5.6. Operations¶
5.6.1. Operators and Operands¶
Now that we can store data in variables, let’s explore how we can generate new data from existing data.
An operator is one or more characters that represents a computation like addition, multiplication, or division. The values an operator works on are called operands.
The following are all legal Python expressions whose meaning is more or less clear:
20 + 32
hour  1
hour * 60 + minute
minute / 60
5 ** 2
(5 + 9) * (15  7)
For example, in the calculation 20 + 32
, the operator is +
and the operands are 20
and 32
.
The symbols +
and 
, and the use of parentheses for grouping, mean in Python what they mean in mathematics. The asterisk (*
) is the symbol for multiplication, and **
is the symbol for exponentiation. Addition, subtraction, multiplication, and exponentiation all do what you expect.
Example
1 2 3 4 5  print(2 + 3)
print(2  3)
print(2 * 3)
print(2 ** 3)
print(3 ** 2)

Console Output
5
1
6
8
9
We use the same terminology as before, stating that 2 + 3
returns the value 5
.
When a variable name appears in the place of an operand, it is replaced with the value that it refers to before the operation is performed. For example, suppose that we wanted to convert 645 minutes into hours. Division is denoted by the operator /
.
Example
1 2 3  minutes = 645
hours = minutes / 60
print(hours)

Console Output
10.75
In summary, operators and operands can be combined to create expressions that are evaluated upon execution. Let’s discuss some specific types of operators
5.6.2. Arithmetic Operators¶
Some of most commonlyused operators are the arithmetic operators, which carry out basic mathematical operations. These behave exactly as you are used to, though the modulus operator (%
) may be new to you.
Operator 
Description 
Example 

Addition ( 
Adds the two operands 

Subtraction ( 
Subtracts the two operands 

Multiplication ( 
Multiplies the two operands 

Division ( 
Divides the first operand by the second 

Modulus ( 
Aka the remainder operator. Returns the integer remainder of dividing the two operands. 

Exponentiation ( 
Calculates the base (first operand) to the exponent (second operand) power, that is, base^{exponent} 

Floor Division ( 
Returns the integral or whole number version of the quotient. 

While the modulus operator (%
) is common in programming, it is not used much
outside of programming. Let’s explore how it works with a few examples.
The %
operator returns the remainder obtained by carrying out integer division of the first operand by the second operand. Therefore, 5 % 3
is 2
because 3 goes into 5 one whole time, with a remainder of 2 left over.
Examples
12 % 4 is 0, because 4 divides 12 evenly (that is, there is no remainder)
13 % 7 is 6
6 % 2 is 0
7 % 2 is 1
The last two examples illustrate a general rule: An integer x is even exactly
when x % 2
is 0
and is odd exactly when x % 2
is 1
.
Note
The value returned by a % b
will be in the range from 0
to b
(not including b
).
Tip
If remainders and the modulus operator seem tricky to you, we recommend getting additional practice at Khan Academy.
5.6.3. Order of Operations¶
When more than one operator appears in an expression, the order of evaluation depends on the rules of precedence. Python follows the same precedence rules for its arithmetic operators that mathematics does.
Parentheses have the highest precedence and can be used to force an expression to evaluate in the order you want. Since expressions in parentheses are evaluated first,
2 * (3  1)
is 4, and(1 + 1) ** (5  2)
is 8. You can also use parentheses to make an expression easier to read, as in(minute * 100) / 60
, even though it doesn’t change the result.Exponentiation has the next highest precedence, so
2 ** 1 + 1
is 3 and not 4, and3 * 1 ** 3
is 3 and not 27. Can you explain why?Multiplication, division, and modulus operators have the same precedence, which is higher than addition and subtraction, which also have the same precedence. So
2 * 3  1
yields 5 rather than 4, and5  2 * 2
is 1, not 6.Operators with the same precedence are evaluated from lefttoright. So in the expression
6  3 + 2
, the subtraction happens first, yielding 3. We then add 2 to get the result 5. If the operations had been evaluated from right to left, the result would have been6  (3 + 2)
, which is 1.
Tip
The acronym PEMDAS can be used to remember order of operations:
P = parentheses
E = exponentiation
M = multiplication
D = division
A = addition
S = subtraction
Note
Due to an historical quirk, an exception to the lefttoright rule is the exponentiation operator **
. A useful hint is to always use parentheses to force exactly the order you want when exponentiation is involved:
1 2 3 4 5  # the rightmost ** operator is applied first
print(2 ** 3 ** 2)
# use parentheses to force the order you want
print((2 ** 3) ** 2)

Console Output
512
64
5.6.4. Check Your Understanding¶
Question
What is the value of the following expression?
16  2 * 5 / 3 + 1
14
24
3
13.666666666666666
Question
What is the output of the code below?
print(1 + 5 % 3)
Question
What is the value of the following expression?
2 ** 2 ** 3 * 3
768
128
12
256