Chapter 1: Integers
Overview of the Chapter
Introduction to Integers
“Integers” in Class 7 Mathematics expands on the concepts introduced in Class 6, delving deeper into operations, properties, and applications of integers. Understanding integers is crucial for progressing in mathematics and solving real-world problems involving whole numbers and their opposites.
What are Integers?
Definition of Integers
Integers: Integers include all positive and negative whole numbers, along with zero. They are represented as {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers do not include fractions or decimals.
Positive and Negative Integers
Positive Integers: Numbers greater than zero, represented without a sign or with a positive sign (+1, +2, +3, …). These are to the right of zero on the number line.
Negative Integers: Numbers less than zero, represented with a negative sign (-1, -2, -3, …). These are to the left of zero on the number line.
Representing Integers on a Number Line
Number Line Concept
Number Line: A straight line with numbers placed at equal intervals, used to represent integers. It extends infinitely in both directions, with zero typically at the center.
Representation
Example: To plot -3, -1, 0, 2, and 4 on a number line:
- Place 0 at the center.
- Place -1 and -3 to the left of zero.
- Place 2 and 4 to the right of zero.
- This visual representation helps in understanding the relative positions and distances between integers.
Number Line
Operations with Integers
Addition of Integers
Same Signs: When adding two integers with the same sign, add their absolute values and keep the common sign.
Example: (+3) + (+5) = +8, (-3) + (-5) = -8. Here, the absolute values are added, and the common sign is retained.
Different Signs: When adding two integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the larger absolute value.
Example: (+7) + (-5) = +2, (-7) + (+5) = -2. Here, the absolute values are subtracted, and the sign of the larger absolute value is retained.
Subtraction of Integers
Changing Signs: Subtracting an integer is the same as adding its opposite (additive inverse).
Example: (+5) – (+3) = (+5) + (-3) = +2, (-5) – (-3) = (-5) + (+3) = -2. Here, subtraction is converted into addition by changing the sign of the subtracted number.
Multiplication of Integers
Same Signs: The product of two integers with the same sign is positive.
Example: (+3) × (+4) = +12, (-3) × (-4) = +12. Both examples result in a positive product.
Different Signs: The product of two integers with different signs is negative.
Example: (+3) × (-4) = -12, (-3) × (+4) = -12. Both examples result in a negative product.
Multiplication by Zero: The product of any integer and zero is always zero.
Example: (+3) × 0 = 0, (-3) × 0 = 0. Multiplying any number by zero results in zero.
Division of Integers
Same Signs: The quotient of two integers with the same sign is positive.
Example: (+12) ÷ (+3) = +4, (-12) ÷ (-3) = +4. Both examples result in a positive quotient.
Different Signs: The quotient of two integers with different signs is negative.
Example: (+12) ÷ (-3) = -4, (-12) ÷ (+3) = -4. Both examples result in a negative quotient.
Division by Zero: Division by zero is undefined. Any number divided by zero does not have a meaning and is not defined in mathematics.
Example: 5 ÷ 0 is undefined.
Properties of Integers
Closure Property
Addition and Multiplication: The sum and product of any two integers are always integers.
Example: 3 + (-2) = 1, 3 × (-2) = -6. This property ensures that performing these operations on integers does not result in a non-integer.
Commutative Property
Addition and Multiplication: Changing the order of the numbers does not change the sum or product.
Example: 3 + (-2) = (-2) + 3, 3 × (-2) = (-2) × 3. This property shows that the sequence of adding or multiplying integers does not affect the result.
Associative Property
Addition and Multiplication: The grouping of numbers does not change the sum or product.
Example: (3 + 4) + 5 = 3 + (4 + 5), (3 × 4) × 5 = 3 × (4 × 5). This property indicates that how integers are grouped in an operation does not affect the outcome.
Distributive Property
Multiplication over Addition: Distributing a multiplication over an addition means multiplying each addend separately and then adding the results.
Example: 3 × (4 + 5) = (3 × 4) + (3 × 5). This property combines both addition and multiplication operations.
Properties of Integers
| Property Name | Example |
|---|---|
| Closure Property | For any two integers a and b, a + b is also an integer. Example: 2 + 3 = 5 |
| Commutative Property | For any two integers a and b, a + b = b + a. Example: 2 + 3 = 3 + 2 |
| Associative Property | For any three integers a, b, and c, (a + b) + c = a + (b + c). Example: (2 + 3) + 4 = 2 + (3 + 4) |
| Distributive Property | For any three integers a, b, and c, a × (b + c) = (a × b) + (a × c). Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) |
Additive Inverse
Definition: For every integer ‘a’, there exists an integer ‘-a’ such that a + (-a) = 0.
Example: The additive inverse of +3 is -3. When these two numbers are added together, the result is zero.
Applications of Integers
Real-life Applications
Temperature: Representing temperatures above and below zero (e.g., +25°C and -10°C).
Financial Transactions: Credits (positive integers) and debits (negative integers) in banking.
Altitude: Heights above sea level (positive integers) and depths below sea level (negative integers).
Conclusion
Summary of the Chapter
The chapter “Integers” provides a comprehensive understanding of integers, their properties, and operations. It explains how to represent integers on a number line and perform basic arithmetic operations with them.
Additional Information
Key Terms and Concepts
Integer: Whole numbers including positive, negative, and zero.
Positive Integer: Numbers greater than zero.
Negative Integer: Numbers less than zero.
Number Line: A line representing numbers at equal intervals.
Additive Inverse: An integer that, when added to a given integer, results in zero.
Important Examples and Cases
Addition and Subtraction:
(+7) + (-5) = +2: Here, the absolute values are subtracted, and the sign of the larger absolute value is retained.
(+5) – (+3) = (+5) + (-3) = +2: Subtraction is converted into addition by changing the sign of the subtracted number.
Multiplication and Division:
(+3) × (-4) = -12: The product of two integers with different signs is negative.
(-12) ÷ (+3) = -4: The quotient of two integers with different signs is negative.
Notable Observations
Integers in Real Life: Understanding integers is crucial for dealing with various real-life situations like temperature, financial transactions, and altitudes.
Properties of Integers: Knowing the properties helps in simplifying mathematical operations involving integers.
FAQs
Integers include all positive and negative whole numbers, along with zero. They are represented as {…, -3, -2, -1, 0, 1, 2, 3, …}.
Positive integers are numbers greater than zero, represented without a sign or with a positive sign (+1, +2, +3, …).
Negative integers are numbers less than zero, represented with a negative sign (-1, -2, -3, …).
Integers are represented on a number line with zero at the center, positive integers to the right of zero, and negative integers to the left of zero.
When adding two integers with the same sign, add their absolute values and keep the common sign.
When adding two integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the larger absolute value.
Subtracting an integer is the same as adding its opposite (additive inverse).
The product of two integers with the same sign is positive.
The product of two integers with different signs is negative.
The product of any integer and zero is always zero.
The quotient of two integers with the same sign is positive.
The quotient of two integers with different signs is negative.
No, division by zero is undefined.
The closure property states that the sum and product of any two integers are always integers.
The commutative property of addition states that changing the order of the numbers does not change the sum.
The commutative property of multiplication states that changing the order of the numbers does not change the product.
The associative property of addition states that the grouping of numbers does not change the sum.
The associative property of multiplication states that the grouping of numbers does not change the product.
The distributive property states that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results.
The additive inverse of an integer ‘a’ is ‘-a’, such that a + (-a) = 0.
Integers are used to represent temperatures above and below zero, such as +25°C and -10°C.
Integers are used to represent credits (positive integers) and debits (negative integers) in banking.
Integers are used to represent heights above sea level (positive integers) and depths below sea level (negative integers).
Yes, (+7) + (-5) = +2, where the absolute values are subtracted, and the sign of the larger absolute value is retained.
Yes, 3 × (4 + 5) = (3 × 4) + (3 × 5), which combines both addition and multiplication operations.
Chapter 1: Integers MCQs
1. What are positive integers?
2. How are integers represented on a number line?
3. What is the sum of (+7) + (-5)?
4. What is the product of (+3) × (-4)?
5. What is the quotient of (+12) ÷ (+3)?
6. What is the result of (-5) – (-3)?
7. What is the closure property of integers in addition?
8. What does the commutative property of multiplication state?
9. What does the associative property of addition state?
10. What does the distributive property of multiplication over addition state?
11. What is the additive inverse of +7?
12. Which property is demonstrated by the equation 3 + (-3) = 0?
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