The number 1 can be formed via the establishment of a ratio of logically related quantities:
A = A
( A / A ) = 1
A = B
( A / B ) = 1
And,
( B / A ) = 1
Furthermore, the product of any number multiplied by 1 is the number itself:
( A / A )( 5 ) = 5
And,
( A )( 5 / A ) = 5
In order to express a metric prefix value in terms of another, we must multiply it by some ratio that is equal to one. In this way, the value being multiplied does not lose its original value; it is only expressed using different terms as follows:
4 quarters = 1 dollar
( 4 quarters / 1 dollar ) = 1
And,
( 1 dollar / 4 quarters ) = 1
We now have two fractions that can be used to convert quarters to dollars or dollars to quarters. Let’s now express 5 dollars in terms of quarters:
( 5 dollars )( 1 ) = 5 dollars
( 5 dollars )( 4 quarters / 1 dollar ) = 20 quarters
Notice how the appropriate conversion fraction contained a denominator expressed in the same terms as the expression being multiplied. This is the appropriate technique to use to express any value in terms of another. We can use the commutative properties of fractions to show why this is the case:
( 5 dollars / 1 dollar )( 4 quarters ) = 20 quarters
( 5 )( 4 quarters ) = 20 quarters
We are now ready to convert metric prefixes from one form to another. Let’s begin by expressing grams ( g ) in terms of kilograms ( kg ):
1,000 g = 1 kg
In order to express a given quantity of grams in terms of kilograms, we first take note of the fact that a kilogram is a larger quantity of mass than a gram:
1 g = 10-3 kg
Any quantity of grams expressed in terms of kilograms will be expressed as some fraction of the larger unit:
Q: How many kilograms are in 5 grams?
A: It takes a small fraction of a kilogram to form an equivalent amount of mass expressed as 1 gram. The appropriate multiplicand and conversion fraction setup is as follows:
( 5 grams )( 1 kg / 1,000 g ) = 5 x 10-3 g
Converting grams to kilograms is rather straightforward. Conversely, expressing a quantity that is smaller than 1 gram in terms of a quantity that is larger than 1 gram can be somewhat tricky:
Q: 1 microgram ( µg ) is one-millionth of a gram. How many kilograms are in 5 micrograms?
A: We must first realize that 1 microgram is one-thousandth of a gram. Furthermore, a kilogram is one thousand times larger than a gram; therefore, 1 microgram is only one-billionth of a kilogram:
1 µg = 10-9 kg
( 5 µg )( 10-9 kg / ug ) = 5 x 10-9 kg
Fortunately, multiple conversion fractions can be used to accomplish the same task. In the previous example, we could just have easily converted micrograms to grams. We would then establish logical relationship between grams in kilograms to arrive at our desired destination:
1 µg = 10-6 g
( 5 µg )( 10-6 g / µg 1 ) = 5 x 10-6 g
And,
1,000 g = 1 kg
( 5 µg )( 10-6 g / µg 1 )( 1 kg / 1,000 g ) = ? kg
( 5 x 10-6 g )( 1 kg / 1,000 g ) = ? kg
( 5 x 10-6 )( 10-3 kg ) = 5 x 10-9 kg
Expressing smaller units in terms of larger ones is more intuitive, because larger units are multiples of smaller units:
Q: How is 5 kilometers expressed in terms of centimeters?
A: 1 km = 1,000 m
( 5 km )( 1,000 m / 1 km ) = 5,000 m
And,
100 cm = 1 m
( 5 km )( 1,000 m / 1 km )( 100 cm / 1 m ) = ? m
( 5 x 1,000 m )( 100 cm / 1 m ) = ? m
( 5 x 1,000 )( 100 cm ) = 500,000 m = 5 x 105 cm
RENAISSANCE PHYSICS 