User:Marty/UCSC Math 110 Fall 2008/Assumed knowledge

=Natural numbers=

You can assume the following basic facts about the natural numbers $\NN = \{0,1,2, \ldots \}$:
 * Natural numbers may be added, multiplied, and exponentiated to yield natural numbers.
 * Every nonempty set of natural numbers has a smallest element.

=Integers=

You can assume the following basic facts about the integers $\ZZ = \{ \ldots, -2,-1,0,1,2,\ldots \}$:
 * Every integer can be expressed as the difference of two natural numbers.
 * Every integer $z$ is either a positive, negative, or zero. If $z$ is negative, then there exists a unique natural number $n$ such that $z = -n$.
 * Integers can be added, subtracted, and multiplied to yield integers.

=Rational numbers=

You can assume the following basic facts about the rational numbers $\QQ$:
 * The rational numbers consist of those real numbers which can be expressed as the quotient of two integers.
 * Every rational number can be expressed in "lowest terms", i.e., as $\pm \frac{a}{b}$, where $a,b \in \NN$, and $GCD(a,b) = 1$.
 * Rational numbers can be added, subtracted, and multiplied to yield rational numbers. Every nonzero rational number has a multiplicative inverse, given by its reciprocal.
 * Between any two distinct rational numbers, a third rational number can be found.

=Real numbers=

You can assume the following basic facts about the real numbers $\RR$:
 * Real numbers can be added, subtracted, and multiplied to yield real numbers.
 * These operations satisfy all familiar arithmetic identities, such as the commutative, associative, and distributive properties, and the identity properties of zero and one. All basic algebraic manipulations can be freely used.
 * Real numbers are ordered; all familiar statements about inequalities can be used, such as "trichotomy" (every real number is positive, negative, or zero), and the fact that the product of positive real numbers is positive.
 * Between any two distinct real numbers, a third real number can be found; in fact, one may find a rational number between any two distinct real numbers.
 * For any real number $r$, there exists a natural number $n$, such that $-n < r < n$.
 * Any nonempty subset of $\RR$ which has an upper bound has a least upper bound.
 * You can use basic analysis, such as limits, derivatives, continuity, and Riemann integration.

Since $\NN \subset \ZZ \subset \QQ \subset \RR$, properties about real numbers also can be applied frequently to natural numbers, integers, and rational numbers.