Problem 75 - Prealgebra

Question

Without pencil and paper or a calculator. Is \(-60\) closer to 0 or \(-100 ?\)

Step-by-Step Solution

Answer

-60 is closer to -100.

Step 1: Identify the Numbers Involved

The numbers involved in the question are -60, 0, and -100.

Step 2: Calculate the Distance to 0

Calculate the absolute value of the difference between -60 and 0. \[ | -60 - 0 | = 60 \] This shows that -60 is 60 units away from 0.

Step 3: Calculate the Distance to -100

Calculate the absolute value of the difference between -60 and -100. \[ | -60 - (-100) | = | -60 + 100 | = 40 \] This shows that -60 is 40 units away from -100.

Step 4: Compare the Distances

Compare the distances calculated in steps 2 and 3. The distance from -60 to 0 is 60, and the distance from -60 to -100 is 40. Since 40 is less than 60, this means -60 is closer to -100.

Key Concepts

Integer Comparison

In mathematics, integer comparison is essential for solving problems involving numbers on a number line. Comparing integers helps us determine which number is larger, smaller, or closer to a specific value, such as zero. It's particularly useful when dealing with real-life scenarios or mathematical exercises.

• To compare any two integers, we look at their position relative to each other on a number line.
• Integers to the right are always larger than those to the left.
• The absolute value tells us how "far" a number is from zero.

When comparing -60 to -100, for example, we see that -60 is to the right of -100 on the number line. Therefore, -60 is closer to zero.
Use integer comparison to establish a clear understanding of hierarchical relationships among numbers, especially when solving problems without calculators or visual aids.

Distance on a Number Line

Understanding distance on a number line is fundamental when evaluating the proximity of numbers, such as whether -60 is closer to zero or -100. Distance is calculated by finding the absolute value of the difference between the numbers.

• The absolute value is always non-negative since it's the "magnitude" without regard to direction.
• To find the distance between two points on the number line, subtract one from the other and take the absolute value of the result.
• For example, to calculate the distance from -60 to 0, compute \( |-60 - 0| = 60 \).
• For the distance from -60 to -100, compute \( |-60 - (-100)| = |-60 + 100| = 40 \).

This shows that -60 is closer to -100 because 40 is less than 60. Understanding this helps solve similar problems mentally, facilitating quick decision-making in related mathematical scenarios.

Negative Numbers

Negative numbers can sometimes cause confusion, especially when considering them in terms of distance and comparison. However, they follow the same basic rules as positive numbers in many ways. Here's a quick refresher on handling negative numbers:

• Negative numbers are values less than zero, found to the left of zero on the number line.
• When comparing negative numbers, the number with the smaller absolute value is actually "greater," or closer to zero.
• To add or subtract negative numbers, remember that subtracting a negative is the same as adding its positive counterpart.
• For example, \(-60 - (-100)\) becomes \(-60 + 100\), which simplifies calculations.

Working with negative numbers becomes intuitive once you understand their properties in relation to zero and each other. This fundamental concept assists in confidently tackling more complex mathematical challenges.