Precedence of Operations

Given an expression involving more than one algebraic operation, there is always the question of which calculation to do first.

For example:

Does:	20 - 6 + 5
mean:	14 + 5 = 19	or	20 - 11 = 9?
Does:	9 + 3 × 4
mean:	12 × 4 = 48	or	9 + 12 = 21?
Does:	4 × 32
mean:	122 = 144	or	4 × 9 = 36?

 

To correctly answer the above examples, we need to know the correct order in which to perform the operations addition, subtraction, multiplication, division, etc...

The Rules of Precedence are a universal convention which tell us to apply operations in the following order:

1. Brackets 2. Powers and Roots 3. Multiplication and Division 4. Addition and Subtraction

When two operations with the same precedence occur together, they are performed from left to right, i.e. in the order that they are read.

So, in the examples above:

20 - 6 + 5 = 14 + 5 = 19

9 + 3 × 4 = 9 + 12 = 21

4 × 3² = 4 × 9 = 36

For the other answers to be correct we would need to use brackets to override the normal precedence.

20 - (6 + 5) = 9

(9 + 3) × 4 = 48

(4 × 3)² = 144

Expressions in brackets must be calculated first. For example

(2 + 3) × (7 - 3) = 5 × 4 = 20

If the brackets are enclosed within other brackets, work must be done from the inside out. For example:

In the expression:

(3 + (7 - (2 + 3 × 6) + 2 × 5) - 7 + 1)

(2 + 3 × 6) = 20 is calculated first, giving

(3 + (7 - 20 + 2 × 5) - 7 + 1)

(7 - 20 + 2 × 5) = -3 is calculated next, giving

(3 + (-3) - 7 + 1)

= 0 - 7 + 1 = -6

In a quotient such as:

the horizontal line acts as brackets for both the numerator and denominator.

For example:

But, if we omit the brackets in the middle expression:

So be careful!

Calculators and computers are programmed to use the rules of precedence. Before you input an expression, ask yourself if brackets are needed to ensure the correct order of operations. If brackets are required and you omit them, then you cannot blame your calculator if the answer you obtained is wrong.

For example, try inputing the expression:

into your calculator without using brackets. What happens?

As you can see, brackets must be used properly in order that the expression you want is correctly interpreted:

- by a calculator or computer
- by another person
- by yourself!

Exercise 4 Rules of Precedence (Part 1)

Exercise 5 Rules of Precedence (Part 2)