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The operator table in the Calculations chapter ends with two operators that you will not find on a calculator keyboard: % and **. This chapter is about those two, and about the kind of problem each one solves.

Operator	Name	What it does	Example	Result
`//`	floor division	how many times one number fits into another	`17 // 5`	`3`
`%`	modulo	what is left over after that	`17 % 5`	`2`
`**`	exponentiation	raising a number to a power	`4 ** 2`	`16`

Modulo: what is left over

% is called modulo, or mod for short. It divides the first number by the second and gives you the remainder: the part that is left over once the second number has gone in as many whole times as it can.

Share 17 sweets between 5 people. Everybody gets 3, which uses up 15, and there are 2 left over. Floor division answers the first half of that, and modulo answers the second half:

print(17 // 5)
print(17 % 5)

3
2

The % sign has nothing to do with percentages here. In Python it always means the remainder.

Is it even?

A number is even when it divides by 2 with nothing left over, so number % 2 is 0 for every even number and 1 for every odd one:

print(8 % 2)
print(7 % 2)

0
1

The same trick works for any number. If number % 3 is 0, the number is in the three times table.

// and % together

Floor division and modulo are a pair: one says how many whole groups there are, the other says how much is left over. Used together, they split an amount into two units. 130 seconds is 2 whole minutes, with 10 seconds left over:

total_seconds = 130

minutes = total_seconds // 60
seconds = total_seconds % 60

print(str(minutes) + " minutes and " + str(seconds) + " seconds")

Change total_seconds and run it again. Try 60, 59 and 3600, and predict each answer before you run it.

Exponentiation: powers

** is called exponentiation. It raises the first number to the power of the second: 4 ** 2 is 4 squared, which is 4 times 4, and 4 ** 3 is 4 cubed, which is 4 times 4 times 4.

print(4 ** 2)
print(4 ** 3)
print(2 ** 10)

16
64
1024

On paper you would write the power small and raised: 4². There is no way to type that into code, so Python writes it as two stars. One star multiplies, two stars raise to a power.

The area of a square is its side squared and the volume of a cube is its side cubed, which makes ** the natural operator for both:

side_length = 3

area = side_length ** 2
volume = side_length ** 3

print("Area is " + str(area))
print("Volume is " + str(volume))

Powers work on floats as well. 2.5 ** 2 is 6.25, and a float to a power is always a float, so 4.0 ** 2 is 16.0, not 16.

Help! My code doesn't work!

Make sure that you check for the following things:

The two stars in ** are written next to each other, with no space between them.
% gives the remainder, not a percentage, and // gives the number of whole times. Check which of the two the question is asking for.
A number typed by the user is wrapped in int() or float() before you calculate with it.
Every number that is concatenated into a print is wrapped in str().