Why 0 ** 0 equals 1 in python
Why does 0 ** 0 equal 1 in Python? Shouldn't it throw an exception, like 0 / 0 does?
python math
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
|
show 4 more comments
Why does 0 ** 0 equal 1 in Python? Shouldn't it throw an exception, like 0 / 0 does?
python math
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
9
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
6
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
8
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
3
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
5
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50
|
show 4 more comments
Why does 0 ** 0 equal 1 in Python? Shouldn't it throw an exception, like 0 / 0 does?
python math
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
Why does 0 ** 0 equal 1 in Python? Shouldn't it throw an exception, like 0 / 0 does?
python math
python math
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
edited Jan 19 '13 at 12:39
Jon Clements♦
101k19178221
101k19178221
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
asked Jan 19 '13 at 12:37
kasperskykaspersky
2,43422042
2,43422042
9
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
6
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
8
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
3
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
5
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50
|
show 4 more comments
9
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
6
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
8
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
3
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
5
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50
9
9
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
6
6
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
8
8
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
3
3
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
5
5
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50
|
show 4 more comments
2 Answers
2
active
oldest
votes
Wikipedia has interesting coverage of the history and the differing points of view on the value of 0 ** 0:
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that
0 ** 0 = 1, until in 1821 Cauchy listed0 ** 0along with expressions like0⁄0in a table of undefined forms. In the 1830s Libri published an unconvincing argument for0 ** 0 = 1, and Möbius sided with him...
As applied to computers, IEEE 754 recommends several functions for computing a power. It defines pow(0, 0) and pown(0, 0) as returning 1, and powr(0, 0) as returning NaN.
Most programming languages follow the convention that 0 ** 0 == 1. Python is no exception, both for integer and floating-point arguments.
edited Jan 3 at 7:22
tripleee
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
add a comment |
consider x^x:
Using limits we can easily get to our solution and rearranging x^x we get :
x^x= exp(log(x^x))
Now , we have from:
lim x->0 exp(log(x^x))= exp(lim x->0 xlog(x)) = exp(lim x->0 log(x)/(x^-1))
Applying L'Hôpital rule , we get :
exp(lim x^-1/(-x^-2)) = exp(lim x->0 -x) = exp(0) = 1=x^x
But according to Wolfram Alpha 0**0 is indeterminate and following explanations were obtained by them :
0^0 itself is undefined. The lack of a well-defined meaning for this
quantity follows from the mutually contradictory facts that a^0 is
always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0
should equal 0. It could be argued that 0^0=1 is a natural definition
since lim_(n->0)n^n=lim_(n->0^+)n^n=lim_(n->0^-)n^n=1.
However, the limit does not exist for general complex values of n. Therefore, the choice of
definition for 0^0 is usually defined to be indeterminate."
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
add a comment |
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2 Answers
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2 Answers
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Wikipedia has interesting coverage of the history and the differing points of view on the value of 0 ** 0:
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that
0 ** 0 = 1, until in 1821 Cauchy listed0 ** 0along with expressions like0⁄0in a table of undefined forms. In the 1830s Libri published an unconvincing argument for0 ** 0 = 1, and Möbius sided with him...
As applied to computers, IEEE 754 recommends several functions for computing a power. It defines pow(0, 0) and pown(0, 0) as returning 1, and powr(0, 0) as returning NaN.
Most programming languages follow the convention that 0 ** 0 == 1. Python is no exception, both for integer and floating-point arguments.
edited Jan 3 at 7:22
tripleee
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
add a comment |
Wikipedia has interesting coverage of the history and the differing points of view on the value of 0 ** 0:
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that
0 ** 0 = 1, until in 1821 Cauchy listed0 ** 0along with expressions like0⁄0in a table of undefined forms. In the 1830s Libri published an unconvincing argument for0 ** 0 = 1, and Möbius sided with him...
As applied to computers, IEEE 754 recommends several functions for computing a power. It defines pow(0, 0) and pown(0, 0) as returning 1, and powr(0, 0) as returning NaN.
Most programming languages follow the convention that 0 ** 0 == 1. Python is no exception, both for integer and floating-point arguments.
edited Jan 3 at 7:22
tripleee
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
add a comment |
Wikipedia has interesting coverage of the history and the differing points of view on the value of 0 ** 0:
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that
0 ** 0 = 1, until in 1821 Cauchy listed0 ** 0along with expressions like0⁄0in a table of undefined forms. In the 1830s Libri published an unconvincing argument for0 ** 0 = 1, and Möbius sided with him...
As applied to computers, IEEE 754 recommends several functions for computing a power. It defines pow(0, 0) and pown(0, 0) as returning 1, and powr(0, 0) as returning NaN.
Most programming languages follow the convention that 0 ** 0 == 1. Python is no exception, both for integer and floating-point arguments.
edited Jan 3 at 7:22
tripleee
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
Wikipedia has interesting coverage of the history and the differing points of view on the value of 0 ** 0:
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that
0 ** 0 = 1, until in 1821 Cauchy listed0 ** 0along with expressions like0⁄0in a table of undefined forms. In the 1830s Libri published an unconvincing argument for0 ** 0 = 1, and Möbius sided with him...
As applied to computers, IEEE 754 recommends several functions for computing a power. It defines pow(0, 0) and pown(0, 0) as returning 1, and powr(0, 0) as returning NaN.
Most programming languages follow the convention that 0 ** 0 == 1. Python is no exception, both for integer and floating-point arguments.
edited Jan 3 at 7:22
tripleee
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
edited Jan 3 at 7:22
tripleee
94.7k13133187
edited Jan 3 at 7:22
tripleee
94.7k13133187
edited Jan 3 at 7:22
tripleee
94.7k13133187
94.7k13133187
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
answered Jan 19 '13 at 12:43
NPENPE
356k67759889
356k67759889
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
add a comment |
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
1
1
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
I always thought that 0 ^ 0 is undefined, just like division by 0, but never thought it could be so tricky. I think the explanation on wikipedia is very good, also explaining why other people think it should be 1, and it also covers the answer to my question, related to Python.
– kaspersky
Jan 19 '13 at 13:05
add a comment |
consider x^x:
Using limits we can easily get to our solution and rearranging x^x we get :
x^x= exp(log(x^x))
Now , we have from:
lim x->0 exp(log(x^x))= exp(lim x->0 xlog(x)) = exp(lim x->0 log(x)/(x^-1))
Applying L'Hôpital rule , we get :
exp(lim x^-1/(-x^-2)) = exp(lim x->0 -x) = exp(0) = 1=x^x
But according to Wolfram Alpha 0**0 is indeterminate and following explanations were obtained by them :
0^0 itself is undefined. The lack of a well-defined meaning for this
quantity follows from the mutually contradictory facts that a^0 is
always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0
should equal 0. It could be argued that 0^0=1 is a natural definition
since lim_(n->0)n^n=lim_(n->0^+)n^n=lim_(n->0^-)n^n=1.
However, the limit does not exist for general complex values of n. Therefore, the choice of
definition for 0^0 is usually defined to be indeterminate."
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
add a comment |
consider x^x:
Using limits we can easily get to our solution and rearranging x^x we get :
x^x= exp(log(x^x))
Now , we have from:
lim x->0 exp(log(x^x))= exp(lim x->0 xlog(x)) = exp(lim x->0 log(x)/(x^-1))
Applying L'Hôpital rule , we get :
exp(lim x^-1/(-x^-2)) = exp(lim x->0 -x) = exp(0) = 1=x^x
But according to Wolfram Alpha 0**0 is indeterminate and following explanations were obtained by them :
0^0 itself is undefined. The lack of a well-defined meaning for this
quantity follows from the mutually contradictory facts that a^0 is
always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0
should equal 0. It could be argued that 0^0=1 is a natural definition
since lim_(n->0)n^n=lim_(n->0^+)n^n=lim_(n->0^-)n^n=1.
However, the limit does not exist for general complex values of n. Therefore, the choice of
definition for 0^0 is usually defined to be indeterminate."
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
add a comment |
consider x^x:
Using limits we can easily get to our solution and rearranging x^x we get :
x^x= exp(log(x^x))
Now , we have from:
lim x->0 exp(log(x^x))= exp(lim x->0 xlog(x)) = exp(lim x->0 log(x)/(x^-1))
Applying L'Hôpital rule , we get :
exp(lim x^-1/(-x^-2)) = exp(lim x->0 -x) = exp(0) = 1=x^x
But according to Wolfram Alpha 0**0 is indeterminate and following explanations were obtained by them :
0^0 itself is undefined. The lack of a well-defined meaning for this
quantity follows from the mutually contradictory facts that a^0 is
always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0
should equal 0. It could be argued that 0^0=1 is a natural definition
since lim_(n->0)n^n=lim_(n->0^+)n^n=lim_(n->0^-)n^n=1.
However, the limit does not exist for general complex values of n. Therefore, the choice of
definition for 0^0 is usually defined to be indeterminate."
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
consider x^x:
Using limits we can easily get to our solution and rearranging x^x we get :
x^x= exp(log(x^x))
Now , we have from:
lim x->0 exp(log(x^x))= exp(lim x->0 xlog(x)) = exp(lim x->0 log(x)/(x^-1))
Applying L'Hôpital rule , we get :
exp(lim x^-1/(-x^-2)) = exp(lim x->0 -x) = exp(0) = 1=x^x
But according to Wolfram Alpha 0**0 is indeterminate and following explanations were obtained by them :
0^0 itself is undefined. The lack of a well-defined meaning for this
quantity follows from the mutually contradictory facts that a^0 is
always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0
should equal 0. It could be argued that 0^0=1 is a natural definition
since lim_(n->0)n^n=lim_(n->0^+)n^n=lim_(n->0^-)n^n=1.
However, the limit does not exist for general complex values of n. Therefore, the choice of
definition for 0^0 is usually defined to be indeterminate."
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
edited Jan 19 '13 at 13:23
Ashwini Chaudhary
179k35325391
179k35325391
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
answered Jan 19 '13 at 12:59
Pj_Pj_
771414
771414
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
add a comment |
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
You also could show that lim x->0 C/x = Inf (where C is some constant), but still, division by zero is undefined. So, C/x, when x->0 isn't the same as C/0. The same principle apply for x^x, you showed it is 1 when x->0, but that doesn't tell anything about the actual 0^0.
– kaspersky
Jan 19 '13 at 18:40
great answer :P
– Mitul Shah
May 8 '14 at 8:59
great answer :P
– Mitul Shah
May 8 '14 at 8:59
add a comment |
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9
Since x^0 = 1...
– Anders Lindahl
Jan 19 '13 at 12:39
6
Because it should equal 1?
– Martijn Pieters♦
Jan 19 '13 at 12:41
8
@AndersLindahl: Calculus teaches us that 0^0 is an indeterminate form. Hence why the OP is asking.
– Michael Foukarakis
Jan 19 '13 at 12:42
3
+1: For enlightening me.
– Abhijit
Jan 19 '13 at 12:50
5
@AndersLindahl, oh please, I could say that 0^x = 0...
– kaspersky
Jan 19 '13 at 12:50