Problem 41 - Prealgebra

Question

Find each of the following absolute values. $$|-231|$$

Step-by-Step Solution

Answer

The absolute value of |-231| is 231.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always non-negative.

Step 2: Apply Absolute Value Definition

For the number -231, we remove the negative sign to find its absolute value, as absolute value considers only the magnitude.Therefore, \(-231 = 231\).

Step 3: Verify the Result

Check that the result is non-negative and represents the magnitude of the original number. Since 231 is positive and represents the magnitude of -231 on the number line, the calculation is correct.

Key Concepts

Number Line

A number line is a visual representation of numbers placed along a straight line. Each point on the number line corresponds to a real number. You can think of it as a road where every point on the line is a particular number. In the center of the line, there's the number zero, and numbers increase as you move to the right, while they decrease as you move to the left.

Understanding the number line is crucial for grasping the concept of absolute value because it helps us visualize distances. The absolute value of a number is simply its distance from zero, irrespective of whether it's to the left (negative) or right (positive) on the number line. For example:

• The point for the number -231 is 231 steps away from zero, just in the opposite direction of positive numbers.
• Positive numbers like 10 are also a certain distance away from zero, in this case, 10 steps.

A number line helps simplify the concept of absolute values by providing a straight path to measure this distance.

Negative Numbers

Negative numbers are those less than zero and they appear on the left side of zero on a number line. Think of them as a reflection of positive numbers, just in the reverse direction. Negative numbers are essential for representing values below a baseline, like temperatures below freezing or debts in financial contexts.

These numbers have signs that differentiate them from positive numbers. But when it comes to measuring absolute value, the negative sign doesn't impact the distance from zero, which is why absolute value always turns out to be a non-negative number.

• Negative numbers can represent a deficit, like owing -5 dollars or an elevator going 5 floors underground when expressed without a minus sign.
• In contrast, this concept of absolute value allows us to treat numbers evenly by focusing only on their size, not direction.

Grasping negative numbers is part of understanding absolute value because they show how values less than zero fit into the broader scope of number measurement.

Magnitude of Numbers

Magnitude refers to the size or amount of something, and in mathematics, it reflects how far a number is from zero on a number line. Magnitude is always positive because it does not take the direction of the number into account, just the size. This is what we refer to when talking about the absolute value of a number.

When you consider a negative number like -231 , its magnitude is exactly 231 . This is similar to considering how heavy something is, without worrying about the direction or way it is measured.

• Magnitude allows us to compare the size of different numbers independent of whether they are positive or negative.
• An absolute value focuses solely on calculating the magnitude, ensuring that all values are treated equally based on their size.

Understanding the magnitude of numbers is essential for solving absolute value problems, as it gives you the exact distance from zero without distractions from the positive or negative signs.