Metric System

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          "nano-meter" (long i, hard c, long o)

µ            "micro-meter" (short a, long o)

mm         "milli-meter" (short i, short i)

cm          "centi-meter" (soft c, short e, short i)

dm         "deci-meter" (soft c ---> 's')

1           "meter" (long e)

dkm       "deka-meter (short e, short a)

hm         "hecto-meter" (hard c ---> 'k')

km         "kilo-meter" see below...

Mm        "mega-meter"

Gm        "giga-meter"

Tm        "tera-meter"

These units are commonly spoke one of two ways

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 light-years. A light-year is the distance light travels in one year. The speed of light is about 3 Х 10⁸ m/s.  In one year there are 31, 557, 600 seconds, which means in one year light travels

3 Х 10⁸ X 31, 557, 600 = 9.46728 X 10¹⁵ m = 9.47 Pm (petameters, next higher from Tm on the chart, but not shown, exponent 15).

So, 1 light-year 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 light-years away. Ee don't say 25,000 Pm away, nor 25 Em (Exa meters, exponent 18), but 25,000 light-years 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 in-depth 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.