Integer Questions

Integer definition

An integer is any whole number that can be positive, negative, or zero. Integers do not have fractional or decimal parts and can be represented on the number line. The set of integers includes:

• Positive integers: 1,2,3,…1, 2, 3, \dots

• Negative integers: −1,−2,−3,…-1, -2, -3, \dots

• Zero: 00

The integers are important because they help represent a wide range of real-life scenarios, from calculating debts to temperatures below freezing. Understanding integers and their operations is a fundamental concept in arithmetic and algebra.

Table of Contents

• Understanding Integer Operation
• Real-Life Examples of Integers
• Integer Properties and Rules
• Solved Integer Questions
• Practice Problems
• Applications of Integers in Real Life
• Conclusion
• Frequently Asked Questions (FAQs)

 

  Understanding Integer Operations

Integers follow specific rules for addition, subtraction, multiplication, and division. Let’s explore these operations in detail.

1. Addition of Integers

Adding integers involves combining two numbers, either both positive or both negative, or one of each. Here are some basic rules:

• Positive + Positive: The sum of two positive integers is always positive.
  Example: 3+4=73 + 4 = 7

• Negative + Negative: The sum of two negative integers is always negative.
  Example: −3+(−4)=−7-3 + (-4) = -7

• Positive + Negative: The sum of a positive and a negative integer depends on the magnitude (absolute value) of the numbers.
  Example: 5+(−3)=25 + (-3) = 2 or −5+3=−2-5 + 3 = -2

• Zero + any integer: Adding zero to any integer leaves the integer unchanged.
  Example: 0+5=50 + 5 = 5 or 0+(−5)=−50 + (-5) = -5

2. Subtraction of Integers

Subtraction of integers can be seen as the addition of a negative integer. Here’s how it works:

• Positive – Positive: If the number being subtracted is smaller, the result is positive; if it's larger, the result is negative.
  Example: 7−3=47 - 3 = 4, but 3−7=−43 - 7 = -4

• Negative – Negative: The rule follows the same principle as positive subtraction.
  Example: −5−(−3)=−5+3=−2-5 - (-3) = -5 + 3 = -2

• Adding the opposite: To subtract an integer, add its opposite.
  Example: 8−(−4)=8+4=128 - (-4) = 8 + 4 = 12

3. Multiplication of Integers

The multiplication of integers follows a few key rules:

• Positive × Positive: The result is positive.
  Example: 3×4=123 × 4 = 12

• Negative × Negative: The result is positive.
  Example: (−3)×(−4)=12(-3) × (-4) = 12

• Positive × Negative: The result is negative.
  Example: 3×(−4)=−123 × (-4) = -12

• Multiplying by zero: Any integer multiplied by zero equals zero.
  Example: 5×0=05 × 0 = 0

4. Division of Integers

Division works similarly to multiplication, with a few differences:

• Positive ÷ Positive: The result is positive.
  Example: 8÷4=28 ÷ 4 = 2

• Negative ÷ Negative: The result is positive.
  Example: (−8)÷(−4)=2(-8) ÷ (-4) = 2

• Positive ÷ Negative: The result is negative.
  Example: 8÷(−4)=−28 ÷ (-4) = -2

• Negative ÷ Positive: The result is negative.
  Example: (−8)÷4=−2(-8) ÷ 4 = -2

• Division by zero: Division by zero is undefined.

 

Real-Life Examples of Integers

Integers are used to represent real-world situations, such as:

• Banking: If you owe money to the bank, it’s represented by a negative integer (e.g., -$50). If you have money in your account, it’s a positive integer.

• Temperature: Temperatures below zero (like in winter) are represented by negative integers, while positive integers represent temperatures above zero.

• Sports Scores: A player’s score may be positive or negative depending on the situation (e.g., points gained or lost).

 

Integer Properties and Rules

Integers follow certain properties that help simplify mathematical operations:

1. Closure Property: The sum, difference, and product of two integers will always be an integer.

2. Commutative Property: The order of addition or multiplication does not affect the result.

• a+b=b+aa + b = b + a

• a×b=b×aa \times b = b \times a

3. Associative Property: The grouping of terms does not affect the result.

• (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

• (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c)

4. Distributive Property: Multiplication distributes over addition.

• a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c

 

Solved Integer Questions

Let’s look at a few solved examples to understand how to approach integer-based problems.

Solved Example 1:

Question: What is (−7)+4(-7) + 4?
Solution: Since the integers have opposite signs, subtract the smaller absolute value from the larger one and use the sign of the larger number.

(−7)+4=−3(-7) + 4 = -3

Solved Example 2:

Question: What is (−12)×3(-12) × 3?
Solution: Since the signs are opposite, the product will be negative.

(−12)×3=−36(-12) × 3 = -36

Solved Example 3:

Question: What is 8÷(−4)8 ÷ (-4)?
Solution: Dividing a positive number by a negative number results in a negative quotient.

8÷(−4)=−28 ÷ (-4) = -2

 

Practice Problems

1. (−9)+6=?(-9) + 6 = ?

2. 14−(−7)=?14 - (-7) = ?

3. (−5)×(−3)=?(-5) × (-3) = ?

4. (−20)÷4=?(-20) ÷ 4 = ?

5. Find the sum of (−7)+(−3)+5(-7) + (-3) + 5.

 

Applications of Integers in Real Life

Integers are not only fundamental to mathematics but also play an important role in solving real-life problems:

1. Financial Calculations: Negative integers are used to represent debts, while positive integers represent assets or earnings.

2. Temperature Measurement: The temperature scale in many parts of the world uses negative numbers to represent temperatures below freezing.

3. Height or Depth: In surveying, negative integers may be used to measure depths below sea level.  

 

Conclusion

Integers are a crucial concept in mathematics, and understanding how to manipulate them using the basic operations - addition, subtraction, multiplication, and division - lays the foundation for more advanced topics. Whether you’re working with financial calculations, measuring temperatures, or solving algebraic equations, integers are an essential tool for solving real-world problems.

 

Frequently Asked Questions (FAQs)

Q1: What is the difference between a positive integer and a negative integer?

Answer: Positive integers are greater than zero, while negative integers are less than zero.

Q2: How do you multiply integers with different signs?

Answer: If the signs are different, the product is always negative.

Q3: What happens if you divide by zero?

Answer: Division by zero is undefined.

Q4: Can you add two negative integers?

Answer: Yes, when you add two negative integers, the sum will always be negative.

By mastering integer questions, you’ll enhance your mathematical skills and gain confidence in handling a wide variety of problems.Learn more at orchids International.