Chapter 6: Integers

Introduction to Integers

What are Integers?

Integers are a set of numbers that include all whole numbers and their negative counterparts. They do not include fractions or decimals. Understanding integers is fundamental for grasping more advanced mathematical concepts.

Importance of Integers

Integers are used in various real-life scenarios such as calculating temperatures, financial transactions, and elevation levels. They form the basis for many mathematical operations and problem-solving techniques.

Representation of Integers

Number Line

A number line is a visual representation of integers. Positive integers are located to the right of zero, while negative integers are located to the left. Zero is the neutral point on the number line.

Positive and Negative Integers

– Positive Integers: Numbers greater than zero (1, 2, 3, …).

– Negative Integers: Numbers less than zero (-1, -2, -3, …).

Operations with Integers

Addition of Integers

– Same Sign: Add the absolute values and keep the common sign.

– Different Signs: Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Subtraction of Integers

– Subtracting an integer is the same as adding its opposite.

– Use the number line to visualize subtraction.

Multiplication of Integers

– Same Sign: The product of two integers with the same sign is positive.

– Different Signs: The product of two integers with different signs is negative.

Division of Integers

– Same Sign: The quotient of two integers with the same sign is positive.

– Different Signs: The quotient of two integers with different signs is negative.

Properties of Integers

Closure Property

Integers are closed under addition, subtraction, multiplication, and division (except division by zero).

Commutative Property

– Addition: (a + b = b + a)

– Multiplication: (a times b = b times a)

Associative Property

– Addition: ((a + b) + c = a + (b + c))

– Multiplication: ((a times b) times c = a times (b times c))

Distributive Property

– (a times (b + c) = (a times b) + (a times c))

Practical Applications

Real-Life Uses of Integers

– Temperature: Measuring temperature in degrees Celsius or Fahrenheit.

– Finance: Calculating profits and losses.

– Elevation: Determining heights above and below sea level.

Activities and Exercises

Practicing Addition and Subtraction

– Using Number Line: Practice adding and subtracting integers using a number line.

– Worksheets: Solve problems involving addition and subtraction of integers.

Practicing Multiplication and Division

– Multiplication Tables: Learn multiplication tables for positive and negative integers.

– Worksheets: Solve problems involving multiplication and division of integers.

Summary

Key Points

– Integers include positive numbers, negative numbers, and zero.

– Operations with integers follow specific rules for addition, subtraction, multiplication, and division.

– Integers have properties such as closure, commutative, associative, and distributive.

Frequently Asked Questions (FAQs)

Common Questions About Integers

1. What are integers?

   – Integers are a set of numbers that include all whole numbers and their negative counterparts.

2. How are integers represented on a number line?

   – Positive integers are to the right of zero, and negative integers are to the left.

3. What is the rule for adding integers with different signs?

   – Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

4. What is the commutative property of integers?   

– For addition: (a + b = b + a), and for multiplication: (a times b = b times a).

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