Lesson 2.1: Integers, Operations, and Order of Operations

Introduction

In this lesson, we will focus on the foundational concepts of integers, operations, and the order of operations, which are critical for solving quantitative reasoning problems on the GMAT. The ability to manipulate integers and execute operations efficiently is essential for success on the test. By mastering these concepts, you will also enhance your mental computation skills, enabling you to approach problems with speed and accuracy.

Learning Objectives:

Understand the properties of integers and the order of operations.
Develop efficient mental computation and estimation techniques.
Identify common arithmetic traps in answer choices.
Compute accurately and quickly without a calculator.
Apply the order of operations correctly under time pressure.

Section 1: Properties of Integers

What are Integers?

Integers are a set of whole numbers that include positive numbers, negative numbers, and zero. Mathematically, the set of integers is represented as:

$$\mathbb{Z} = \{...,-3,-2,-1,0,1,2,3,...\}$$

Key Properties of Integers

Closure Property: The sum or product of two integers is always an integer.

Example: $2 + 3 = 5$ and $3 \times 4 = 12$ both yield integers.

Commutative Property: The order in which you add or multiply integers does not change the result.

Example: $5 + 2 = 2 + 5$ and $3 \times 4 = 4 \times 3$.

Associative Property: The way in which integers are grouped when adding or multiplying does not affect the final result.

Example: $(1 + 2) + 3 = 1 + (2 + 3)$.

Identity Property: There are unique integers that do not change the number when added or multiplied. For addition, the identity is $0$, and for multiplication, it is $1$.

Example: $5 + 0 = 5$ and $7 \times 1 = 7$.

Distributive Property: Multiplication distributes across addition.

Example: $a(b + c) = ab + ac$.

Worked Example 1

Problem: Using the properties of integers, simplify the expression $3 \times (4 + 5) - 6$.

Step 1: Apply the distributive property to simplify the expression:

$3 \times 4 + 3 \times 5 - 6$.

Step 2: Calculate the products:

$12 + 15 - 6$.

Step 3: Perform the addition and subtraction:

$27 - 6 = 21$.

Answer: The simplified expression equals $21$.

Common Misconceptions

Negative Integers: Some students forget that negative integers are also included in the set of integers. Thus, $-1$ is an integer, while $1.5$ is not.
Zero: Students often overlook zero as an integer, which can lead to errors in calculations and problem-solving.

Section 2: Operations with Integers

Basic Operations

The basic operations that can be performed with integers are addition, subtraction, multiplication, and division. Understanding how to execute these operations correctly is vital for solving arithmetic problems.

Addition: The act of combining two integers.

Example: $2 + 3 = 5$.

Subtraction: The process of finding the difference between two integers.

Example: $5 - 3 = 2$.

Multiplication: Repeated addition of an integer a specified number of times.

Example: $3 \times 4 = 3 + 3 + 3 + 3 = 12$.

Division: Splitting an integer into a specified number of equal parts.

Example: $12 \div 3 = 4$. This means that 12 is divided into 3 equal parts, each part being 4.

Order of Operations (PEMDAS)

To correctly solve mathematical expressions with multiple operations, you need to follow the order of operations known as PEMDAS:

Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Worked Example 2

Problem: Simplify the expression $4 + 2 \times (3^2 - 1)$ following the order of operations.

Step 1: Solve the expression within the parentheses:

$3^2 = 9$, so $9 - 1 = 8$.

Step 2: Substitute back into the expression:

$4 + 2 \times 8$.

Step 3: Execute the multiplication:

$2 \times 8 = 16$.

Step 4: Finally, do the addition:

$4 + 16 = 20$.

Answer: The simplified expression equals $20$.

Common Mistakes with Order of Operations

Ignoring parentheses, leading to incorrect calculations because operations may be performed in the wrong order.
Performing all multiplications and divisions before any additions and subtractions instead of working from left to right.

Section 3: Efficient Mental Computation and Estimation

Mental Computation Strategies

In the GMAT exam, time is crucial. Efficient mental calculation can help you avoid using a calculator and save time. Here are some strategies:

Rounding: Round numbers to make calculations easier, then adjust the result back to reflect the rounding.
Breaking Down Problems: Split complex problems into simpler parts that are easier to compute mentally.
Estimating: When exact answers are not necessary, an estimate can help you narrow down answer choices quickly.

Worked Example 3

Problem: Estimate the total cost of 47 items priced at $19.99 each.

Step 1: Round $19.99$ to $20$ for easier calculation.

Step 2: Multiply: $47 \times 20 = 940$.

Step 3: The estimated total cost is around $940$.

Common Estimation Mistakes

Overlooking the need for rounding when dealing with complex multiplication or division.
Focusing too much on precision when an estimate suffices can waste time.

Conclusion

Mastering the properties of integers and the order of operations is essential for success in quantitative reasoning on the GMAT. By practicing these skills, you will enhance your ability to perform quick mental computations, make accurate estimates, and solve problems effectively under time constraints. Instead of relying solely on a calculator, you will have the confidence to tackle arithmetic challenges head-on.

Study Notes

Integers include negative numbers, positive numbers, and zero.
Properties of integers: closure, commutative, associative, identity, and distributive properties.
Follow the order of operations (PEMDAS) to solve expressions correctly.
Use mental computation strategies such as rounding and breaking down problems for efficient calculations.
Practice estimating answers when precision is not necessary.