Problem 77 - Prealgebra

Question

Without pencil and paper or a calculator. Is \(-10\) closer to \(-20\) or \(20 ?\)

Step-by-Step Solution

Answer

-10 is closer to -20 than 20.

Step 1: Understanding the Problem

We need to determine which number -10 is closer to: -20 or 20 . This involves calculating the distance of -10 from both numbers and comparing them.

Step 2: Calculate the Distance to -20

The distance from -10 to -20 is calculated using the absolute difference: \[ |-10 + 20| = 10 \].

Step 3: Calculate the Distance to 20

The distance from -10 to 20 is calculated using the absolute difference: \[ |-10 - 20| = 30 \].

Step 4: Compare the Distances

Compare the two distances calculated: 10 (to -20 ) and 30 (to 20 ). Since 10 < 30 , -10 is closer to -20 .

Key Concepts

Integer Distance

When we talk about the "integer distance" between two numbers, we're discussing how far apart these numbers are on the number line. Integer distance is always a non-negative number. The beauty of using a number line is that it provides a visual way of seeing that distance. To figure out this distance, you use the concept of absolute value. The formula for finding the distance between two numbers, say "a" and "b," is \(|a - b|\). This absolute value eliminates any negative signs, as distance is inherently a positive concept. For instance, in the exercise, we determined the distance from \(-10\) to \(-20\) by computing \(|-10 + 20| = 10\). This tells us that \(-10\) is actually 10 units away from \(-20\) on the number line.

Negative Numbers

Negative numbers are those that are less than zero, and they hold an important place on the number line. On a number line, zero is the midpoint, and negative numbers are to the left while positive numbers are to the right. Despite being less than zero, negative numbers can be just as significant as positive numbers when calculating distances and solving problems.

• For example, paired with absolute values, they help in determining distances.
• Negative signs just help identify direction on the number line but not distance.

In this context, when asked whether \(-10\) is closer to \(-20\) or \20\, recognizing negative numbers is crucial as it dictates movement to the left of zero on a number line.

Number Line

A number line is a straightforward, yet powerful tool for understanding our numeric system. It is essentially a line with numbers placed in their sequential order. The concept of a number line helps us visualize numbers and understand their relationships, including integer distances.

• Zero is typically at the center, with positive numbers to the right and negative numbers to the left.
• Distances can be visualized easily, making it simpler to determine which numbers are closer together.

In our example, using a number line allows us to physically see that \(-10\) is closer to \(-20\) than it is to \20\. Number lines show us these relationships at a glance, being a great aid in both simple math problems and more complex mathematical concepts.