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The concept of an real irrational power [duplicate]
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What Is Exponentiation?
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One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times:
$$x^n=underbracexcdot xcdotdotsbcdot x_n$$
We can also get the idea of a rational power, $x^fracpq=sqrt[q]x^p$ , looking for the $q$-th root of a number, where $q$ is a integer, so it is the same idea as a power of a natural number.
But how can we understand a power of a irrational number $x^r$? Of course we can define it as the limit of the power of the rational number who is tending to the number $r$, but is there a better more intuitive way of explaining this, staying with the number $r$, instead of its approximation?
algebra-precalculus exponentiation
asked Jul 22 at 23:23
David
184
marked as duplicate by dxiv, Xander Henderson, Ross Millikan, Mark S., amWhy Jul 23 at 0:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |Â
up vote
2
down vote
favorite
This question already has an answer here:
What Is Exponentiation?
11 answers
One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times:
$$x^n=underbracexcdot xcdotdotsbcdot x_n$$
We can also get the idea of a rational power, $x^fracpq=sqrt[q]x^p$ , looking for the $q$-th root of a number, where $q$ is a integer, so it is the same idea as a power of a natural number.
But how can we understand a power of a irrational number $x^r$? Of course we can define it as the limit of the power of the rational number who is tending to the number $r$, but is there a better more intuitive way of explaining this, staying with the number $r$, instead of its approximation?
algebra-precalculus exponentiation
asked Jul 22 at 23:23
David
184
marked as duplicate by dxiv, Xander Henderson, Ross Millikan, Mark S., amWhy Jul 23 at 0:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
4
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
This question already has an answer here:
What Is Exponentiation?
11 answers
One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times:
$$x^n=underbracexcdot xcdotdotsbcdot x_n$$
We can also get the idea of a rational power, $x^fracpq=sqrt[q]x^p$ , looking for the $q$-th root of a number, where $q$ is a integer, so it is the same idea as a power of a natural number.
But how can we understand a power of a irrational number $x^r$? Of course we can define it as the limit of the power of the rational number who is tending to the number $r$, but is there a better more intuitive way of explaining this, staying with the number $r$, instead of its approximation?
algebra-precalculus exponentiation
asked Jul 22 at 23:23
David
184
This question already has an answer here:
What Is Exponentiation?
11 answers
One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times:
$$x^n=underbracexcdot xcdotdotsbcdot x_n$$
We can also get the idea of a rational power, $x^fracpq=sqrt[q]x^p$ , looking for the $q$-th root of a number, where $q$ is a integer, so it is the same idea as a power of a natural number.
But how can we understand a power of a irrational number $x^r$? Of course we can define it as the limit of the power of the rational number who is tending to the number $r$, but is there a better more intuitive way of explaining this, staying with the number $r$, instead of its approximation?
This question already has an answer here:
What Is Exponentiation?
11 answers
algebra-precalculus exponentiation
asked Jul 22 at 23:23
David
184
edited Jul 24 at 20:53
asked Jul 22 at 23:23
David
184
asked Jul 22 at 23:23
David
184
asked Jul 22 at 23:23
David
184
184
marked as duplicate by dxiv, Xander Henderson, Ross Millikan, Mark S., amWhy Jul 23 at 0:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by dxiv, Xander Henderson, Ross Millikan, Mark S., amWhy Jul 23 at 0:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
4
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24
add a comment |Â
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
4
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
4
4
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24
add a comment |Â
1 Answer
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I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via:
$$a^r = exp left(r log(a)right)$$
where $exp(u)$ denotes the exponential function $u mapsto e^u$ and $log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $emapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $log x$ as the integral $int_1^x frac1t dt$, and then define $exp x$ as the inverse function of $log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $log$ and $exp$ are defined, $a^r$ can be defined as above.
answered Jul 22 at 23:36
mweiss
16.8k23267
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via:
$$a^r = exp left(r log(a)right)$$
where $exp(u)$ denotes the exponential function $u mapsto e^u$ and $log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $emapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $log x$ as the integral $int_1^x frac1t dt$, and then define $exp x$ as the inverse function of $log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $log$ and $exp$ are defined, $a^r$ can be defined as above.
answered Jul 22 at 23:36
mweiss
16.8k23267
add a comment |Â
up vote
2
down vote
accepted
I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via:
$$a^r = exp left(r log(a)right)$$
where $exp(u)$ denotes the exponential function $u mapsto e^u$ and $log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $emapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $log x$ as the integral $int_1^x frac1t dt$, and then define $exp x$ as the inverse function of $log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $log$ and $exp$ are defined, $a^r$ can be defined as above.
answered Jul 22 at 23:36
mweiss
16.8k23267
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via:
$$a^r = exp left(r log(a)right)$$
where $exp(u)$ denotes the exponential function $u mapsto e^u$ and $log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $emapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $log x$ as the integral $int_1^x frac1t dt$, and then define $exp x$ as the inverse function of $log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $log$ and $exp$ are defined, $a^r$ can be defined as above.
answered Jul 22 at 23:36
mweiss
16.8k23267
I don't know if you'd consider it "intuitive" or not, but the usual way to define an irrational power of a (positive) real number is via:
$$a^r = exp left(r log(a)right)$$
where $exp(u)$ denotes the exponential function $u mapsto e^u$ and $log$ denotes the logarithm base $e$.
Now, I hear you ask, isn't this circular, in that $emapsto e^u$ an example of precisely the kind of exponentiation the question is asking about? And the answer to that is: we can define $log x$ as the integral $int_1^x frac1t dt$, and then define $exp x$ as the inverse function of $log x$. Neither of those definitions requires any restriction on what type of number $x$ is. Once $log$ and $exp$ are defined, $a^r$ can be defined as above.
answered Jul 22 at 23:36
mweiss
16.8k23267
answered Jul 22 at 23:36
mweiss
16.8k23267
answered Jul 22 at 23:36
mweiss
16.8k23267
answered Jul 22 at 23:36
mweiss
16.8k23267
16.8k23267
add a comment |Â
add a comment |Â
You might want to start by reading It ain't no repeated addition (and the follow-ups). Basically, while the multiplication of two natural numbers can be seen as repeated addition, this is not really the right mathematical definition. Similarly, while exponentiation can be seen as repeated multiplication, this isn't really right. More typically, exponentiation is defined via a Taylor series expansion or via a differential equation.
– Xander Henderson
Jul 22 at 23:28
4
Duplicate of What Is Exponentiation?. See also What does $2^x$ really mean when $x$ is not an integer?
– dxiv
Jul 22 at 23:29
"how can we understand a power of a[n] irrational number $x^r$?" Perhaps you meant to ask how we understand an irrational power of a (positive real) number $x$. In any case one can define it by an approximating sequence.
– hardmath
Jul 22 at 23:30
@dxiv I was looking more for an explaination for real powers, so I tought that "what is exponentiation" is for all powers including complex powers. but a friend of mine added another answer . So now it adresses my question good enough...
– David
Jul 29 at 21:24