Problem 19 - Elementary Algebra

Question

For the following problems, next to each real number, note all collections to which it belongs by writing \(N\) for natural numbers, \(W\) for whole numbers, \(Z\) for integers, \(Q\) for rational numbers, Ir for irrational numbers, and \(R\) for real numbers. Some numbers may require more than one letter. $$49.125125125 \ldots$$

Step-by-Step Solution

Answer

Answer: The repeating decimal 49.125125125... belongs to the sets of rational numbers (Q) and real numbers (R).

Step 1: Determine if the given number is rational or irrational

To identify if the given number is rational or irrational, recall that rational numbers can be expressed as a fraction of two integers (numerator and denominator), where the denominator is not equal to zero. Observe if we can rewrite the decimal \(49.125125125 \ldots\) as a fraction. Given that the number repeats itself, we can rewrite it as a fraction and therefore, it is a rational number.

Step 2: Assign the number to the sets it belongs to

Since we have determined that the number is a rational number, it automatically belongs to the set of rational numbers (\(Q\)). Additionally, all rational numbers are considered real numbers, so the number also belongs to the set of real numbers (\(R\)). So, the number \(49.125125125 \ldots\) can be assigned to these sets: \(Q\) and \(R\).

Key Concepts

Rational Numbers

Rational numbers are a fundamental concept in mathematics that allow us to express quantities accurately in fractional terms. A number is considered rational if it can be written as a fraction \( \frac{a}{b} \), where both \(a\) and \(b\) are integers, and \(b eq 0\). This includes integers themselves because they can be expressed with a denominator of one (e.g., \(5 = \frac{5}{1}\)).
When encountering a decimal number, like \(49.125125125\ldots\), identifying whether it's rational involves looking at its decimal representation. If it repeats or terminates, it is a rational number. Repeating decimals, such as \(49.125125125\ldots\), have an infinite sequence that cycles and can be converted into fractions. This means they fit into the family of rational numbers. Here's how:

• The decimal repeats, indicating periodicity.
• This periodicity allows us to convert the number into a fraction by algebraic methods.

Therefore, if you see repeating patterns in a decimal, you can confidently categorize it as rational.

Number Sets

Understanding number sets is crucial in identifying where various numbers belong. In mathematics, numbers are organized into different sets based on their properties. Here's a quick overview:

• Natural Numbers (\(N\)) include positive integers starting from one: \(1, 2, 3, \ldots\)
• Whole Numbers (\(W\)) extend natural numbers by including zero: \(0, 1, 2, 3, \ldots\)
• Integers (\(Z\)) encompass positive and negative whole numbers, including zero: \(-3, -2, -1, 0, 1, 2, 3, \ldots\)
• Rational Numbers (\(Q\)) can be written as a fraction \(\frac{a}{b}\), where \(b eq 0\). They include integers, finite decimals, and repeating decimals.
• Irrational Numbers (Ir) are numbers that cannot be written as a simple fraction. Their decimal expansions are non-repeating and non-terminating.
• Real Numbers (\(R\)) include both rational and irrational numbers, covering all possible numbers on the number line.

Identifying number sets helps in visualizing and understanding where a number like \(49.125125125\ldots\) sits; it belongs to both the rational and real number sets.

Decimal Numbers

Decimal numbers are a way to represent numbers with a base-ten system, incorporating a fractional component after a decimal point. They are versatile and commonly used in mathematical expressions and real-world applications.
Decimals come in two forms: terminating and repeating:

• Terminating decimals come to an end, such as \(0.75\) or \(2.5\).
• Repeating decimals continue indefinitely with a repeating pattern, like \(0.666\ldots\) or \(49.125125125\ldots\).

Converting decimals to fractions involves identifying their type:

• For terminating decimals, simply use the place value of the last digit (e.g., \(0.75 = \frac{75}{100} = \frac{3}{4}\)).
• For repeating decimals, use algebraic methods to set up equations that isolate the repeating part, enabling conversion into a fraction (e.g., \(49.125125125\ldots\) can be represented as a fraction by recognizing its repeating nature).

Decimals allow for precise representation and are an integral part of working with rational numbers.