Mixing
Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and â„�
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and â„�
So far, I have only found this:
http://andrusia.com/math/preliminaries/ExponentiationTheorems.pdf
But I think this only works for exponents that are members of ℕ, and I don't even know if what is offered in that document is an explanation more than a proof, since they should be proven using induction and not ellipsis "..." . Do you consider these are correct proofs?
On the other hand I haven't been able to find over the internet the proofs for exponents members of ℤ, ℚ, ℙ, and ℠(where ℙ = â„Âℚ, to clarify what ℙ is).
Do you know where can these proofs be found?
Thanks in advance.
algebra-precalculus proof-writing exponential-function exponentiation
edited Jul 19 at 11:08
Fytch
159117
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
 |Â
show 2 more comments
up vote
2
down vote
favorite
Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and â„�
So far, I have only found this:
http://andrusia.com/math/preliminaries/ExponentiationTheorems.pdf
But I think this only works for exponents that are members of ℕ, and I don't even know if what is offered in that document is an explanation more than a proof, since they should be proven using induction and not ellipsis "..." . Do you consider these are correct proofs?
On the other hand I haven't been able to find over the internet the proofs for exponents members of ℤ, ℚ, ℙ, and ℠(where ℙ = â„Âℚ, to clarify what ℙ is).
Do you know where can these proofs be found?
Thanks in advance.
algebra-precalculus proof-writing exponential-function exponentiation
edited Jul 19 at 11:08
Fytch
159117
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
2
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
2
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
1
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30
 |Â
show 2 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and â„�
So far, I have only found this:
http://andrusia.com/math/preliminaries/ExponentiationTheorems.pdf
But I think this only works for exponents that are members of ℕ, and I don't even know if what is offered in that document is an explanation more than a proof, since they should be proven using induction and not ellipsis "..." . Do you consider these are correct proofs?
On the other hand I haven't been able to find over the internet the proofs for exponents members of ℤ, ℚ, ℙ, and ℠(where ℙ = â„Âℚ, to clarify what ℙ is).
Do you know where can these proofs be found?
Thanks in advance.
algebra-precalculus proof-writing exponential-function exponentiation
edited Jul 19 at 11:08
Fytch
159117
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and â„�
So far, I have only found this:
http://andrusia.com/math/preliminaries/ExponentiationTheorems.pdf
But I think this only works for exponents that are members of ℕ, and I don't even know if what is offered in that document is an explanation more than a proof, since they should be proven using induction and not ellipsis "..." . Do you consider these are correct proofs?
On the other hand I haven't been able to find over the internet the proofs for exponents members of ℤ, ℚ, ℙ, and ℠(where ℙ = â„Âℚ, to clarify what ℙ is).
Do you know where can these proofs be found?
Thanks in advance.
algebra-precalculus proof-writing exponential-function exponentiation
edited Jul 19 at 11:08
Fytch
159117
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
edited Jul 19 at 11:08
Fytch
159117
edited Jul 19 at 11:08
Fytch
159117
edited Jul 19 at 11:08
Fytch
159117
159117
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
asked Jul 19 at 3:17
Daniel Bonilla Jaramillo
38819
38819
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
2
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
2
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
1
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30
 |Â
show 2 more comments
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
2
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
2
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
1
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
2
2
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
2
2
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
1
1
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30
 |Â
show 2 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2856220%2fwhere-can-i-find-the-proofs-of-the-properties-of-exponentiation-when-powers-are%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
so $mathbb P$ is the irrational numbers?
– spaceisdarkgreen
Jul 19 at 3:22
Yes; I always have problems giving irrational numbers a sign that anybody can understand.
– Daniel Bonilla Jaramillo
Jul 19 at 3:25
2
Yes, it is odd that there is no standard notation for them. You can say $mathbb Rsetminus mathbb Q.$ Actually, here you don't really need to mention them... the function for $x,yinmathbb R$ will agree with the function for $x,yinmathbb Q$ on $mathbb Q$... so asking about $mathbb R$ is really just asking about how to extend to the irrationals.
– spaceisdarkgreen
Jul 19 at 3:28
2
Before you can prove it for non-naturals you must define it for non-naturals. Usually these are easily proven by applying the definition. Example if $a =frac pqin mathbb Q$ then $x^a= (sqrt[q]x)^p$ and the proofs follow.
– fleablood
Jul 19 at 3:29
1
The negative reals and the non-negative rules behave differently under exponent.
– Q the Platypus
Jul 19 at 3:30