Problem 55 - College Algebra

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 is an irrational number.

Step 1: Understand the Problem

We need to determine the type of number that represents the circumference of a quarter, which is calculated by multiplying the diameter by \(\pi\). The diameter given is 0.955 inches.

Step 2: Write the Formula for Circumference

The formula for the circumference \(C\) of a circle is given by \(C = \pi \times \text{diameter}\). For the quarter, this becomes \(C = \pi \times 0.955\).

Step 3: Analyze the Factors

The diameter 0.955 is a rational number, as it can be expressed as a fraction. However, \(\pi\) is an irrational number, meaning it cannot be expressed as a simple fraction.

Step 4: Determine the Nature of the Product

The product of a rational number (0.955) and an irrational number (\(\pi\)) results in an irrational number. This is because multiplying a non-zero rational number with an irrational number gives an irrational product.

Key Concepts

Understanding Irrational Numbers

Irrational numbers are a unique set of numbers that can't be written as a simple fraction. Unlike rational numbers, they do not have a repeating or terminating decimal representation. The most famous example of an irrational number is pi (\(\pi\)).
\(\pi\), which is approximately 3.14159, goes on forever without repeating. This means you can't find two whole numbers that divide evenly to create \(\pi\).
Some key features of irrational numbers include:

• Non-terminating decimal expansions.
• Non-repeating sequences of numbers.
• Can't be accurately represented by a fraction.

When you mix irrational numbers like \(\pi\) with other number types (except zero), such as in mathematical operations like multiplication, the result is typically irrational as well. This characteristic explains why the product of 0.955, a rational number, and \(\pi\) is irrational. The circumference of the quarter in this exercise is, therefore, an irrational number.

Exploring Rational Numbers

Rational numbers are conveniently predictable. These are numbers that can be expressed as the quotient of two integers (a fraction), where the denominator is not zero. Whole numbers, fractions, and integers are all rational numbers.
For instance, the number 0.955 in our exercise is rational because it can be expressed as a fraction: \(\frac{955}{1000}\).
Rational numbers have several properties:

• They have either terminating or repeating decimals.
• They can be positive, negative, or zero.
• When multiplied or divided by other rationals (except zero), they remain rational.

In the context of the U.S. Mint quarter, the diameter is a rational number since it can easily be converted into a fraction and thus fits well within the framework of rational numbers.

The U.S. Mint Quarter

The U.S. Mint is responsible for producing the nation’s coinage, including the well-known quarter. A quarter's diameter is 0.955 inches, which fits into precise specifications for consistency in currency.
This dimension is crucial since it helps ensure that machines processing coins (like vending machines) accept them uniformly.
The following are some interesting facts about the U.S. Quarter:

• Made primarily from copper and nickel.
• Weighs about 5.67 grams.
• Originally introduced in 1796, with the current design evolving over the years.

Understanding these characteristics, along with the mathematical calculations involving its dimensions, allows us to grasp how measurements, like its circumference calculated using \(\pi\), play a role in defining its unique identity in currency circulation.