Problem 54 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-2-29.285\)

Step-by-Step Solution

Answer

The result is -31.285.

Step 1: Understand the Problem

We need to perform subtraction involving negative numbers. The expression is \(-2 - 29.285\). This means we are subtracting 29.285 from \(-2\).

Step 2: Rewrite the Expression

Subtracting a positive number from a negative number is the same as adding the negative of that number. Hence, \(-2 - 29.285\) can be rewritten as \(-2 + (-29.285)\).

Step 3: Add the Two Negative Numbers

When you add two negative numbers, you add their absolute values and keep the negative sign in front. \(-2\) has an absolute value of 2, and \(-29.285\) has an absolute value of 29.285. Thus, we calculate 2 + 29.285 = 31.285. The result is -31.285.

Step 4: Conclusion

The final answer is \(-31.285\), which is the result of adding the absolute values of the initial negative number and the second value, and then keeping the negative sign in front.

Key Concepts

Negative Numbers

Negative numbers are values that represent a position below zero on the number line. They are denoted with a "-" sign. Understanding negative numbers is crucial, as they often represent deficits or losses. In arithmetic, they are treated with specific rules. For instance, when dealing with negative numbers in subtraction, a subtraction from a negative is akin to adding a larger deficit.

• Negative numbers extend the number line below zero.
• Operations with negative numbers can often be rewritten for easier manipulation.
• For example, subtracting a positive number from a negative can be seen as adding two negative numbers.

By visualizing negative numbers on a number line, you can better understand how they interact with positive numbers and with each other.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering its direction. This concept is vital when adding or subtracting negative numbers because it helps simplify the arithmetic by focusing on magnitude alone.

• The absolute value is always non-negative.
• Mathematically, the absolute value of a number \(x\) is denoted as \(|x|\).
• For example, the absolute value of \(-2\) is 2, because it is 2 units away from zero.

Understanding absolute values allows you to strip away negative signs temporarily, making it easier to perform operations like addition of negative numbers, as shown in the step-by-step solution.

Arithmetic Operations

Arithmetic operations form the foundation of many mathematical concepts, including addition, subtraction, multiplication, and division. When solving problems involving subtraction of decimals with negative numbers, these operations become intertwined. It's important to understand how these operations affect negative numbers differently than positive numbers.

• Subtraction of positives and negatives can often be rewritten using addition rules.
• When you add two negative numbers, you actually add their absolute values and keep the result negative.
• Understanding these operations helps in rewriting expressions for simpler calculation.

In the example of subtracting 29.285 from \(-2\), converting the operation into addition helps clarify the process, turning subtraction into a more straightforward addition of two absolute values, followed by restoring the negative sign to the result.