Negative Exponents and Scientific Notation
- Having successfully defined 0 as an exponent, it seems
reasonable (to a mathematitian) to define negative
integers as exponents next.
As before, we look to the quotient rule as our guide. To
decide how to define a negative exponent, look at a quotient in
which the power on the bottom is bigger than the power on the
Now look at the same problem, but try applying the quotient
rule to it instead:
Thus, it makes sense to say that
Definition (Negative Integers as Exponents)
For any real number a ( a 0) and any integer n,
- Note that there is no negative sign on the right-hand
side of the equation in the definition above.
- This definition also tells us that (Why?)
- The usual rules for powers hold with this definition.
- Scientists often deal with very large (such as
93,000,000,000,000) and very small numbers (such as
0.00000000000113). To help deal with it, they have a
special notation, called scientific notation
which they use to write these numbers.
Definition (Scientific Notation)
Scientific notation for a number is an
expression which looks like N Ã—10 n, where 1 N <10 and n is any integer.
This means that any number written properly in scientific
notation will have a single nonzero digit before the decimal