Negative Integers Lesson: Addition, Subtraction, Multiplication, and Division

Negative integers are numbers less than zero. They play a crucial role in mathematics and appear in various real-life scenarios, such as temperatures below freezing, depths below sea level, or bank withdrawals. Understanding negative integers helps students build a strong foundation in arithmetic and prepares them for advanced mathematical concepts.

Understanding Negative Integers

Negative integers are whole numbers with a minus sign in front of them. They are the opposite of positive integers.

• Examples: -1, -5, -12, -100
• They appear on the left side of zero on the number line
• Zero is neither positive nor negative
• Used to represent losses, deficits, and values below a baseline

Number Line Representation:

... -4  -3  -2  -1  0  1  2  3  4 ...

Each step to the left decreases the value, and each step to the right increases it.

Adding and Subtracting Negative Integers

Addition and subtraction of negative integers can be confusing at first, but following specific rules makes the process easier.

Addition Rules:

• Positive + Positive = Positive
  Example: 3 + 5 = 8

• Negative + Negative = Negative
  • Add the absolute values and keep the negative sign
    Example: -3 + (-2) = -5

• Positive + Negative (or vice versa)
  • Subtract the smaller absolute value from the larger
  • Use the sign of the number with the greater absolute value
    Example: -7 + 4 = -3
    6 + (-9) = -3

Subtraction Rules:

Subtracting a number is the same as adding its opposite.

• Change the subtraction sign to addition and flip the sign of the number being subtracted.
• Then apply the rules of addition.

Examples:

• 5 - (-3) → 5 + 3 = 8
• -6 - 2 → -6 + (-2) = -8
• -4 - (-5) → -4 + 5 = 1

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Integers: Positive and Negative Numbers

Multiplying and Dividing Negative Integers

Multiplication and division of integers follow predictable patterns based on the number of negative signs involved.

Multiplication Rules:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

Examples:

• (-3) × (-4) = 12
• (-2) × 5 = -10
• 6 × (-7) = -42

Division Rules:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

Examples:

• (-20) ÷ (-5) = 4
• 16 ÷ (-4) = -4
• (-30) ÷ 6 = -5

Real-Life Applications of Negative Integers

Negative integers are used in a variety of practical situations:

• Temperatures: Below-zero readings like -5°C or -10°F
• Finance: Representing losses or debts, e.g., -₹500
• Elevation: Measuring depth below sea level
• Sports: Point deductions or scoring deficits
• Stock Market: Indicating drop in stock prices

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Math Quiz-Multiplying Negative Numbers

Comparing Negative and Positive Integers

Negative numbers are always smaller than zero and any positive number.

Comparison Rules:

• A negative number is always less than a positive number.
  Example: -2 < 3
• Among two negative numbers, the one with a smaller absolute value is greater.
  Example: -3 > -7

Helpful Tip: On the number line, numbers farther to the left are smaller.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction.

• Denoted by vertical bars: | -7 | = 7
• Always a positive number

Use in Operations: When subtracting or comparing integers, absolute value helps determine magnitude without considering the sign.

Word Problems Involving Negative Integers

Understanding how to translate real-world situations into integer operations is key.

Example 1:

A hiker descends 100 meters, then climbs 40 meters.
Represent this with integers:

• Descent: -100
• Ascent: +40
  Total elevation change: -100 + 40 = -60 meters

Example 2:

The temperature dropped from 3°C to -5°C.
Change = -5 - 3 = -8°C
There was an 8-degree drop.

Summary Table of Integer Operations

Common Errors and Misconceptions

• Treating subtraction of negative as subtraction again
  • Mistake: 8 - (-2) = 6
  • Correction: 8 - (-2) = 8 + 2 = 10

• Forgetting sign rules in multiplication or division
  • Mistake: -3 × -4 = -12
  • Correction: -3 × -4 = 12

• Wrong sign choice after adding mixed signs
  • Mistake: -6 + 4 = 2
  • Correction: -6 + 4 = -2 (since -6 has greater absolute value)

• Mixing up absolute value and actual value
  • | -8 | is 8, not -8

Mastering negative integers requires a strong understanding of signs, rules, and real-world logic. By consistently applying these principles and visualizing operations using the number line, one can accurately solve problems involving negative numbers across various mathematical topics.

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Negative Integers Quiz 1