pandas focuses on tabular data structures and when doing the operations (addition, subtraction etc.) it looks at the labels - not positions.
Consider the following DataFrame:
df = pd.DataFrame(np.random.randn(5, 3), index=list('abcde'), columns=list('xyz'))
Here, df[1:] is:
df[1:]
Out:
x y z
b 1.003035 0.172960 1.160033
c 0.117608 -1.114294 -0.557413
d -1.312315 1.171520 -1.034012
e -0.380719 -0.422896 1.073535
And df[:-1] is:
df[:-1]
Out:
x y z
a 1.367916 1.087607 -0.625777
b 1.003035 0.172960 1.160033
c 0.117608 -1.114294 -0.557413
d -1.312315 1.171520 -1.034012
If you do df[1:] / df[:-1] it will divide row b's by row b's, row c's by row c's and row d's by row d's. For row a and e, it will not be able to find corresponding rows in the other DataFrame (either in the first one or in the second one) so it will return nan:
df[1:] / df[:-1]
Out:
x y z
a NaN NaN NaN
b 1.0 1.0 1.0
c 1.0 1.0 1.0
d 1.0 1.0 1.0
e NaN NaN NaN
If you just want to do element-wise division ignoring the labels, accessing the underlying numpy array by .values for one of the frames is a way of telling pandas to ignore labels. Since numpy arrays don't have labels, pandas will just do element-wise operations:
df[1:]/df[:-1].values
Out:
x y z
b 0.733258 0.159028 -1.853749
c 0.117252 -6.442482 -0.480515
d -11.158359 -1.051357 1.855018
e 0.290112 -0.360981 -1.038223