Problem 6 - Algebra and Trigonometry

Question

The absolute value of the difference between \(a\) and \(b\) is (geometrically) the _____ between them on the real number line.

Step-by-Step Solution

Answer

Distance

Step 1: Understanding Absolute Value

To solve this, recall that the absolute value, denoted as \(|x|\), of a number \(x\) is the distance of \(x\) from zero on the real number line. The distance is always non-negative.

Step 2: Difference Between Two Numbers

When considering the difference between two numbers \(a\) and \(b\), the expression \(a-b\) gives the algebraic difference between them, which can be positive, negative, or zero.

Step 3: Applying Absolute Value to Difference

The absolute value of the difference, \(|a-b|\), is the distance on the real number line between the points \(a\) and \(b\). This expression evaluates the true, non-negative distance without regard to direction.

Key Concepts

Distance on the Real Number Line

Visualize the real number line as a long horizontal line that stretches infinitely in both directions. Every point on this line corresponds to a real number. The distance on this line measures how far apart two points (or numbers) are from one another. This concept is crucial because it allows us to compare the location of one number to another. To find the distance between numbers, we use the absolute value. This guarantees that the measured distance is always a non-negative value. For example, to find the distance between 2 and 5 on the real number line, calculate the absolute value of their difference: \[ |2 - 5| = 3 \] This means that the distance between 2 and 5 is 3 units, regardless of direction.

Algebraic Difference

The algebraic difference between two numbers, say \(a\) and \(b\), is simply expressed as \(a-b\). This subtraction gives us a number that can be positive, negative, or zero, depending on the relative sizes of \(a\) and \(b\).

• If \(a > b\), then \(a-b\) will be positive.
• If \(a = b\), then \(a-b\) will be zero.
• If \(a < b\), then \(a-b\) will be negative.

The algebraic difference shows us the direction and how one number compares with another on the real number line, with positive values indicating \(a\) is to the right of \(b\), and negative values indicating \(a\) is to the left of \(b\).

Non-Negative Distance

Whenever we want to determine purely how far two numbers are from each other, we care only about the magnitude of this distance, not its direction. This is where absolute value becomes important because it transforms the algebraic difference between two numbers into a non-negative number.

• The absolute value, denoted as \(|a-b|\), effectively ignores whether \(a\) is greater or smaller than \(b\).
• No matter how positive or negative the algebraic difference is, the absolute value ensures we get a clear measure of distance, that is always a positive or zero value.

In short, using the absolute value provides a definite measure of how far apart two numbers are on the real number line without any negative signs influencing the result. This helps in various applications in both mathematics and real-world scenarios where only the distance between numbers is relevant.