Factorials about decimals
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How do you get the factorial of a decimal number using a pen and paper if it is possible?
Example: Find the factorial of $0.5!$
factorial gamma-function
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up vote
1
down vote
favorite
How do you get the factorial of a decimal number using a pen and paper if it is possible?
Example: Find the factorial of $0.5!$
factorial gamma-function
What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
How do you get the factorial of a decimal number using a pen and paper if it is possible?
Example: Find the factorial of $0.5!$
factorial gamma-function
How do you get the factorial of a decimal number using a pen and paper if it is possible?
Example: Find the factorial of $0.5!$
factorial gamma-function
edited Aug 20 at 11:24
drhab
87.8k541119
87.8k541119
asked Aug 20 at 11:21
MMJM
225
225
What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46
add a comment |Â
What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46
What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46
add a comment |Â
2 Answers
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up vote
1
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You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to:
$$
Gamma(1.5) = int_0^infty sqrt x e^-x dx
$$
and it turns out to be $frac12sqrtpi$.
add a comment |Â
up vote
0
down vote
You can use Stirling's approximation, but it won't work well for small numbers like $frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! approx sqrt2pi nBigl(frac neBigr)^n.$$
For example, $12.7!approx 2.8616times10^9$ and the Stirling formula gives around $2.843times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+frac12$ your best bet is just to know that $$frac12! = frac12sqrtpi$$
and then use the rule that $(x+1)! = (x+1)x!$ so for example $frac32! = frac32cdot frac12! = frac34sqrtpi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $xmapsto frac1x!$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to:
$$
Gamma(1.5) = int_0^infty sqrt x e^-x dx
$$
and it turns out to be $frac12sqrtpi$.
add a comment |Â
up vote
1
down vote
accepted
You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to:
$$
Gamma(1.5) = int_0^infty sqrt x e^-x dx
$$
and it turns out to be $frac12sqrtpi$.
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to:
$$
Gamma(1.5) = int_0^infty sqrt x e^-x dx
$$
and it turns out to be $frac12sqrtpi$.
You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to:
$$
Gamma(1.5) = int_0^infty sqrt x e^-x dx
$$
and it turns out to be $frac12sqrtpi$.
answered Aug 20 at 11:31
Arthur
101k793176
101k793176
add a comment |Â
add a comment |Â
up vote
0
down vote
You can use Stirling's approximation, but it won't work well for small numbers like $frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! approx sqrt2pi nBigl(frac neBigr)^n.$$
For example, $12.7!approx 2.8616times10^9$ and the Stirling formula gives around $2.843times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+frac12$ your best bet is just to know that $$frac12! = frac12sqrtpi$$
and then use the rule that $(x+1)! = (x+1)x!$ so for example $frac32! = frac32cdot frac12! = frac34sqrtpi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $xmapsto frac1x!$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
add a comment |Â
up vote
0
down vote
You can use Stirling's approximation, but it won't work well for small numbers like $frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! approx sqrt2pi nBigl(frac neBigr)^n.$$
For example, $12.7!approx 2.8616times10^9$ and the Stirling formula gives around $2.843times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+frac12$ your best bet is just to know that $$frac12! = frac12sqrtpi$$
and then use the rule that $(x+1)! = (x+1)x!$ so for example $frac32! = frac32cdot frac12! = frac34sqrtpi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $xmapsto frac1x!$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You can use Stirling's approximation, but it won't work well for small numbers like $frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! approx sqrt2pi nBigl(frac neBigr)^n.$$
For example, $12.7!approx 2.8616times10^9$ and the Stirling formula gives around $2.843times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+frac12$ your best bet is just to know that $$frac12! = frac12sqrtpi$$
and then use the rule that $(x+1)! = (x+1)x!$ so for example $frac32! = frac32cdot frac12! = frac34sqrtpi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $xmapsto frac1x!$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
You can use Stirling's approximation, but it won't work well for small numbers like $frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! approx sqrt2pi nBigl(frac neBigr)^n.$$
For example, $12.7!approx 2.8616times10^9$ and the Stirling formula gives around $2.843times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+frac12$ your best bet is just to know that $$frac12! = frac12sqrtpi$$
and then use the rule that $(x+1)! = (x+1)x!$ so for example $frac32! = frac32cdot frac12! = frac34sqrtpi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $xmapsto frac1x!$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
edited Aug 20 at 11:49
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MJD
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What do you call the factorial of a decimal number ? Do you refer to the $Gamma$ function ? If yes you have to calculate the integral defined by $Gamma$.
â tmaths
Aug 20 at 11:24
A quick google search of "factorial real number" lead to this question. So I think it's a duplicate
â F.Carette
Aug 20 at 11:28
The other question asks for the definition; this question asks for methods of calculating numeric values with pen and paper.
â MJD
Aug 20 at 11:46