proof of properties of the exponential
This proof will build up to the results in three easy steps. First, they will be proven for integer exponents, then for rational exponents, and finally for real exponents. For simplicity, I have assumed that the exponents are positive; it is easy enough to derive the reults for negative exponents by taking reciprocals.
Integer exponents This first case is rather trivial. Although one could make the proofs formal and rigorous by using induction or infinite descent, there is no need to go to such extremities except as an exercise in formal logic, so simple verbal indications should suffice.
Additivity: This is a consequence of assocuiativity of multiplication – multiplied by itself times can be rebracketed as multiplied by itself times multiplied by multiplied by itself times.
Monotonicity: If and for positive integers , then . Applying this fact repeatedly to shows that when and is a positive integer.
Continuity: This is irrelevant since the integers are discrete.
Rational exponents For rational exponent , may be defined as the Dedekind cut
For this to be well-defined, three conditions need to be verified:
1) It must not depend on the choice of and as long as
Any pair of integers such that may uniquely be expressed as where and are relatively prime. The monotonicity property implies that if and only if .
2) If and then .
By transitivity, . By monotonicity, it follows that .
3) At most one rational number can not belong to either or .
Given a rational number , it is only possible for a rational number to satisfy neither nor if . So for a rational number to belong to neither set, it must be the case that . Suppose that there were two rational numbers such that and . Then . If , either or . Either way, motonicity implies that , so . Hence at most one rational number belongs to neither set, so one has a well-defined Dedekind cut which defines when is a rational number.
Homogeneity: The Dedekind cuts defining and are
respectively. By the homogeneity property for integer exponents, if and , then . Likewise, if and , then . By the definition of multiplication for Dedekind cuts, it follows that for rational exponents .
Additivity: Write and over a common denominator: and . Then and are determined by the Dedekind cuts
respectively. If and , then by additivity for integer exponents. Likewise, if and , then . By the definition of multiplication for Dedekind cuts, it follows that for rational exponents and .
Monotonicity: Suppose that . Write and over a common denominator: and . Then . Then and are determined by the Dedekind cuts
respectively. If , then since by the law of monotonicity for integer exponents. Likewise, , then . Hence, by the definition of “greater than” for Dedekind cuts, .
Continuity: Because of the additivity property, it suffices to prove that . By monotonicity, it suffices to prove that . Suppose that . Write . The considerations of last paragraph show that one can restrict attention to the case . Let . By simple algebra, one has
By monotonicity, . Hence, . Repeating this line of reasoning times, it follows that, if , then . Hence , so .
If is real, define where is a sequence of rational numbers such that . The limit is well-defined and does not depend on the choice of sequence because of the monotonicity and continuity properties for rational exponents. The homogeneity, additivity, monotonicity, and homogeneity properties for real exponents follow from the properties for rational exponents by standard theorems on limits.
|Title||proof of properties of the exponential|
|Date of creation||2013-03-22 14:34:17|
|Last modified on||2013-03-22 14:34:17|
|Last modified by||rspuzio (6075)|