Problem 36 - College Algebra with Modeling and Visualization

Question

Between any two real numbers \(a\) and \(b\) there is always another real number. How could such a number be found?

Step-by-Step Solution

Answer

Use the formula \(c = \frac{a + b}{2}\) to find a number between \(a\) and \(b\).

Step 1: Identify Two Real Numbers

Let's choose two real numbers, such as \(a = 2\) and \(b = 5\). These represent any generic real numbers between which we need to find another real number.

Step 2: Understand the Concept

The statement implies that for any two numbers \(a\) and \(b\), it is always possible to find a number \(c\) such that \(a < c < b\). This is based on the property of the density of real numbers.

Step 3: Locate a Midpoint

To find a number \(c\) between \(a\) and \(b\), consider the midpoint formula. The midpoint \(c\) can be calculated as \(c = \frac{a + b}{2}\).

Step 4: Calculate Midpoint for Given Numbers

Substitute \(a = 2\) and \(b = 5\) into the midpoint formula: \(c = \frac{2 + 5}{2} = \frac{7}{2} = 3.5\).

Step 5: Verify the Solution

Verify that \(c = 3.5\) satisfies \(a < c < b\) for our choice of \(a = 2\) and \(b = 5\): \(2 < 3.5 < 5\), confirming the solution is correct.

Key Concepts

Midpoint Formula

The midpoint formula is a useful concept in mathematics that helps us find a number exactly halfway between two given points. Imagine you have two points on a number line, labeled as \(a\) and \(b\). The formula to find their midpoint \(c\) is given by:\[c = \frac{a + b}{2}\]This formula is simple yet powerful. It works by essentially averaging the two numbers, thereby giving the number right in the middle. For example, let's consider the numbers 2 and 5. If you plug these into the formula, you get:\[c = \frac{2 + 5}{2} = 3.5\]Thus, 3.5 is the point directly between 2 and 5 on the number line. This concept is vital in various mathematical problems and helps establish other principles such as finding points within intervals.

Density of Real Numbers

The principle of the density of real numbers is a fascinating aspect of the real number system. It indicates that between any two real numbers, there are infinitely many other real numbers. This property underscores the continuous nature of real numbers on the number line. Here's what this means in practical terms:

• If you have any two real numbers, say 2 and 5, you can always find another number between them.
• In fact, there isn't just one; there are infinitely many. For example, numbers like 3, 4, and 3.5 are all between 2 and 5.
• This property applies to any pair of real numbers, no matter how close they are to each other.

This intrinsic density is what makes calculus and other advanced areas of mathematics possible, as it allows us to think about limits and continuity.

Inequalities

Inequalities are a crucial tool in mathematics to compare values to determine their relative sizes. They use symbols like \(<\), \(>\), \(\leq\), and \(\geq\), helping us express that one number is less than or greater than another.When you calculate a midpoint, inequalities help verify that the number you've found indeed lies between your chosen numbers. For the example situation where we found the midpoint of 2 and 5 to be 3.5, we use the inequality:\[2 < 3.5 < 5\]This confirms:

• 3.5 is greater than 2, which is shown by \(2 < 3.5\).
• 3.5 is less than 5, which is shown by \(3.5 < 5\).

Inequalities help ensure accuracy in calculations and provide clear, mathematical proof that one value is indeed placed between two others.