In this chapter, you will learn
- —Understand the concept of integers (positive, negative, and zero)
- —Compare and order integers using the number line
- —Perform addition and subtraction of integers
- —Perform multiplication and division of integers
- —Apply properties of integers in problem-solving
- —Understand and use absolute value
What are Integers?
Integers are a set of numbers that include all whole numbers (positive and negative) and zero.
Definition: Integers = {..., -3, -2, -1, 0, 1, 2, 3, ...}
The symbol Z is used to represent the set of integers.
Types of Integers:
- Positive Integers: All natural numbers greater than zero: 1, 2, 3, 4, ...
- Negative Integers: Numbers less than zero: -1, -2, -3, -4, ...
- Zero: Neither positive nor negative: 0
Real-Life Applications:
- Temperature: 5°C above zero (positive) vs. -3°C below zero (negative)
- Bank Account: Money deposited (+) vs. money withdrawn (-)
- Altitude: Height above sea level (+) vs. depth below sea level (-)
- Profit and Loss: Profit is positive, loss is negative
Exam Tip
Integers form the basis of most Class 7 mathematics. Make sure you understand the number line representation and can compare any two integers.
Common Mistake
Students sometimes think negative numbers are 'opposite' of positive in a strict sense. Remember: -5 is not the opposite of 5 in subtraction; -5 is the additive inverse.
Number Line and Comparison of Integers
A number line is a horizontal line with integers placed at equal intervals. It helps us visualize integers and compare them easily.
Rules for Comparing Integers:
- Every positive integer > every negative integer
- Zero > every negative integer and Zero < every positive integer
- When comparing two positive integers, the larger number is greater
- When comparing two negative integers, the number closer to zero is greater (e.g., -2 > -5)
Examples of Comparison:
- 5 > -3 (positive > negative)
- -2 > -5 (closer to zero)
- 0 > -7 (zero > negative)
- -1 > -10 (closer to zero)
Exam Tip
Comparing negative integers is tricky. Always remember: the closer to zero, the greater the number. -1 is greater than -100!
Common Mistake
Students often confuse the magnitude with the value. -10 has a larger magnitude than -1, but -1 is actually greater.
Addition of Integers
There are three cases when adding integers:
Case 1: Both integers are positive
Add the numbers normally. The result is positive.
Example: 5 + 3 = 8
Case 2: Both integers are negative
Add the absolute values and put a negative sign to the result.
Example: (-5) + (-3) = -8
Case 3: One positive, one negative
Subtract the smaller absolute value from the larger and use the sign of the larger absolute value.
Example 1: 5 + (-3) = 2
Example 2: (-5) + 3 = -2
Number Line Method: Start from the first number. For addition, move right (positive) or left (negative) by the second number.
Exam Tip
Use the number line method if you're unsure about signs. It's a visual way to check your answers.
Common Mistake
Students often forget to change the sign when adding negative numbers. Remember: +(-3) means subtract 3.
Subtraction of Integers
Key Rule: To subtract an integer, add its opposite (additive inverse).
a - b = a + (-b)
Examples:
- 7 - 3 = 7 + (-3) = 4
- 5 - (-2) = 5 + 2 = 7
- (-3) - 2 = (-3) + (-2) = -5
- (-5) - (-3) = (-5) + 3 = -2
Key Points:
- Subtracting a positive number moves left on the number line
- Subtracting a negative number moves right on the number line (becomes addition)
- The most common mistake is forgetting to change the sign when subtracting a negative number
Exam Tip
Always convert subtraction to addition of the opposite. This reduces errors significantly.
Common Mistake
Forgetting to change the sign: 5 - (-3) ≠ 5 - 3. It equals 5 + 3 = 8, not 2!
Multiplication and Division of Integers
Sign Rules for Multiplication and Division:
Multiplication Sign Rules:
- Positive × Positive = Positive (e.g., 4 × 3 = 12)
- Negative × Negative = Positive (e.g., (-4) × (-3) = 12)
- Positive × Negative = Negative (e.g., 4 × (-3) = -12)
- Negative × Positive = Negative (e.g., (-4) × 3 = -12)
- Any integer × 0 = 0
Division Sign Rules:
The same sign rules apply as multiplication!
- 12 ÷ 3 = 4 (+ ÷ + = +)
- (-12) ÷ (-3) = 4 (- ÷ - = +)
- 12 ÷ (-3) = -4 (+ ÷ - = -)
- (-12) ÷ 3 = -4 (- ÷ + = -)
- 0 ÷ a = 0 (a ≠ 0)
- a ÷ 0 = undefined (Cannot divide by zero)
Quick Tip: Count the negative signs. If even number of negatives, result is positive. If odd number, result is negative.
Exam Tip
The sign rules for multiplication and division are identical. Use this to remember both easily.
Common Mistake
Students often make errors with multiple multiplications. Example: (-2) × 3 × (-4) = ? (Answer: 24, because two negative signs = positive)
Properties of Integers
Integers follow important mathematical properties that are used in problem-solving:
Closure Property: The sum, difference, and product of any two integers is always an integer.
Example: 5 + (-3) = 2 ✓ (all are integers)
Commutative Property:
- For addition: a + b = b + a (e.g., 5 + 3 = 3 + 5)
- For multiplication: a × b = b × a (e.g., 4 × 2 = 2 × 4)
Associative Property:
- For addition: (a + b) + c = a + (b + c)
- For multiplication: (a × b) × c = a × (b × c)
Distributive Property: a × (b + c) = (a × b) + (a × c)
Example: 3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18
Identity Elements:
- Additive Identity: 0 (a + 0 = a)
- Multiplicative Identity: 1 (a × 1 = a)
Exam Tip
Properties are often used implicitly in problem-solving. Understanding them helps you solve complex expressions correctly.
Common Mistake
Subtraction and division are NOT commutative or associative. Don't apply these properties to them!
Chapter Summary
Integers are fundamental to all mathematics. This chapter covers:
- Definition and Types: Positive, negative, and zero integers
- Comparison: Using number lines and understanding that -1 > -10
- Addition: Three cases depending on signs of numbers
- Subtraction: Converting to addition of opposite (a - b = a + (-b))
- Multiplication and Division: Sign rules (positive × negative = negative)
- Properties: Closure, Commutative, Associative, Distributive, Identity
Exam Focus: Operations on integers, sign rules, solving word problems using integers, properties of operations.