Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify the number as to type. (For example, \(\frac{1}{2}\) is rational and real, whereas \(\sqrt{5}\) is irrational and real.) $$ 2 . \overline{421} $$

Step-by-Step Solution

Answer

The number \(2.\overline{421}\) is a rational and real number.

Step 1: Identify the given number

We are given the decimal number: \(2.\overline{421}\). The horizontal line above the numbers 421 indicates that the pattern repeats infinitely.

Step 2: Determine if the number is rational or irrational

A rational number can be expressed as a fraction of the form \(\frac{p}{q}\), where both \(p\) and \(q\) are integers and \(q\) is not equal to zero. Since the given number has a repeating pattern (421), it is a rational number. We can find the corresponding fraction form of the repetition: \(\frac{421}{999}\).

Step 3: Classify the number as real or non-real

All rational and irrational numbers are considered real numbers. Since we determined that the given number is rational, it must be a real number as well.

Step 4: State the final classification

The given number, \(2.\overline{421}\), is a rational and real number.

Key Concepts

Real Numbers

Real numbers are an essential part of mathematics that encompass a wide range of values. They include all the numbers that are used in everyday mathematics, such as whole numbers, fractions, and decimals. Real numbers can be visualized on an infinite number line that stretches from negative infinity to positive infinity.

To fully understand real numbers, it's important to know that they are comprised of two main categories: rational numbers and irrational numbers.

• Rational Numbers: These are numbers that can be expressed as a fraction, \( \frac{p}{q} \), where both \( p \) and \( q \) are integers and \( q eq 0 \). For example, \( \frac{1}{2} \) and \( 5 \) are rational numbers.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction fall into this category. An example is \( \sqrt{2} \), whose decimal form continues without repeating.

Real numbers are crucial because they provide a complete and continuous numerical framework for mathematics and they ensure that most mathematical solutions can be expressed within this set.

Decimal Expansion

Decimal expansion refers to how we express numbers with decimal points. This is particularly important when dealing with non-integer real numbers.

There are three main types of decimal expansions:

• Terminating Decimals: These have a finite number of digits after the decimal point, such as \( 0.25 \).
• Repeating Decimals: These have one or more recurring digits after the decimal point. The number \( 2.\overline{421} \) is an example, where "421" repeats infinitely.
• Non-repeating, Non-terminating Decimals: These decimals go on forever without a repeat pattern. Numbers like \( \pi \) are expressed in this way.

Understanding decimal expansion is key to determining the nature of a given number, especially in differentiating between rational and irrational numbers. Numbers with repeating or terminating decimals are rational, while others are generally irrational.

Number Classification

Number classification is a fundamental part of mathematics that helps in organizing numbers based on their characteristics and properties.

With our example, \( 2.\overline{421} \), it's classified into two categories:

• Rational Numbers: The repeating decimal \( 2.\overline{421} \) indicates it can be expressed as a fraction \( \frac{421}{999} \), making it a rational number.
• Real Numbers: Since rational numbers are part of the broad set of real numbers, \( 2.\overline{421} \) is also real.

Number classification extends beyond just rational and real numbers to include other types such as: integers, whole numbers, and natural numbers. Each category helps in identifying specific properties and uses of numbers. By classifying numbers correctly, mathematicians can accurately decipher equations and functions from the simplest to the most complex.