Problem 58 - Prealgebra

Question

For each pair of numbers, choose the number that is closest to 0. $$0.1 \text { and } 0.01$$

Step-by-Step Solution

Answer

0.01 is closer to 0.

Step 1: Understand the Goal

We need to find which of the two numbers, \( 0.1 \) or \( 0.01 \), is closer to zero. This means we are comparing their absolute values since they are both positive.

Step 2: Find the Absolute Values

Since both numbers are positive, their absolute values are equal to the numbers themselves. Thus, \( |0.1| = 0.1 \) and \( |0.01| = 0.01 \).

Step 3: Compare the Absolute Values

We compare \( 0.1 \) and \( 0.01 \). Since \( 0.01 < 0.1 \), \( 0.01 \) is closer to zero than \( 0.1 \).

Step 4: Conclusion

The number that is closer to zero is \( 0.01 \).

Key Concepts

Comparing Numbers

When it comes to comparing numbers, the primary task is to determine which number is greater or smaller. In the context of finding numbers closer to zero, we compare their absolute values. Thus, we are essentially comparing the size of the numbers without considering the sign, which is especially useful when dealing with both negative and positive numbers.

• For positive numbers, the absolute value is the number itself.
• For negative numbers, the absolute value is their positive counterpart.

In our example, the numbers 0.1 and 0.01 are both positive, so we compare these numbers directly. Simply observe their magnitude. Since 0.01 is smaller than 0.1, 0.01 is closer to zero.

Positive Numbers

Positive numbers are greater than zero and are typically found to the right of zero on a number line. They have no negative sign, and their absolute values are the same as the numbers themselves.

• Examples of positive numbers include 1, 2.5, and 0.001.
• For any positive number, its position on the number line will always be to the right of zero.

When comparing positive numbers to determine their proximity to zero, it is straightforward. The number with the smaller magnitude is closer to zero. For instance, between 0.1 and 0.01, since 0.01 has a smaller magnitude, it is closer to zero.

Number Line Comparison

A number line is a visual tool to help you understand the position of numbers relative to each other. It extends infinitely in both directions, with zero placed in the center. Positive numbers are marked to the right, while negative ones lie to the left. Using a number line, you can easily compare which numbers are closer to zero.

• On a number line, numbers that are closer to zero will have a smaller distance from it.
• It provides a clear and visual way to compare different numbers by observing their positions.

For example, on the number line, 0.01 is closer to the zero point than 0.1, illustrating that 0.01 is smaller in magnitude and hence closer to zero. This visual method aids in solidifying the abstract concept of numerical distance, making it easier to compare values effectively.