Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-0.4+(-0.9)$$

Step-by-Step Solution

Answer

The answer is -1.3

Step 1: Understand the Numbers

Recognize that both numbers are negative decimals. (-0.4) and (-0.9) are both negative, so their sum is also negative.

Step 2: Find the Sum

Add the absolute values of the numbers together. Thus, |(-0.4)| + |(-0.9)| = 0.4 + 0.9 = 1.3

Step 3: Assigning Negative Sign

The sum of two negative numbers is also negative, assign a negative sign to the answer. The sum, hence results as -1.3

Key Concepts

Decimal Addition

Adding decimal numbers involves aligning the decimal points to ensure precision. Whether the numbers are positive or negative, the process remains the same. When adding decimals:

• Line up the numbers so that the decimal points are aligned vertically.
• Add zeros if necessary to ensure that both numbers have the same number of decimal places.
• Add the numbers as you would with whole numbers, column by column.

For example, with -0.4 and -0.9, you align the decimals and add 0.4 to 0.9, resulting in 1.3. The key is to focus on the digits without being distracted by negative signs until the end. This clarity helps especially when dealing with negative numbers. Remember to keep the decimals aligned throughout your calculation.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering the direction. This means the absolute value is non-negative, regardless of whether the original number is positive or negative.

• Notation: The absolute value of a number \( x \) is denoted as \( |x| \).
• For example, \( |-0.4| = 0.4 \) and \( |-0.9| = 0.9 \).

In the context of adding negative numbers, absolute values make it easier to focus on the magnitude of the numbers. When you see an exercise like \(-0.4 + (-0.9)\), you first find the absolute values, which are 0.4 and 0.9. This approach simplifies the process, turning attention away from the negative signs until the final step.

Sum of Negative Numbers

The sum of negative numbers keeps the negative sign. This happens because when you add two negative quantities, you are essentially increasing the magnitude of negativity.

• Think of it as owing money: If you owe \(0.4 and then owe an additional \)0.9, you now owe $1.3.
• Graphically, moving to the left on a number line confirms the result is still negative.

In mathematical terms, when both numbers are negative, their absolute values are added, and the sum is assigned a negative sign. In the equation \(-0.4 + (-0.9) = -1.3\), you're adding 1.3 in negative terms. This understanding helps clarify why the answer remains negative. Always reapply the negative sign after determining the total from absolute values.