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Chapter 1: Problem 17

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

Short Answer

Expert verified
The answer is -1.3

Step by step solution

01

Understand the Numbers

Recognize that both numbers are negative decimals. (-0.4) and (-0.9) are both negative, so their sum is also negative.
02

Find the Sum

Add the absolute values of the numbers together. Thus, |(-0.4)| + |(-0.9)| = 0.4 + 0.9 = 1.3
03

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(50 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(80 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminant removed increases.

Simplify: \(-8-2-(-5)+11 .\) (Section \(1.6,\) Example 3 )

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Determine whether this inequality is true or false: \(19 \geq-18 .\) (Section 1.3, Example 7)

Find this sum, indicated by a question mark. \(3(-3)=(-3)+(-3)+(-3)=?\)

Exercises \(150-152\) will help you prepare for the material covered in the next section. In each exercise, an expression with an exponent is written as a repeated multiplication. Find this product, indicated by a question mark. $$(-6)^{2}=(-6)(-6)=?$$
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