Problem 60 - Prealgebra

Question

Which number can only be classified as a rational number? \(\mathbf{F}-2 \quad \mathbf{G} \frac{1}{2}\) \(\begin{array}{lll}\mathbf{H} & \sqrt{2} & \mathbf{J} 2\end{array}\)

Step-by-Step Solution

Answer

The only number classified solely as a rational number is \( \frac{1}{2} \) (option G).

Step 1: Define Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). This includes integers, fractions, and finite or repeating decimals.

Step 2: Evaluate Each Option

Assess each option to determine if it can be represented as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers. * **F: -2** - \(-2\) is an integer, and therefore a rational number because it can be written as \( \frac{-2}{1} \).* **G: \( \frac{1}{2} \)** - This is already a fraction, so it is a rational number.* **H: \( \sqrt{2} \)** - \( \sqrt{2} \) is an irrational number because it cannot be expressed as a fraction of two integers.* **J: 2** - \(2\) is an integer, and therefore a rational number because it can be written as \( \frac{2}{1} \).

Step 3: Identify Unique Option

Out of the options assessed, identify the number that only fits the category of a rational number, without also being an integer. Since options F and J are both integers and rational numbers, and option H is irrational, the unique rational number that is not an integer is option G, \( \frac{1}{2} \).

Key Concepts

Prealgebra Concepts

Prealgebra is the foundation for understanding mathematical concepts at a higher level. It introduces you to the basic building blocks of math that are crucial for further studies. One of the key aspects of prealgebra is working with different types of numbers and understanding how they are classified.

• Numbers: In prealgebra, you start dealing with various forms of numbers, including whole numbers, integers, fractions, and decimals.
• Operations: You learn the fundamental operations such as addition, subtraction, multiplication, and division, particularly focusing on how they apply to different numbers.
• Conceptualization: It's essential to develop a strong grasp of number sense, which helps you in both numerical and problem-solving tasks later on.

Grasping these concepts paves the way for being comfortable with numbers, helping you advance to more complex mathematical topics.

Fractions and Decimals

Fractions and decimals are two ways to express numbers that are not whole. They are incredibly important in mathematics, and understanding them allows you to interpret, compare, and compute with numbers effectively.

• Fractions: A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, in the fraction \(\frac{1}{2}\), 1 is the numerator and 2 is the denominator.
• Decimals: Decimals are another way of representing fractions, especially those with denominators that are powers of ten. For example, \(0.5\) is equal to \(\frac{1}{2}\).
• Conversion: You can convert between fractions and decimals. Understanding how to do this is crucial, as it helps in flexible problem-solving.

Getting comfortable with these forms of numbers ensures you can work efficiently and accurately, especially in real-world applications where such data is often used.

Integer Classification

Integers are a specific set of numbers that include whole numbers as well as their negative counterparts. They form a significant part of number theory and general mathematics study. Understanding how to classify integers in the context of rational and irrational numbers is essential.

• Definition: Integers include the positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero.
• Classification: It's important to know that all integers are rational numbers. This means any integer can be written as a fraction \(\frac{p}{q}\) where \( q eq 0\). For instance, the integer -2 can be expressed as \(\frac{-2}{1}\).
• Distinction: Not all rational numbers are integers. For example, \(\frac{1}{2}\) is a rational number but not an integer, because it represents a part less than a whole one.

By classifying integers within the broader category of rational numbers, you build a comprehensive understanding of how numbers interact and relate to one another.