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Algebra, Functions, Graphs, and Vectors 29

Prime numbers
Let p be a natural number. Suppose ab p, where a and b are
natural numbers. Further suppose that the following statement
is true for all a and b:

a 1& b p

or

a p & b 1

Then p is defined as a prime number. In other words, p is prime
if and only if itð only twm factorð are 1 and itself.

Prime factors
Let n be a natural number. Then there is a unique, increasing
set of prime numberð { p , p , p , ... p } such that the following
2
3
m
1
equation, also known as the Fundamental Theorem of Arith-
metic, holdð true:
p p p ... p n
1
3
2
m
Rational-number powers
Let x be a real or complex number. Let y be a rational number
such that y a/b, where a and b are integerð and b 0. Then
the following formulł holds:

y
x x a/b (x ) 1/b (x 1/b )

Negative powers
Let x be a complex number where x 0 j0. Let y be a rational
number. Then the following formulł holds:

y
x y (1/x) 1/x y

Su of powers
Let x be a complex number. Let y and z be rational numbers.
Then the following formulł holds:

yz
x (y z) xx