What Is The Rule When Adding And Subtracting Integers

What Are the Rules When Adding and Subtracting Integers? A Comprehensive Guide

Understanding how to add and subtract integers is fundamental to mastering arithmetic and algebra. Integers are whole numbers, including zero, and their negative counterparts. While seemingly simple, the rules governing integer operations can be tricky if not fully grasped. This comprehensive guide will break down the rules, provide examples, and offer strategies to ensure you confidently tackle any integer addition and subtraction problem.

Understanding Integers: Positive, Negative, and Zero

Before diving into the rules, let's solidify our understanding of integers. Integers can be visualized on a number line:

• Positive Integers: These are numbers greater than zero (e.g., 1, 2, 3, 100, 1000). They are often represented without a plus sign (+), although it's perfectly acceptable to include it (+1, +2, +3).

• Negative Integers: These are numbers less than zero (e.g., -1, -2, -3, -100, -1000). They are always preceded by a minus sign (-).

• Zero: Zero (0) is neither positive nor negative; it's the neutral point on the number line.

The Rules of Integer Addition

The rules for adding integers depend on the signs of the numbers being added:

1. Adding Two Positive Integers:

This is the simplest case. You simply add the numbers together.

Example: 3 + 5 = 8

2. Adding Two Negative Integers:

Add the absolute values (the numbers without their signs) of the integers. The result will be negative.

Example: -3 + (-5) = -8 (Think: 3 + 5 = 8, but since both are negative, the result is -8)

3. Adding a Positive and a Negative Integer:

This requires a bit more thought. There are two approaches:

• Method 1: Subtraction: Find the difference between the absolute values of the integers. The sign of the result will be the same as the integer with the larger absolute value.

  Example 1: 5 + (-3) = 2 (Think: 5 - 3 = 2; 5 is larger, so the result is positive) Example 2: -5 + 3 = -2 (Think: 5 - 3 = 2; 5 is larger, so the result is negative)

• Method 2: Number Line: Visualize the integers on a number line. Start at the first integer and move to the right if adding a positive number, and to the left if adding a negative number.

  Example 1: For 5 + (-3), start at 5 and move 3 units to the left (because we are adding a negative number), landing on 2. Example 2: For -5 + 3, start at -5 and move 3 units to the right (because we are adding a positive number), landing on -2.

The Rules of Integer Subtraction

Subtraction of integers is closely related to addition. The key is to rewrite the subtraction problem as an addition problem.

The "Add the Opposite" Rule:

The most effective way to subtract integers is to change the subtraction problem into an addition problem by adding the opposite (additive inverse) of the second integer.

Example 1: 5 - 3 = 5 + (-3) = 2 Example 2: 5 - (-3) = 5 + 3 = 8 Example 3: -5 - 3 = -5 + (-3) = -8 Example 4: -5 - (-3) = -5 + 3 = -2

Breaking it Down:

• Subtracting a positive integer: This is the same as adding a negative integer.
• Subtracting a negative integer: This is the same as adding a positive integer. This is where many students make mistakes. Subtracting a negative is like adding a positive. Think of it as removing a debt – it improves your situation, so it's like an addition!

Practical Examples and Applications

Let’s delve into more complex examples to solidify your understanding.

Example 5: (-12) + 7 + (-5) - 10

Solution: First, rewrite subtractions as additions of opposites:

(-12) + 7 + (-5) + (-10)

Now, group the negative and positive integers:

(-12) + (-5) + (-10) + 7

Add the negative integers: -12 + (-5) + (-10) = -27

Finally, add the result to the positive integer: -27 + 7 = -20

Therefore, (-12) + 7 + (-5) - 10 = -20

Example 6: A submarine is at a depth of -250 meters. It ascends 100 meters. What is its new depth?

Solution: The submarine's initial depth is -250 meters. Ascending 100 meters means adding 100 meters to its depth:

-250 + 100 = -150 meters

The submarine's new depth is -150 meters.

Example 7: The temperature was -5°C in the morning. By afternoon, it increased by 8°C. What was the afternoon temperature?

Solution: The initial temperature is -5°C. An increase of 8°C means adding 8°C:

-5 + 8 = 3°C

The afternoon temperature was 3°C.

Strategies for Success

• Use a Number Line: Visualizing integers on a number line can be incredibly helpful, especially when starting out.

• Rewrite Subtractions as Additions: Make this your default approach. It eliminates confusion about subtracting negatives.

• Break Down Complex Problems: Tackle complex problems step-by-step, grouping similar integers (positive with positive, negative with negative) before performing the addition.

• Practice Regularly: The key to mastering integer addition and subtraction is consistent practice. Work through various examples, including word problems, to build your confidence and understanding.

Advanced Concepts and Extensions

Once you've mastered the basics, you can explore more advanced concepts involving integers:

• Order of Operations (PEMDAS/BODMAS): Remember to follow the order of operations when dealing with expressions involving multiple operations (parentheses/brackets, exponents/orders, multiplication and division, addition and subtraction).

• Integer Multiplication and Division: The rules for multiplication and division of integers also build upon the concepts of positive and negative numbers.

• Algebraic Expressions: Integer operations are the foundation for solving algebraic equations and inequalities.

Conclusion

Mastering integer addition and subtraction is a crucial step in your mathematical journey. By understanding the rules, applying the "add the opposite" strategy, and practicing regularly, you'll build a solid foundation for more advanced mathematical concepts. Remember to visualize, break down complex problems, and utilize a number line if needed. With consistent effort, you'll confidently tackle any integer addition and subtraction problem you encounter.