Step 1 uses the plus operator, which attempts to add a scalar value to each value of each column of the DataFrame. As the columns are all numeric, this operation works as expected. There are some missing values in each of the columns but they stay missing after the operation.

Mathematically, adding .005 should be enough so that the floor division in the next step correctly rounds to the nearest whole percentage. The trouble appears because of the inexactness of floating point numbers:

>>> .045 + .005
0.049999999999999996

There is an extra .00001 added to each number to ensure that the floating point representation has the first four digits the same as the actual value. This works because the maximum precision of all the points in the dataset is four decimal places.

Step 2 applies the floor division operator, //, to all the values in the DataFrame. As we are dividing by a fraction, in essence, it is multiplying each value by 100 and truncating any decimals. Parentheses are needed around the first part of the expression, as floor division has higher precedence than addition. Step 3 uses the division operator to return the decimal to the correct position.

In step 4, we reproduce the previous steps with the round method. Before we can do this, we must again add an extra .00001 to each DataFrame value for a different reason from step 1. NumPy and Python 3 round numbers that are exactly halfway between either side to the even number. This ties to the even (http://bit.ly/2x3V5TU) technique is not usually what is formally taught in schools. It does not consistently bias numbers to the higher side (http://bit.ly/2zhsPy8).

It is necessary here to round up so that both DataFrame values are equal. The equals method determines if all the elements and indexes between two DataFrames are exactly the same and returns a boolean.