Problem 18 - 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. $$86.3333 \ldots$$

Step-by-Step Solution

Answer

A) Natural Numbers B) Whole Numbers C) Integers D) Rational Numbers E) Irrational Numbers F) Real Numbers

Step 1: Check if the number is a Natural Number

A natural number is a positive integer (1, 2, 3, etc.). Since \(86.3333 \ldots\) is a decimal number, it cannot be a natural number.

Step 2: Check if the number is a Whole Number

A whole number is a non-negative integer (0, 1, 2, etc.). Since \(86.3333 \ldots\) is still a decimal, it cannot be a whole number.

Step 3: Check if the number is an Integer

An integer is a whole number that can be positive, negative, or zero (-2, -1, 0, 1, 2, etc.). Since \(86.3333 \ldots\) is a decimal, it cannot be an integer.

Step 4: Check if the number is a Rational Number

A rational number is a number that can be expressed as the quotient or fraction \(\frac{a}{b}\) of two integers, where \(a\) and \(b\) are integers and \(b \neq 0\). Since we can express \(86.3333 \ldots\) as a fraction, for example: \(\frac{259}{3}\), it is a rational number. So, we can write \(Q\) as one of the collections it belongs to.

Step 5: Check if the number is an Irrational Number

An irrational number is a number that cannot be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers. Since we found that \(86.3333 \ldots\) can be expressed as a fraction in step 4, it cannot be an irrational number.

Step 6: Check if the number is a Real Number

A real number is any number that can be placed on the number line. Both rational and irrational numbers are included in the set of real numbers. Since \(86.3333 \ldots\) is a rational number, it is also a real number. Thus, we can write \(R\) as one of the collections it belongs to.

Step 7: Final Answer

The number \(86.3333 \ldots\) belongs to the collections \(Q\) for rational numbers, and \(R\) for real numbers.

Key Concepts

Rational Numbers

Rational numbers are a fascinating group within mathematics. They are numbers that can be expressed in the form of a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and importantly, \(b\) cannot be zero. This is because dividing by zero is undefined in mathematics. Rational numbers include both positive and negative numbers and even fractions like \(\frac{1}{2}\) or \(\frac{-4}{3}\). Another interesting aspect of rational numbers is their decimal representation:

• If the decimal repeats, such as 0.333... or 1.666..., it's a rational number.
• If the decimal terminates, for example, 0.75 or 1.5, it's also a rational number.

In the example of 86.3333..., even though the decimal seems endless, it repeats, allowing it to be written as the fraction \(\frac{259}{3}\). Thus, 86.3333... is a rational number.

Number Classifications

Classifying numbers is like organizing them into various categories based on shared properties. This helps us understand their distinct characteristics and use them properly in mathematical operations.Here are the main classifications of numbers:

• Natural Numbers (N): All positive integers starting from 1, like 1, 2, 3, and so on.
• Whole Numbers (W): Like natural numbers, but they also include 0, so 0, 1, 2, 3, etc.
• Integers (Z): This set includes all whole numbers and their negative counterparts, such as -2, -1, 0, 1, 2.
• Rational Numbers (Q): Already discussed, including numbers that can be expressed as a fraction.
• Irrational Numbers (Ir): Numbers that cannot be expressed as a fraction, like \(\sqrt{2}\) or \(\pi\).
• Real Numbers (R): A broad category that includes both rational and irrational numbers.

Understanding these classifications allows us to see how numbers connect and where each number fits in the grand scheme of mathematics.

Number Sets

Number sets are essential in mathematics because they group numbers that share common properties, which simplifies the study and understanding of these numbers. The concept of number sets helps in:

• Organizing numbers for easy reference.
• Clearly defining boundaries and distinctions between different types of numbers.
• Assisting in mathematical problem-solving by knowing which operations can be applied to which number sets.

The most commonly referenced number sets include:

• Natural Numbers (N)
• Whole Numbers (W)
• Integers (Z)
• Rational Numbers (Q)
• Irrational Numbers (Ir)
• Real Numbers (R)

For example, understanding that the real numbers encompass everything from rational to irrational numbers helps comprehend the entire span of the number line. This makes real numbers fundamental in mathematical understanding. Knowing these sets allows us to manipulate and use numbers accurately in all mathematical contexts.