Problem 150 - Algebra and Trigonometry

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. There exist rational numbers that are zero or negative.

Step 1: Understanding Rational Numbers

A rational number is any number that can be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \) is not equal to zero. Rational numbers include all integers, fractions, and decimals that repeat or terminate. They can be positive, negative, or zero.

Step 2: Analyzing the Statement

The statement is, 'Some rational numbers are not positive,' meaning that there exist rational numbers that are either zero or negative.

Step 3: Verifying the Statement

It's known that rational numbers can be negative (for example, \( -\frac{2}{3} \)) or zero (0 is rational as it can be expressed as \( \frac{0}{1} \)), so the statement is true.

Key Concepts

Understanding Positive and Negative Numbers

Positive and negative numbers are foundational concepts in mathematics that help us describe quantities. A positive number is greater than zero, and it's often used to represent things like elevation above sea level or a gain in finances. On the other hand, a negative number is less than zero, indicating situations like debts or a temperature below freezing.

In the context of rational numbers, both positive and negative values are possible. For example, the fraction \( \frac{3}{4} \) is a positive rational number, while \( -\frac{3}{4} \) is a negative one.

Key points to remember:

• A number is positive if it's greater than zero.
• A number is negative if it's less than zero.
• Zero is neither positive nor negative; it serves as the dividing line between positive and negative numbers.

This understanding is crucial when dealing with rational numbers, as it helps determine whether a particular number is above or below zero.

All About Integers

Integers are the whole numbers that can be positive, negative, or zero. They do not include fractions or decimals, which makes them unique within the category of rational numbers.

A set of integers includes numbers like -3, -2, -1, 0, 1, 2, 3, and so on. These numbers are commonly used for counting, ordering, and representing quantities in day-to-day life.

Important details about integers:

• "Positive integers" refer to all the whole numbers greater than zero (e.g., 1, 2, 3).
• "Negative integers" refer to all the whole numbers less than zero (e.g., -1, -2, -3).
• The integer zero is unique, as it's considered neither negative nor positive.

Integers are a key component of rational numbers, since any integer \( n \) can be expressed as a fraction \( \frac{n}{1} \). This showcases their role in the broader category of rational numbers.

Decoding Fractions

Fractions represent a part of a whole and are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) the denominator. Fractions are a core part of rational numbers and can signify smaller units of any quantity.

Fractions can either be:

• Proper fractions: Where the numerator is less than the denominator, such as \( \frac{3}{4} \), which always indicates a value less than 1.
• Improper fractions: These have numerators greater than or equal to denominators, like \( \frac{5}{3} \) or \( \frac{8}{8} \). Improper fractions can also be expressed as mixed numbers, combining a whole number and a proper fraction, such as \( 1\frac{2}{3} \).
• Equivalent fractions: Though they look different, they represent the same value, such as \( \frac{1}{2} \) and \( \frac{2}{4} \).

Fractions can also be positive or negative, allowing for a vast range of values even between whole numbers. Understanding fractions as parts of rational numbers is critical for many mathematical applications.