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 inductionMathworldPlanetmath 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.

Homogeneity: This is a simple consequence of commutativity of multiplicationPlanetmathPlanetmathxy multiplied by itself p times can be rewritten as x multiplied by itself p times y multiplied by itself p times.

Additivity: This is a consequence of assocuiativity of multiplication – x multiplied by itself p+q times can be rebracketed as x multiplied by itself p times multiplied by x multiplied by itself q times.

Monotonicity: If a<b and c<d for positive integers a,b,c,d, then ac<bd. Applying this fact repeatedly to x<y shows that xp<y<p when x<y and p is a positive integer.

Continuity: This is irrelevant since the integers are discrete.

Rational exponents For rational exponent p=m/n, xp 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 m and n as long as p=m/n

Any pair of integers (m,n) such that p=m/n may uniquely be expressed as (km,kn) where m and n are relatively prime. The monotonicity property implies that yn<xm if and only if yn<xm.

2) If yn<xm and zn>xm then y<z.

By transitivity, yn<zn. By monotonicity, it follows that y<z.

3) At most one rational number can not belong to either {yyn<xm} or {zzn>xm}.

Given a rational number r, it is only possible for a rational number q to satisfy neither q<r nor q> if q=r. So for a rational number r to belong to neither set, it must be the case that rn=xm. Suppose that there were two rational numbers such that r1n=xm and r2n=xm. Then r1n=r2n. If r1r2, either r1<r2 or r1>r2. Either way, motonicity implies that r1n=r2n, so r1=r2. Hence at most one rational number belongs to neither set, so one has a well-defined Dedekind cut which defines xp when p is a rational number.

Homogeneity: The Dedekind cuts defining xp and yp are




respectively. By the homogeneity property for integer exponents, if u1n<xm and u2n<ym, then (u1u2)n<(xy)m. Likewise, if v1n>xm and v2n>ym, then (v1v2)n<(xy)m. By the definition of multiplication for Dedekind cuts, it follows that xpyp=(xy)p for rational exponents p.

Additivity: Write p and q over a common denominator: p=m/k and q=n/k. Then xp and xq are determined by the Dedekind cuts




respectively. If u1k<xm and u2k<xn, then (u1u2)k<xm+n by additivity for integer exponents. Likewise, if v1k>xm and v2k>xn, then (v1v2)k>xm+n. By the definition of multiplication for Dedekind cuts, it follows that xpxq=xp+q for rational exponents p and q.

Monotonicity: Suppose that p<q. Write p and q over a common denominator: p=m/k and q=n/k. Then m<n. Then xp and xq are determined by the Dedekind cuts




respectively. If u1k<xm, then u1k<xn since xm<xn by the law of monotonicity for integer exponents. Likewise, v2k>xn, then v2k>xm. Hence, by the definition of “greater than” for Dedekind cuts, xp<xq.

Continuity: Because of the additivity property, it suffices to prove that limp0xp=1. By monotonicity, it suffices to prove that limnx1/2n=1. Suppose that x>1. Write x=1+y. The considerations of last paragraph show that one can restrict attention to the case 0<y<1/4. Let x1/2=1+z. By simple algebra, one has


By monotonicity, z>0. Hence, zy/2. Repeating this line of reasoning n times, it follows that, if x1/2n=1+zn, then zny/2n. Hence limnzn=0, so limnx1/2n=1.

Real exponents

If p is real, define xp=limnxrn where rn is a sequenceMathworldPlanetmath of rational numbers such that limnrn=p. The limit is well-defined and does not depend on the choice of sequence rn 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 theoremsMathworldPlanetmath on limits.

Title proof of properties of the exponential
Canonical name ProofOfPropertiesOfTheExponential
Date of creation 2013-03-22 14:34:17
Last modified on 2013-03-22 14:34:17
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Proof
Classification msc 26A03