Problem 76 - Prealgebra

Question

Without pencil and paper or a calculator. Is \(-20\) closer to 0 or \(-30 ?\)

Step-by-Step Solution

Answer

-20 is closer to -30.

Step 1: Understand the Number Line

Imagine the number line with 0, -20, and -30 marked. The number line helps visualize the distances between these points.

Step 2: Determine Distance from -20 to 0

Calculate the absolute distance from -20 to 0 as |-20 - 0| = 20.

Step 3: Determine Distance from -20 to -30

Calculate the absolute distance from -20 to -30 as |-20 - (-30)| = 10.

Step 4: Compare Distances

Compare the distances: 20 (from -20 to 0) and 10 (from -20 to -30). Since 10 is less than 20, -20 is closer to -30.

Key Concepts

Understanding Absolute Distance

Absolute distance is the measurement of how far apart two points are on a number line, regardless of their direction. For example, the absolute distance between any two points on a number line can be calculated by finding the difference between these points and taking the absolute value of that difference. This means you disregard the negative or positive sign and focus solely on the actual space between them.

In our example, to find how far -20 is from 0, you calculate the absolute distance |-20 - 0|, which is 20. Similarly, to find how far -20 is from -30, you calculate |-20 - (-30)|, which simplifies to 10. Both measurements ignore the direction, showing you the exact space between the numbers as a positive number, making it easier to compare.

Dealing with Negative Numbers

Negative numbers are numbers that appear to the left of zero on a number line. They represent values less than zero and often denote a lack of something, such as debt or below freezing temperatures. Understanding how to work with negative numbers is crucial when solving problems involving directions and distances.

When looking at a number line and identifying negative numbers, it’s important to think about how far these numbers are apart from each other or zero, always considering that as you move leftwards, numbers decrease. In our example, both -20 and -30 are negative numbers, positioned to the left of zero. This illustrates that numbers further left on the number line are actually less than numbers to their right, even if they "look bigger" due to their absolute value. Making sense of this can help clarify how number lines work with negative values.

Comparing Distances on a Number Line

Comparing distances on a number line is a straightforward process once you understand how to calculate absolute distances and interpret negative numbers.

• First, calculate the absolute distance of each number from a point of interest, such as zero. -20's absolute distance from 0 is 20.
• Next, do the same comparison with another number, like -30. Compare these values.

In this exercise, when you compare 20 (distance from -20 to 0) with 10 (distance from -20 to -30), you see that 10 is the smaller number, indicating a shorter distance between -20 and -30.

This means -20 is closer to -30 than it is to 0. This concept allows you to make decisions based on which number is effectively "closer" in terms of distance, not just on sight but through computation. By understanding how to handle negative values accurately, you can make precise comparisons even without a calculator or additional tools.