Problem 27 - Elementary Algebra

Question

Find the sums. \((-11)+(-8)\)

Step-by-Step Solution

Answer

Answer: The sum of the two negative numbers (-11) and (-8) is -19.

Step 1: 1. Identify the numbers to be added

We are given two negative numbers, \((-11)\) and \((-8)\), to be added together.

Step 2: 2. Add the numbers together

To find the sum of the two negative numbers, we can simply add them together: \((-11)+(-8) = -11 - 8\)

Step 3: 3. Perform the addition

Continuing with the addition process, we subtract 8 from -11: \(-11 - 8 = -19\)

Step 4: 4. Write the final result

The sum of \((-11)\) and \((-8)\) is \(-19\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are often used to represent values that are missing, debts, temperatures below freezing, and many more. They are represented with a minus sign (-) in front of the number. For example,

• -11 means 11 units below zero.
• -8 signifies 8 units less than zero.

When dealing with negative numbers, it is important to understand that as you move further into negative numbers (e.g., from -8 to -11), you are actually decreasing in value, moving farther away from zero. This often trips up students who need to switch their understanding of 'addition' as 'moving to the right' like with positive numbers. Instead, adding negative numbers might often seem like a subtraction, but it actually means you're just moving left on the number line.

Addition Properties

The properties of addition are foundational rules that help simplify and make calculations easier. Understanding these can help you manage numbers with confidence, even when negatives are involved. Here are some of the key properties:

• Commutative Property: This property states that the order in which we add numbers does not change the sum. For example, \(a + b = b + a\), so \((-11) + (-8) = (-8) + (-11)\).
• Associative Property: This property indicates that the way numbers are grouped in addition does not affect the sum. For example, \((a + b) + c = a + (b + c)\).
• Identity Property of Zero: Adding zero to any number does not change its value. For example, \(a + 0 = a\).

When adding negative numbers, these properties still hold true, offering a structured way to approach these calculations by grouping or reordering terms where needed.

Arithmetic Operations

Arithmetic operations are the basic computations we perform in mathematics, which include addition, subtraction, multiplication, and division. Understanding each operation fully is key to handling problems involving numbers, including negatives. Let's review:

• Addition with Negatives: Adding two negative numbers is like adding two positive counterparts and then adding a minus sign to the result. For example, adding \((-11) + (-8)\) is like adding 11 + 8, but the sum becomes negative.
• Subtraction: When subtracting numbers, think of it as adding from the other direction on the number line.
• Multiplication and Division: These operations with negative numbers have rules that dictate if the result is positive or negative,- Negative × Negative = Positive- Negative × Positive = Negative- Negative ÷ Negative = Positive- Negative ÷ Positive = Negative

By mastering the operations, especially addition with negative numbers, you will be well-equipped to tackle a wide range of mathematical problems and understand how negative values affect calculations.