Order of Operations ~ Lovers of Math

Friday, June 29, 2012

Order of Operations

4:05:00 PM Unknown No comments

Order of Operations:

Introduction

Look at and evaluate the following expression:
$2 + 4 \times 7 - 1 = ?$
How many different ways can we interpret this problem, and how many different answers could someone possibly find for it?
The simplest way to evaluate the expression is simply to start at the left and work your way across:

$& 2 + 4 \times 7 - 1\\ &= 6 \times 7 - 1\\ &= 42 - 1\\ &= 41$

This is the answer you would get if you entered the expression into an ordinary calculator. But if you entered the expression into a scientific calculator or a graphing calculator you would probably get 29 as the answer.

In mathematics, the order in which we perform the various operations (such as adding, multiplying, etc.) is important. In the expression above, the operation of multiplication takes precedence over addition, so we evaluate it first. Let’s re-write the expression, but put the multiplication in brackets to show that it is to be evaluated first.

$2 + (4 \times 7) - 1 = ?$

First evaluate the brackets: $4 \times 7 = 28$ . Our expression becomes:

$2 + (28) - 1 = ?$

When we have only addition and subtraction, we start at the left and work across:

$& 2 + 28 - 1\\ &= 30 - 1\\ &= 29$

Algebra students often use the word “PEMDAS” to help remember the order in which we evaluate the mathematical expressions: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.

Order of Operations

Evaluate expressions within Parentheses (also all brackets $[ \ ]$ and braces { }) first.
Evaluate all Exponents (terms such as $3^2$ or $x^3$ ) next.
Multiplication and Division is next - work from left to right completing both multiplication and division in the order that they appear.
Finally, evaluate Addition and Subtraction - work from left to right completing both addition and subtraction in the order that they appear