Milli is smaller than hecto, so there must be more in the end,
therefore move the decimal point to the right.
Unit
Conversions (Unit Analysis)
More
indepth treatment here
First I will define an equivalent fraction. An equivalent fraction is a
fraction whose numerator and denominator name quantities with their units that
name the same measurement. For example, 100 cm and 1 meter name the same length.
If we made a fraction out of these two quantities then we are naming the same
length, and this length divided by itself equals 1. Therefore,
if we multiply one quantity by an equivalent fraction, we don't alter
that quantity except to change its units and corresponding magnitude.
100 cm = 1 m, so an equivalent fraction is
100cm / 1m, another is 1m / 100cm
There
is a systematic way to do unit conversions and it is called
unit
analysis.
The
idea is to start with the given measurement and continue to multiply by
equivalent fractions until the desired unit results. (Sometimes
this procedure will require a few attempts.)
Ok
then, to do this we need to remember how to multiply fractions,
canceling common factors between the numerators and the denominators.
In
the example below, the 7s cancel, leaving the 3 and the 5.
One
more example: Convert 3 yards to decimeters.
Here's
a formal description of our problem and solution that may be useful
to you.
goal: convert English units to metric units
strategy: use unit analysis
to get from yards to dm
method:
find equivalent fractions to multiply in sequence to reach dm
implementation: multiply
fractions by hand canceling common factors and common units until
the result is obtained
Now,
does this answer make sense? Remember a yard and a meter measure
lengths that are almost equal. 10 decimeters is a 1 meter and 1 yd
is almost 1 meter. We had 3 yds, so approximately 30 decimeters. We
calculated 27.43 decimeters, so we're good.