Problem 114 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some rational numbers are not positive.

Step-by-Step Solution

Answer

The statement 'Some rational numbers are not positive' is true.

Step 1: Recognize the properties of rational numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b}\), where 'a' and 'b' are integers and 'b' is not equal to zero. They include positive numbers, negative numbers, and zero.

Step 2: Evaluate the statement

Approach this statement: 'Some rational numbers are not positive'. It is clear that if a fraction has a negative numerator or negative denominator (but not both), it results in a negative rational number, which is not a positive number. Additionally, zero is a rational number and it's neither positive nor negative. Therefore, this statement is true.

Step 3: Structuring the final verdict

Having recognized that rational numbers can be negative or zero as well, it is concluded that the statement 'Some rational numbers are not positive' is indeed true.

Key Concepts

Properties of Rational Numbers

Rational numbers are versatile members of the numerical family. They can be expressed in the form of a fraction \( \frac{a}{b} \), where 'a' is the numerator, 'b' is the denominator, and both 'a' and 'b' are integers. An important condition for a rational number is that the denominator 'b' cannot be zero.

Key properties to remember about rational numbers include:

• They can be added, subtracted, multiplied, and divided (except by zero) without leaving the category of rational numbers.
• They may appear as repeating or terminating decimals. For example, \( \frac{1}{3} = 0.333... \) or \( \frac{1}{4} = 0.25 \).
• Rational numbers include positive numbers, negative numbers, and zero.

Understanding these properties can make solving math problems easier, as you can anticipate and interact with rational numbers in a consistent manner.

Positive and Negative Numbers

Rational numbers can be either positive or negative. Positive rational numbers are greater than zero and are typically represented as fractions with both positive numerators and denominators. Negative rational numbers are less than zero and must always involve either a negative numerator or negative denominator, but never both. If both are negative, the negatives cancel out and the number becomes positive.

Here's how to determine their sign:

• If only the numerator is negative, the overall fraction is negative, such as \( \frac{-2}{3} \).
• If only the denominator is negative, the fraction remains negative, e.g., \( \frac{4}{-5} \).
• If both numerator and denominator are negative, like \( \frac{-6}{-8} \), the rational number is positive, simplifying to \( \frac{3}{4} \).

It's crucial to note that negative rational numbers are not positive, confirming the statement 'Some rational numbers are not positive' is indeed true.

Zero as a Rational Number

Zero is a unique and interesting rational number. It can be expressed in the form \( \frac{0}{b} \), where 'b' is any non-zero integer. Zero stands out because it is neither positive nor negative. Instead, it serves as the neutral element on the number line, dividing positive and negative numbers.

Some aspects to remember about zero include:

• Zero multiplied by any rational number results in zero, reflecting its fundamental neutral property.
• Zero cannot be used as a denominator in a fraction, as division by zero is undefined.
• Zero plays a crucial role in balancing equations and is often referred to as the 'identity element' for addition.

Thus, understanding zero as a rational number helps clarify that rational numbers can indeed be neither positive nor negative, reinforcing the truth of the statement about rational numbers not being exclusively positive.