Basic to the understanding of unit conversion is the understanding of
equivalence, the understanding of the multiplicative identity of 1, and how the
two are related.
To say that two quantities are equivalent, is to say that both quantities
measure the same thing or both quantities measure the same value. For example,
we could say that a person is 5 feet tall or we could say that the person is 60
inches tall. Since 5 feet and 60 inches both represent the height of the same
person, these expressions are equivalent.
Most individuals would agree that there
are 12 inches in 1 foot, and most have seen this relationship written as 12
inches = 1 foot. This relationship is an equation because there is an equal sign
separating the left side from the right side. Does this equal sign mean that the
left side of the equation is identical to the right side of the equation? Before
you answer yes, let's make some observations: The number 12 and the word inches
are located to the left of the equal sign, and the number 1 and the word foot
are located to the right of the equal sign. Anyone can see that the left side of
the equation does not look exactly like the right side of the equation.
Therefore, the left and right sides of the equation are not identical.
How then can we say that the left side is equal to the right side? We can say
equal without saying identical. What we are saying is that the left side is
equivalent to the right side. Since the 12 inches measures the same length as
does the 1 foot, the two quantities are equivalent.
Note that 1 mile is equivalent to 5,280 feet because 1 mile measures the same
distance as does 5,280 feet, and the relationship can be written 1 mile = 5,280
feet. Note that 4 quarts is equivalent to 1 gallon because 4 quarts measures the
same amount of liquid as does 1 gallon, and the relationship can be written 4
quarts = 1 gallon.
Multiplicative identity of 1:
In the real number system, the multiplicative identity is 1. This means that
you can multiply any real number by 1 and not change the original number. For
3 * 1 = 3
* 1 = 27/41, and
(x2 + 2x + 7) * 1 = (x2
+ 2x + 7)
Since division is another form of multiplication, we can also divide an
expression by 1 without changing the value of the original number. For example,
3/1 = 3
÷ 1 = 27/41, and
(x2 + 2x + 7) ÷ 1 = (x2
+ 2x + 7)
When we combine the concept of equivalence with the multiplicative identity,
we have a powerful tool, a tool that is used in unit conversion. We use the
concept of equivalence to show that the number 1 has many faces.
We know that a fraction has a value of 1 when the numerator equals the
denominator. Now we can say that a fraction has a value of 1 when the numerator
and denominator are equivalent.
Let's illustrate how this works. If we divide both sides of the equation
12" = 1' by 1', we have
the new equation 12" ÷ 1' = 1'.
Note that the fraction has a value of 1 and the numerator and denominator are
not identical. However, the numerator and denominator are equivalent
because they measure the same thing.
This review will provide examples and problems for various topics of units of