# 2.11. 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 mathematical operators that mathematics does.

1. 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: in this case, the parentheses don’t change the result, but they reinforce that the expression in parentheses will be evaluated first.

2. Exponentiation has the next highest precedence, so 2**1+1 is 3 and not 4, and 3*1**3 is 3 and not 27. Can you explain why?

3. Multiplication and both division 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, and 5-2*2 is 1, not 6.

4. Operators with the same precedence are evaluated from left-to-right. In algebra we say they are left-associative. 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 been 6-(3+2), which is 1.

Note

Due to some historical quirk, an exception to the left-to-right left-associative rule is the exponentiation operator **. A useful hint is to always use parentheses to force exactly the order you want when exponentiation is involved:

Note

This is a second way that parentheses are used in Python. The first way you’ve already seen is that () indicates a function call, with the inputs going inside the parentheses. How can Python tell when parentheses specify to call a function, and when they are just forcing the order of operations for ambiguous operator expressions?

The answer is that if there’s a an expression to the left of the parentheses that evaluates to a function object, then the parentheses indicate a function call, and otherwise not. You will have to get used to making the same inference when you see parentheses: is this a function call, or just specifying precedence?

data-11-1: What is the value of the following expression:

16 - 2 * 5 // 3 + 1

• 14
• Using parentheses, the expression is evaluated as (2*5) first, then (10 // 3), then (16-3), and then (13+1).
• 24
• Remember that * has precedence over -.
• 3
• Remember that // has precedence over -.
• 13.667
• Remember that // does integer division.

Here is an animation for the above expression:

16 - 2 * 5 // 3 + 1