Why is exponentiation applied right to left?
Question:
I am reading an Intro to Python textbook and came across this line:
Operators on the same row have equal precedence and are applied left to right, except for exponentiation, which is applied right to left.
I understand most of this, but I do not understand why they say exponentiation is applied right to left. They do not provide any examples either. Also, am I allowed to ask general questions like this, or are only problem solving questions preferred?
Answers:
from http://docs.python.org/reference/expressions.html
Operators in the same box group left to right (except for comparisons, including tests, which all have the same precedence and chain from left to right — see section Comparisons — and exponentiation, which groups from right to left).
>>> 2 ** 2 ** 2
16
>>> 2 ** 2 ** 2 ** 2
65536
>>> (2 ** 2 ** 2) ** 2
256
For the middle case 2 ** 2 ** 2 ** 2, this are the intermediate steps –
- broken down to
2 ** (2 ** (2 ** 2))
2 ** (2 ** (4)) # progressing right to left2 ** (16) # this is 2 to the power 16
which finally evals to 65536
Hope that helps!
This explanation seems quite clear to me. Let me show you an example that might enlighten this :
print 2 ** 2 ** 3 # prints 256
If you would read this from left to right, you would first do 2 ** 2, which would result in 4, and then 4 ** 3, which would give us 64.
It seems we have a wrong answer. 🙂
However, from right to left…
You would first do 2 ** 3, which would be 8, and then, 2 ** 8, giving us 256 !
I hope I was able to enlighten this point for you. 🙂
EDIT : Martijn Pieters answered more accurately to your question, sorry. I forgot to say it was mathematical conventions.
The ** operator follows normal mathematical conventions; it is right-associative:
In the usual computer science jargon, exponentiation in mathematics is right-associative, which means that xyz should be read as x(yz), not (xy)z. In expositions of the BODMAS rules that are careful enough to address this question, the rule is to evaluate the top exponent first.
and from Wikipedia on the Order of Operations:
If exponentiation is indicated by stacked symbols, the usual rule is to work from the top down, because exponention is right-associative in mathematics.
So 2 ** 3 ** 4 is calculated as 2 ** (3 ** 4) (== 2417851639229258349412352) not (2 ** 3) ** 4 (== 4096).
This is pretty universal across programming languages; it is called right-associativity, although there are exceptions, with Excel and MATLAB being the most notable.
Power operator, exponentiation, is handled differently across applications and languages.
If it has LEFT associativity then 2^3^4 = (2^3)^4 = 4096.
If it has RIGHT associativity then 2^3^4 = 2^(3^4) = 2417851639229260000000000.
In Excel, Matlab, Apple Numbers and more others exponentiation has LEFT associativity.
In Python, Ruby, Google Sheets, … – RIGHT associativity.
Here is a vast list of how different languages and apps handle exponentiation: Exponentiation Associativity and Standard Math Notation
I am reading an Intro to Python textbook and came across this line:
Operators on the same row have equal precedence and are applied left to right, except for exponentiation, which is applied right to left.
I understand most of this, but I do not understand why they say exponentiation is applied right to left. They do not provide any examples either. Also, am I allowed to ask general questions like this, or are only problem solving questions preferred?
from http://docs.python.org/reference/expressions.html
Operators in the same box group left to right (except for comparisons, including tests, which all have the same precedence and chain from left to right — see section Comparisons — and exponentiation, which groups from right to left).
>>> 2 ** 2 ** 2
16
>>> 2 ** 2 ** 2 ** 2
65536
>>> (2 ** 2 ** 2) ** 2
256
For the middle case 2 ** 2 ** 2 ** 2, this are the intermediate steps –
- broken down to
2 ** (2 ** (2 ** 2)) 2 ** (2 ** (4)) # progressing right to left2 ** (16) # this is 2 to the power 16
which finally evals to65536
Hope that helps!
This explanation seems quite clear to me. Let me show you an example that might enlighten this :
print 2 ** 2 ** 3 # prints 256
If you would read this from left to right, you would first do 2 ** 2, which would result in 4, and then 4 ** 3, which would give us 64.
It seems we have a wrong answer. 🙂
However, from right to left…
You would first do 2 ** 3, which would be 8, and then, 2 ** 8, giving us 256 !
I hope I was able to enlighten this point for you. 🙂
EDIT : Martijn Pieters answered more accurately to your question, sorry. I forgot to say it was mathematical conventions.
The ** operator follows normal mathematical conventions; it is right-associative:
In the usual computer science jargon, exponentiation in mathematics is right-associative, which means that xyz should be read as x(yz), not (xy)z. In expositions of the BODMAS rules that are careful enough to address this question, the rule is to evaluate the top exponent first.
and from Wikipedia on the Order of Operations:
If exponentiation is indicated by stacked symbols, the usual rule is to work from the top down, because exponention is right-associative in mathematics.
So 2 ** 3 ** 4 is calculated as 2 ** (3 ** 4) (== 2417851639229258349412352) not (2 ** 3) ** 4 (== 4096).
This is pretty universal across programming languages; it is called right-associativity, although there are exceptions, with Excel and MATLAB being the most notable.
Power operator, exponentiation, is handled differently across applications and languages.
If it has LEFT associativity then 2^3^4 = (2^3)^4 = 4096.
If it has RIGHT associativity then 2^3^4 = 2^(3^4) = 2417851639229260000000000.
In Excel, Matlab, Apple Numbers and more others exponentiation has LEFT associativity.
In Python, Ruby, Google Sheets, … – RIGHT associativity.
Here is a vast list of how different languages and apps handle exponentiation: Exponentiation Associativity and Standard Math Notation