The process by which metric prefixes are converted from one value to another is based upon the following premise:
1 x 1 = 1
This premise can be easily applied to statements of truth that involve variables:
a = a
( a / a ) = 1
b = b
( b / b ) = 1
( 1 )( 1 ) = ( a / a )( b / b ) = 1
The commutative properties of fractions allows the following operations to be valid:
( a / a )( b / b ) = 1
And,
( a / b )( b / a ) = 1
In the latter example, notice how the numerator value ( a ) of the first fraction is the denominator value of the second fraction in the multiplication. Take note of the fact that the following statements are true as well:
( a / b )( b ) = a
And,
( a )( b / a ) = b
In each of the two previous examples, a standalone value was multiplied by a fraction to obtain an equivalent value that is expressed in new terms. In order for this process to be valid, the units of the standalone value must appear in the denominator of the fraction it is multiplied by. Furthermore, there must be a mathematically logical relationship between the numerator and denominator values of the conversion fraction that is used:
Q: If 2a = b, how can it be used as a conversion fraction?
A: ( 2a / b ) = 1
And,
( b / 2a ) = 1
Q: Which conversion fraction must be used to convert a value of ( 5a ) to its equivalent b-value?
A: ( 5a )( b / 2a ) = ( 5 )( a / a )( b / 2 ) = ( 5b / 2 )
Q: In order to convert ( 5b / 2 ) to its equivalent value expressed in terms of ( a ), which fraction must be used?
A: ( 5b / 2 )( 2a / b ) = ( 5 )( b / b )( 2a / 2 ) = ( 5 )( b / b )( 2 / 2 )( a ) = 5a
The appropriate symmetry of conversion fractions is easier to notice with familiar numbers being multiplied by one another:
20 = 20
( 2 )( 10 ) = ( 5 )( 4 )
( 2 / 4 )( 10 ) = ( 1 / 2 )( 10 ) = 5
In order for the number ( 10 ) to retain its value, the numerator and denominator of the conversion fraction must be equal:
2 x 2 = 4
( 10 )( [ 2 x 2 ] / 4 ) = 10
10 x 1 = 10
We will see this technique used over and over again to convert metric prefixes to equivalent terms, and in general, we will use conversion fractions over and over again to solve problems across all scientific disciplines.
RENAISSANCE PHYSICS 