Proof real to the power of realAxiomatizing the (positive-based) exponential functionMy first simple direct proof (very simple theorem on real numbers). Please mark/grade.Dedekind cuts and circularityDo we need to prove that $b-a = b + (-a)$, where $a,b$ are real numbers?Trench Real Analysis real numbers problemUnique real number proofProving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.Real and Non-real Numbers; Value of Zero?Laws of Indices with real exponentsReal Analysis Real Number Inequality ProofIs using the fact that multiplication and addition are closed in an ordered field to prove that there are no “gaps” in the real number line circular?
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Proof real to the power of real
Axiomatizing the (positive-based) exponential functionMy first simple direct proof (very simple theorem on real numbers). Please mark/grade.Dedekind cuts and circularityDo we need to prove that $b-a = b + (-a)$, where $a,b$ are real numbers?Trench Real Analysis real numbers problemUnique real number proofProving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.Real and Non-real Numbers; Value of Zero?Laws of Indices with real exponentsReal Analysis Real Number Inequality ProofIs using the fact that multiplication and addition are closed in an ordered field to prove that there are no “gaps” in the real number line circular?
$begingroup$
Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?
real-analysis exponentiation real-numbers
edited 7 mins ago
José Carlos Santos
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
$endgroup$
add a comment |
$begingroup$
Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?
real-analysis exponentiation real-numbers
edited 7 mins ago
José Carlos Santos
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
$endgroup$
$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
2
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago
add a comment |
$begingroup$
Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?
real-analysis exponentiation real-numbers
edited 7 mins ago
José Carlos Santos
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
$endgroup$
Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?
real-analysis exponentiation real-numbers
real-analysis exponentiation real-numbers
edited 7 mins ago
José Carlos Santos
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
edited 7 mins ago
José Carlos Santos
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
edited 7 mins ago
José Carlos Santos
192k24148265
edited 7 mins ago
José Carlos Santos
192k24148265
edited 7 mins ago
José Carlos Santos
192k24148265
192k24148265
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
asked 8 hours ago
topologicalmagiciantopologicalmagician
40210
40210
$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
2
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago
add a comment |
$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
2
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago
$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
2
2
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally,
$$
a^p = exp(p ln a) .
$$
If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.
This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
$endgroup$
add a comment |
$begingroup$
Obviously, you must defined first the meaning of $a^p$, when $a,pinmathbb R$ and $pneq0$. But I have never seen a definition which defines, say, $(-1)^sqrt2$.
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
$endgroup$
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally,
$$
a^p = exp(p ln a) .
$$
If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.
This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
$endgroup$
add a comment |
$begingroup$
The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally,
$$
a^p = exp(p ln a) .
$$
If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.
This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
$endgroup$
add a comment |
$begingroup$
The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally,
$$
a^p = exp(p ln a) .
$$
If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.
This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
$endgroup$
The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally,
$$
a^p = exp(p ln a) .
$$
If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.
This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
edited 7 hours ago
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
answered 7 hours ago
Ethan BolkerEthan Bolker
50.5k558128
50.5k558128
add a comment |
add a comment |
$begingroup$
Obviously, you must defined first the meaning of $a^p$, when $a,pinmathbb R$ and $pneq0$. But I have never seen a definition which defines, say, $(-1)^sqrt2$.
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
$endgroup$
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
add a comment |
$begingroup$
Obviously, you must defined first the meaning of $a^p$, when $a,pinmathbb R$ and $pneq0$. But I have never seen a definition which defines, say, $(-1)^sqrt2$.
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
$endgroup$
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
add a comment |
$begingroup$
Obviously, you must defined first the meaning of $a^p$, when $a,pinmathbb R$ and $pneq0$. But I have never seen a definition which defines, say, $(-1)^sqrt2$.
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
$endgroup$
Obviously, you must defined first the meaning of $a^p$, when $a,pinmathbb R$ and $pneq0$. But I have never seen a definition which defines, say, $(-1)^sqrt2$.
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
answered 8 hours ago
José Carlos SantosJosé Carlos Santos
192k24148265
192k24148265
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
add a comment |
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
Oh, I see. But is last expression you wrote real or not?? And why? May you please elaborate?
$endgroup$
– topologicalmagician
8 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
$begingroup$
As I wrote, it depends upon how you define it.
$endgroup$
– José Carlos Santos
7 hours ago
add a comment |
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$begingroup$
Related.
$endgroup$
– Cameron Buie
7 hours ago
2
$begingroup$
I hope you’re specifying that $a$ be a positive real. Otherwise, you’re in serious trouble.
$endgroup$
– Lubin
7 hours ago
$begingroup$
@Lubin, that’s the point. I’m trying to see if it holds for any real a.
$endgroup$
– topologicalmagician
7 hours ago
$begingroup$
It’s pretty much meaningless for negative real numbers, isn’t it ?
$endgroup$
– Lubin
7 hours ago