proof of properties of the exponential

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.

Homogeneity: This is a simple

consequence

of

commutativity

of

multiplication

–

x

⁢

y

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

a

⁢

c

<

b

⁢

d

. Applying this fact repeatedly to

x

<

y

shows that

x

p

<

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

,

x

p

may be defined as the

Dedekind cut

(

{

y

∈

ℚ

∣

y

n

<

x

m

}

,

{

z

∈

ℚ

∣

z

n

>

x

m

}

)

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

(

k

⁢

m

′

,

k

⁢

n

′

)

where

m

′

and

n

′

are relatively prime. The monotonicity

property

implies that

y

n

<

x

m

if and only if

y

n

′

<

x

m

′

.

2) If

y

n

<

x

m

and

z

n

>

x

m

then

y

<

z

.

By

transitivity

,

y

n

<

z

n

. By monotonicity, it follows that

y

<

z

.

3) At most one rational number can not belong to either

{

y

∈

ℚ

∣

y

n

<

x

m

}

or

{

z

∈

ℚ

⁢

∣

z

n

>

⁢

x

m

}

.

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

r

n

=

x

m

. Suppose that there were two rational numbers such that

r

1

n

=

x

m

and

r

2

n

=

x

m

. Then

r

1

n

=

r

2

n

. If

r

1

≠

r

2

, either

r

1

<

r

2

or

r

1

>

r

2

. Either way, motonicity implies that

r

1

n

=

r

2

n

, so

r

1

=

r

2

. Hence at most one rational number belongs to neither set, so one has a well-defined Dedekind cut which defines

x

p

when

p

is a rational number.

Homogeneity: The Dedekind cuts defining

x

p

and

y

p

are

(

{

u

∈

ℚ

∣

u

n

<

x

m

}

,

{

v

∈

ℚ

∣

v

n

>

x

m

}

)

and

(

{

u

∈

ℚ

∣

u

n

<

y

m

}

,

{

v

∈

ℚ

∣

v

n

>

y

m

}

)

respectively. By the homogeneity property for integer exponents, if

u

1

n

<

x

m

and

u

2

n

<

y

m

, then

(

u

1

⁢

u

2

)

n

<

(

x

⁢

y

)

m

. Likewise, if

v

1

n

>

x

m

and

v

2

n

>

y

m

, then

(

v

1

⁢

v

2

)

n

<

(

x

⁢

y

)

m

. By the

definition

of multiplication for Dedekind cuts, it follows that

x

p

⁢

y

p

=

(

x

⁢

y

)

p

for rational exponents

p

.

Additivity: Write

p

and

q

over a common

denominator

:

p

=

m

/

k

and

q

=

n

/

k

. Then

x

p

and

x

q

are determined by the Dedekind cuts

(

{

u

∈

ℚ

∣

u

k

<

x

m

}

,

{

v

∈

ℚ

∣

v

k

>

x

m

}

)

and

(

{

u

∈

ℚ

∣

u

k

<

x

n

}

,

{

v

∈

ℚ

∣

v

k

>

x

n

}

)

respectively. If

∣

u

1

k

<

x

m

and

u

2

k

<

x

n

, then

(

u

1

⁢

u

2

)

k

<

x

m

+

n

by additivity for integer exponents. Likewise, if

∣

v

1

k

>

⁢

x

m

and

v

2

k

>

x

n

, then

(

v

1

⁢

v

2

)

k

>

x

m

+

n

. By the definition of multiplication for Dedekind cuts, it follows that

x

p

⁢

x

q

=

x

p

+

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

x

p

and

x

q

are determined by the Dedekind cuts

(

{

u

∈

ℚ

∣

u

k

<

x

m

}

,

{

v

∈

ℚ

∣

v

k

>

x

m

}

)

and

(

{

u

∈

ℚ

∣

u

k

<

x

n

}

,

{

v

∈

ℚ

∣

v

k

>

x

n

}

)

respectively. If

∣

u

1

k

<

x

m

, then

u

1

k

<

x

n

since

x

m

<

x

n

by the law of monotonicity for integer exponents. Likewise,

∣

v

2

k

>

⁢

x

n

, then

v

2

k

>

x

m

. Hence, by the definition of “greater than” for Dedekind cuts,

x

p

<

x

q

.

Continuity: Because of the additivity property, it suffices to prove that

lim

p

→

0

⁡

x

p

=

1

. By monotonicity, it suffices to prove that

lim

n

→

∞

⁡

x

1

/

2

n

=

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

x

1

/

2

=

1

+

z

. By

simple algebra

, one has

z

=

y

-

z

2

2

By monotonicity,

z

>

0

. Hence,

z

≤

y

/

2

. Repeating this line of reasoning

n

times, it follows that, if

x

1

/

2

n

=

1

+

z

n

, then

z

n

≤

y

/

2

n

. Hence

lim

n

→

∞

⁡

z

n

=

0

, so

lim

n

→

∞

⁡

x

1

/

2

n

=

1

.

Real exponents

If

p

is real, define

x

p

=

lim

n

→

∞

⁡

x

r

n

where

r

n

is a

sequence

of rational numbers such that

lim

n

→

∞

⁡

r

n

=

p

. The limit is well-defined and does not depend on the choice of sequence

r

n

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

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

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