Units and Equal Signs
The equality sign is usually has the "forward and
backward"
reflective property that a = b when and only when b = a. Here may read
the equality sign to mean the same as or is equivalent to. For
writing
4 pennies + 3 pennies = 7 pennies
gives and comes from
7 pennies = 4 pennies + 3 pennies
But with a change of units we may write
3 apples + 4
bananas = 7 fruits
( 3 apples
+ 4 bananas yields 7 fruits)
and
2 apples + 5
oranges = 7 fruits
The equal sign here has a give interpretation and not the same as
interpretation.
The change of units here does not allow to conclude with out further
information that
7 fruits = or is the same as 3 apples + 4 bananas
because we might also have
7 fruits = 2 apples + 5 oranges
In discussing fruits of different types, 3 apples + 4
bananas = 7 fruits is equivalent to 7 fruits = 3 apples + 4
bananas when and only when we are situation where all individual
fruits appearing are considered equivalent when and only when the
equality sign means same value as not the stronger meaning, same
as.
In mathematics, the equality sign more often that not indicates same
value instead of the same as role. It is correct to use the
equality sign with both interpretations, but the give interpretation is not
used, or should be used with great caution. The first two roles of the
equal sign can be used forwards and backwards, while the last "gives
use" may only be used in a forward direction.
|