Problem 55 - College Algebra with Corequisite Support

Question

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by \(\pi\). Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Step-by-Step Solution

Answer

The circumference of the quarter is an irrational number.

Step 1: Identify the Given Information

Begin by identifying the information given in the problem. The diameter of the quarter is provided as 0.955 inches. We need to determine if the circumference is a whole number, a rational number, or an irrational number.

Step 2: Apply the Circumference Formula

To find the circumference of the quarter, use the formula for the circumference of a circle, which is \( C = \pi \times d \), where \( d \) is the diameter. Substitute the diameter value to get \( C = \pi \times 0.955 \).

Step 3: Classify the Nature of \( \pi \)

Recall that \( \pi \) is an irrational number. This means it cannot be expressed as a simple fraction and its decimal representation is non-repeating and non-terminating.

Step 4: Determine the Nature of the Product

The product of an irrational number (\( \pi \)) and a rational number (0.955, which can be expressed as \( \frac{955}{1000} \)) is always an irrational number. Therefore, \( \pi \times 0.955 \) is irrational.

Step 5: Conclude the Solution

Since \( \pi \times 0.955 \) results in an irrational number, the circumference of the quarter is not a whole number or a rational number, but an irrational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers. In other words, a rational number can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). This means that rational numbers include fractions, whole numbers (since they can be expressed with a denominator of 1), and any number with a finite decimal expansion.

• A number like 0.5 is rational because it can be written as \( \frac{1}{2} \).
• Similarly, 0.955 is rational, as it can be represented as \( \frac{955}{1000} \).

Rational numbers are straightforward to understand because their decimal representation ends after a certain point or starts repeating.
It is important to note that the set of rational numbers includes both positive and negative numbers.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction of two integers. Their decimal expansions are non-repeating and non-terminating, meaning they go on forever without forming a repeating pattern.

• Examples include \( \pi \), \( \sqrt{2} \), and the number \( e \).
• Each of these numbers cannot be precisely written as a fraction, which makes them irrational.

An interesting property is that the product of a rational number and an irrational number results in an irrational number, as seen with the circumference calculation within this exercise.
When you multiply \( \pi \) (an irrational number) by 0.955 (a rational number), the result is irrational, showing the interesting relationship between these types of numbers.

Circle Circumference

The circumference of a circle is the total distance around it. To calculate this, we use the formula \( C = \pi \times d \), where \( C \) represents the circumference, \( \pi \) is a constant (approximately 3.14159), and \( d \) is the diameter of the circle.

• The diameter is a line going through the center and touching two points on the edge of the circle.
• Using this formula is crucial to find answers to real-world geometry problems, like determining the measurement around circular objects.

The nature of \( \pi \) introduces complexity. Since \( \pi \) is an irrational number, it impacts the way we view measurements involving circles, making them commonly irrational numbers.
For a quarter, with a diameter of 0.955 inches, substituting in the formula \( C = \pi \times 0.955 \) shows how the irrational nature of \( \pi \) results in the circumference being irrational.