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 mathematics. Integers encompass all whole numbers, both positive and negative, including zero. While addition and subtraction of positive whole numbers might seem straightforward, incorporating negative numbers introduces a new layer of complexity that requires specific rules and a solid grasp of the number line. This comprehensive guide will break down these rules, providing examples and strategies to help you confidently tackle integer arithmetic.

Understanding the Number Line

Before delving into the rules, it's crucial to visualize integers using the number line. The number line is a visual representation of numbers, extending infinitely in both positive and negative directions. Zero is positioned at the center, positive integers to the right, and negative integers to the left.

Visualizing Positive and Negative Integers

• Positive Integers: These are numbers greater than zero (e.g., 1, 2, 3, 100, etc.). On the number line, they are located to the right of zero.
• Negative Integers: These are numbers less than zero (e.g., -1, -2, -3, -100, etc.). On the number line, they are located to the left of zero.
• Zero: Zero is neither positive nor negative; it's the point of origin on the number line.

Understanding the number line provides a visual context for understanding addition and subtraction of integers, making the processes more intuitive.

Adding Integers: The Rules and Techniques

Adding integers involves combining two or more numbers. The rules depend on the signs of the integers being added.

Rule 1: Adding Integers with the Same Sign

When adding integers with the same sign (both positive or both negative), you simply add their absolute values and keep the common sign. The absolute value of a number is its distance from zero, regardless of its sign.

Example 1: Adding Positive Integers

5 + 3 = 8 (Both positive, so add the values and keep the positive sign)

Example 2: Adding Negative Integers

(-5) + (-3) = -8 (Both negative, so add the absolute values (5+3=8) and keep the negative sign)

Rule 2: Adding Integers with Different Signs

When adding integers with different signs (one positive and one negative), you subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value.

Example 3: Positive and Negative Integers

8 + (-3) = 5 (Subtract the smaller absolute value (3) from the larger absolute value (8), resulting in 5. Keep the positive sign because 8 has a larger absolute value.)

Example 4: Negative and Positive Integers

(-8) + 3 = -5 (Subtract the smaller absolute value (3) from the larger absolute value (8), resulting in 5. Keep the negative sign because 8 has a larger absolute value.)

Using the Number Line for Addition

The number line provides a visual method for adding integers. Start at the first integer on the number line. If you're adding a positive integer, move to the right. If you're adding a negative integer, move to the left. The final position on the number line represents the sum.

Example 5: Visualizing Addition on the Number Line

Let's visualize 5 + (-3) using the number line. Start at 5. Since we're adding -3 (a negative integer), move three units to the left. You'll land on 2, which is the sum.

Subtracting Integers: The Rules and Techniques

Subtracting integers involves finding the difference between two numbers. A key concept in integer subtraction is the relationship between subtraction and addition. Subtracting a number is the same as adding its opposite (additive inverse).

Rule 1: The Additive Inverse

The additive inverse of a number is its opposite. For example:

• The additive inverse of 5 is -5.
• The additive inverse of -5 is 5.
• The additive inverse of 0 is 0.

Rule 2: Subtracting Integers

To subtract an integer, add its additive inverse. This converts subtraction problems into addition problems, allowing you to apply the rules of integer addition.

Example 6: Subtracting Positive Integers

8 - 3 = 8 + (-3) = 5 (Subtract 3 by adding its additive inverse, -3)

Example 7: Subtracting Negative Integers

8 - (-3) = 8 + 3 = 11 (Subtract -3 by adding its additive inverse, 3)

Example 8: Subtracting a Larger Number from a Smaller Number

3 - 8 = 3 + (-8) = -5 (Subtract 8 by adding its additive inverse, -8)

Example 9: Subtracting a Negative Number from a Negative Number

(-3) - (-8) = (-3) + 8 = 5 (Subtract -8 by adding its additive inverse, 8)

Using the Number Line for Subtraction

Similar to addition, you can use the number line for subtraction. Start at the first integer. If you're subtracting a positive integer, move to the left. If you're subtracting a negative integer, move to the right (because subtracting a negative is the same as adding a positive).

Example 10: Visualizing Subtraction on the Number Line

Let's visualize 5 - 3 using the number line. Start at 5. Since we're subtracting 3 (a positive integer), move three units to the left. You'll land on 2, which is the difference.

Combining Addition and Subtraction

Many problems involve a combination of both addition and subtraction. In these cases, it's helpful to rewrite the expression to only involve addition, using the additive inverse rule for subtraction.

Example 11: Combining Addition and Subtraction

5 + (-3) - 2 + (-7)

First, rewrite the subtractions as additions:

5 + (-3) + (-2) + (-7)

Now, add the numbers:

5 + (-3) + (-2) + (-7) = 5 + (-12) = -7

Remember to work from left to right, or group numbers with the same signs together for easier calculation.

Solving Real-World Problems with Integers

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

• Temperature: Temperatures can be both positive and negative. For example, adding a temperature increase or subtracting a temperature decrease.
• Finance: Positive integers represent income or deposits, while negative integers represent expenses or withdrawals. Calculating your bank balance requires adding and subtracting integers.
• Elevation: Elevations above sea level are positive, and elevations below sea level are negative. Calculating changes in altitude involves integer addition and subtraction.
• Scorekeeping: Many games use a scoring system that can involve both positive and negative points.

Practice Problems

To solidify your understanding, try these practice problems:

1. -12 + 7 = ?
2. 15 + (-8) = ?
3. -6 - 4 = ?
4. -9 - (-5) = ?
5. 10 + (-5) - 3 + 2 = ?
6. -20 + 12 - (-8) + 6 = ?
7. A submarine is at -250 meters. It ascends 100 meters. What is its new depth?
8. The temperature was 5 degrees Celsius. It dropped 8 degrees. What is the new temperature?

Conclusion

Adding and subtracting integers might initially seem daunting, but by understanding the rules, utilizing the number line for visualization, and consistently practicing, you can master these fundamental arithmetic operations. Remember that subtracting an integer is equivalent to adding its opposite, simplifying complex expressions. By applying these principles, you'll confidently navigate mathematical problems involving integers and their applications in various real-world scenarios. Continuous practice and problem-solving are key to building a strong foundation in integer arithmetic.