Comparing Values Absolute Value |-0.073| Vs 10.731

When dealing with mathematical comparisons, it's crucial to have a solid understanding of concepts like absolute value and how to compare decimal numbers. In this article, we will delve into a specific problem that involves comparing the absolute value of a negative decimal with a positive decimal. We'll break down the steps, explain the underlying principles, and ultimately arrive at the correct solution. This comprehensive guide aims to not only answer the question but also to enhance your overall understanding of numerical comparisons.

Breaking Down the Problem

The question at hand presents us with the following comparison:

|-0.073| □ 10.731

Our task is to determine which of the given statements accurately compares the values on either side of the square. The statements are:

A. 0.073 < 0.73 B. 0.073 < -0.73 C. 0.073 > 0.73 D. 0.073 = 0.73

To correctly address this, we need to first understand what absolute value means and how it affects the number we're dealing with. Then, we'll compare the resulting value with the decimal number on the other side of the comparison.

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. It is denoted by two vertical bars surrounding the number, like this: |x|. The absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. For example:

• |5| = 5
• |-5| = 5

In essence, the absolute value strips away the sign of the number, leaving us with its magnitude. This concept is crucial for solving our problem.

Applying Absolute Value to Our Problem

In our given problem, we have |-0.073|. Applying the concept of absolute value, we find that:

|-0.073| = 0.073

This means we are now comparing 0.073 with 10.731. The original comparison now looks like this:

0. 073 □ 10.731

Now, we need to evaluate the options provided to see which one correctly fills the square.

Comparing Decimal Numbers

To compare decimal numbers, we look at the digits from left to right, starting with the ones place, then the tenths place, the hundredths place, and so on. If the digits in a certain place value are different, the number with the larger digit in that place is the larger number. If the digits are the same, we move to the next place value.

In our case, we are comparing 0.073 and 10.731. Clearly, 10.731 is a much larger number because it has a '1' in the tens place, while 0.073 has a '0' in the ones place. Therefore, 0.073 is less than 10.731.

Evaluating the Options

Now that we understand the absolute value and how to compare decimals, let's evaluate the given options:

A. 0.073 < 0.73: This statement compares 0.073 with 0.73. While both are positive decimals less than 1, 0.73 is larger than 0.073 because the tenths place in 0.73 (which is 7) is greater than the tenths place in 0.073 (which is 0). This option is relevant as it involves the correct absolute value but doesn't solve our original problem's comparison with 10.731 directly.

B. 0.073 < -0.73: This statement is incorrect. A positive number (0.073) cannot be less than a negative number (-0.73). Negative numbers are always less than positive numbers. This option is clearly not the correct answer.

C. 0.073 > 0.73: This statement is also incorrect. As we discussed in option A, 0.073 is less than 0.73. The greater than (>) sign is inappropriate here.

D. 0.073 = 0.73: This statement is incorrect. 0.073 and 0.73 are distinct decimal numbers with different values. The equal sign (=) is not suitable for this comparison.

Identifying the Correct Comparison

After determining the absolute value of -0.073, which is 0.073, we need to compare it with 10.731. None of the options A, B, C, or D directly compare 0.073 with 10.731. However, by understanding the relationship between 0.073 and the values in the options, we can deduce the correct comparison.

Option A states 0.073 < 0.73, which is true. This tells us that 0.073 is a small value compared to 0.73. Since 10.731 is significantly larger than both 0.073 and 0.73, the correct relationship for our original problem is 0.073 < 10.731. Although this option doesn't explicitly state the comparison with 10.731, it lays the groundwork for understanding the magnitude of 0.073.

The Correct Answer and Why

While none of the options directly provide the comparison 0.073 < 10.731, option A (0.073 < 0.73) is the most relevant. It correctly establishes that 0.073 is a small value. Considering the original problem, the correct comparison should be:

0. 073 < 10.731

This is because 0.073 is significantly smaller than 10.731. 10.731 is a number greater than 10, while 0.073 is a fraction of 1. Therefore, 0.073 is undoubtedly less than 10.731.

Why Other Options Are Incorrect

To further solidify our understanding, let's reiterate why the other options are incorrect:

• Option B (0.073 < -0.73): A positive number can never be less than a negative number.
• Option C (0.073 > 0.73): 0.073 is smaller than 0.73, not greater.
• Option D (0.073 = 0.73): These are two distinct values; they are not equal.

Key Takeaways

• Absolute Value: The absolute value of a number is its distance from zero and is always non-negative.
• Comparing Decimals: Compare digits from left to right, starting with the largest place value.
• Number Magnitude: Understand the relative size of numbers to make accurate comparisons.

Conclusion

In conclusion, while the provided options don't directly state the comparison between 0.073 and 10.731, understanding the underlying mathematical principles allows us to deduce that 0.073 is less than 10.731. The most relevant option, A (0.073 < 0.73), helps establish the small magnitude of 0.073, making it clear that it is significantly less than 10.731. This exercise underscores the importance of grasping fundamental concepts like absolute value and decimal comparisons in mathematics.

By breaking down the problem into smaller parts, understanding the definitions and principles involved, and carefully evaluating each option, we can confidently arrive at the correct comparison. This approach not only solves the immediate problem but also enhances our overall mathematical reasoning and problem-solving skills.