Algebra/Order of Operations

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Algebra
Algebra
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The Order of Operations is used when doing expressions with more than one operation (e.g., ×, +, -). These are rules so you only get one answer all the time.

Example: When faced with 4+2 \times 3, how do you proceed?

There are two ways:

4 + 2 \times 3 = (4 + 2) \times 3

4 + 2 \times 3 = 6 \times 3

4 + 2 \times 3= 18

or

4 + 2 \times 3 = 4 + (2 \times 3)

4 + 2 \times 3 = 4 + 6

4 + 2 \times 3 = 10

This is confusing, so which is correct? (Parentheses, "(" and ")" are used to show what to do first)

In order to communicate using mathematical expressions we must agree on an order of operations so that each expression has only one value.

For the above example all mathematicians agree the correct answer is 10.

You're probably wondering what this order is.

The Standard Order of Operations[edit]

Evaluate expressions in this order.

  • Parentheses or Brackets (evaluate what's inside them)
  • Exponents
  • Multiplication and/or division from left to right
  • Addition and/or subtraction from left to right

An Easy Way of Remembering[edit]

Use this memory tool to help remember the order! Please Excuse My Dear Annoying Sister It is also commonly called by its acronym, PEMDAS.

An alternative form of this is; Brackets Indices Division or Multiplication Addition or Subtraction (BIDMAS).

Yet another way of remembering this is Brackets Orders Division Multiplication Addition Subtraction (BODMAS)

or Bring Our Dear Mother Along Saturday

Examples[edit]

Order of Operations - Examples
Expression Evaluation Operation
4 × 2 + 1 = 4 × 2 + 1 Multiplication
= 8 + 1 Addition
= 9
12 - 9 ÷ 3 = 12 - 9 ÷ 3 Division
= 12 - 3 Subtraction
= 9
2 × 9 ÷ 3 = 2 × 9 ÷ 3 Left to Right
= 18 ÷ 3 division
= 6
9 ÷ 3 × 3 = 9 ÷ 3 × 3 Left to Right
= 3 ×3 multiplication
= 9
3 + 12 ÷ (5 - 2) = 3 + 12 ÷ (5 - 2) Parentheses
= 3 + 12 ÷ 3 Division
= 3 + 4 Addition
= 7
7 × 10 - (2 × 4)2 = 7 × 10 - (2 × 4)2 Parentheses
= 7 × 10 - 82 Exponents
= 7 × 10 - 64 Multiplication
= 70 - 64 Subtraction
= 6

Practice Problems[edit]

  1. Evaluate the numerical expression  2+4*3
  2. Evaluate the numerical expression  2*4+3
  3. Evaluate the numerical expression  (2+4)*3
  4. Evaluate the numerical expression  9^2 + 1 -7*(8+4)/2

Practice Answers

Playing with Mathematics[edit]

To get yourself thinking about this, try this simple mathematical game:

Take the numbers 1 through 10 on the left side of an equation, and pick a number for the right side.

Example: 1 2 3 4 5 6 7 8 9 10 = 1

Now put operators between those numbers. Only use parentheses when necessary.

Example: 1 + 2 - 3 + 4 - 5 + 6 + 7 + 8 - 9 - 10 = 1

Change the number on the right-hand side. Can you generate an expression for this number? If not, can you prove why not?

Does this change if you change the order of the numbers?