Addition and Subtraction of Integers

Integers are a set of numbers that includes all whole numbers and their negative counterparts. In simple terms, integers are numbers that do not have a fractional part and can be positive, negative, or zero. Examples of integers include -3, 0, 4, and 7. Understanding the addition and subtraction of integers is fundamental in math, especially when students begin exploring different mathematical concepts beyond counting numbers. This guide will provide a detailed explanation of how to add and subtract integers.

Understanding integers

Before we get into the addition and subtraction of integers, it is important to establish a clear understanding of what integers are.

• Positive integers: These are numbers greater than zero (e.g., 1, 2, 3, 4, 5,…).
• Negative integers: These are numbers less than zero (e.g., -1, -2, -3, -4, -5,…).
• Zero: Zero is considered an integer, but it is neither positive nor negative.

Addition of integers

Adding integers can be simple if we follow some rules based on the signs of the numbers. Let's explore these rules and apply them in different scenarios. There are two main cases to consider when adding integers:

1. Adding integers with the same sign

When adding two integers with the same sign, you simply add their absolute values (ignoring the sign) and add their common sign to the result.

Example:

• 5 + 3 = 8 (both positive, so answer is positive)
• -4 + (-2) = -6 (both negative, so answer is negative)

+3 + +2 = +5 -3 + -2 = -5

2. Adding integers with different signs

When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and add the sign of the number with the larger absolute value to the result.

Example:

• -6 + 4 = -2 (sign of 6 is negative and it is greater)
• 7 + (-3) = 4 (sign of 7 is positive and it is greater)

+6 + -4 = +2 -6 + +4 = -2

If the absolute values are equal, the result will be zero:

• 5 + (-5) = 0

Subtraction of integers

Subtracting can be a little tricky, but there is a simple rule that involves reversing the sign of the number being subtracted and then adding it. Essentially, subtracting a number is the same as adding its opposite.

1. Subtracting integers with the same sign

When subtracting integers with the same sign, convert the subtraction to addition of opposite integers, then follow the rules for addition of integers.

Example:

• 7 - 3 = 4 (Convert to 7 + (-3) = 4)
• -5 - (-2) = -3 (convert to -5 + 2 = -3)

+7 - +3 = +4 -7 - -3 = -4

2. Subtracting integers with different signs

Change the subtraction operation to addition and then add the opposite of the other integer.

Example:

• -4 - 7 = -11 (convert to -4 + (-7) = -11)
• 6 - (-2) = 8 (convert to 6 + 2 = 8)

-4 - +7 = -11 +6 - -2 = +8

The number line: A visual aid

The number line is a great tool for illustrating integer addition and subtraction. Below is an illustration of a number line that shows how these operations are performed.

<---|---|---|---|---|---|---|---|---|---> -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Using a number line:
To add +2 to +3: We start from +3, jump 2 places to the right. Result is +5.
To subtract +2 from +4: Start from +4, jump 2 places to the left. Result is +2.

Properties of integer addition and subtraction

1. Exchangeable assets

This property states that the order in which two numbers are added does not change the sum:

a + b = b + a

• Example: 5 + (-7) = -7 + 5 = -2

2. Associative property

This property states that when three or more numbers are added, grouping the numbers does not change the sum:

(a + b) + c = a + (b + c)

• Example: (-3 + 2) + 4 = -3 + (2 + 4)

3. Identity property

This property states that any number will remain the same even if it is greater than zero:

a + 0 = a

• Example: 8 + 0 = 8

Practice problems

Here are some practice problems to help you further solidify the concepts of adding and subtracting integers. Try solving these yourself and check your answers below.

• -8 + 3 = ?
• 9 + (-4) = ?
• -5 - 4 = ?
• 7 - (-2) = ?

Solving practice problems

1. -8 + 3 = -5
2. 9 + (-4) = 5
3. -5 - 4 = -9
4. 7 - (-2) = 9

Conclusion

Through exploring adding and subtracting integers, we gain a deeper understanding of how these interactions work and how they are applied in real-world situations. Mastering these concepts is important for advancing in mathematics and solving complex problems. Remember to practice using number lines for visualization and revisit these rules whenever needed. With practice and patience, you can strengthen your skills in handling integers efficiently.

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