Problem 18 - Intermediate Algebra

Question

From the list \(0,14, \frac{2}{3}, \pi, \sqrt{7},-\frac{11}{14}\), \(2.34,-19, \frac{55}{8},-\sqrt{17}, 3.2 \overline{1}\), and \(-2.6\), identify each of the following. The nonpositive integers

Step-by-Step Solution

Answer

0 and -19 are the nonpositive integers.

Step 1: Understand Nonpositive Integers

Nonpositive integers include all integers that are either zero or negative. This set of numbers consists of zero and all negative whole numbers (-1, -2, -3, ...). It is important to note that fractions, decimal numbers, and irrational numbers cannot be nonpositive integers even if they are negative or zero.

Step 2: Identify Integer Candidates

From the list provided, focus on values that appear to be whole numbers or their negative forms: - 0 - -19. We will verify their status as integers and whether they meet the nonpositive condition.

Step 3: Verify Integer and Nonpositive Conditions

Now, determine if the selected numbers are nonpositive integers: - **0:** This is a nonpositive integer as it is not greater than zero and is a whole number. - **-19:** This is a nonpositive integer as it is a whole number and less than zero.

Step 4: Conclusion

The numbers 0 and -19 are the only items from the list that satisfy the conditions of being both nonpositive and integers.

Key Concepts

Nonpositive Integers

Nonpositive integers are an important concept in mathematics. These numbers include all integers that are either zero or negative. Thus, the set of nonpositive integers includes zero and negative whole numbers like

• 0
• -1
• -2
• -3
• ...

It's crucial to remember that even if a number is negative or zero, it cannot be classified as a nonpositive integer if it is not a whole number. This means that fractions, decimals, and irrational numbers do not qualify as nonpositive integers. For example, while - \(-\frac{11}{14}\)- and \(-\sqrt{17}\) are negative, they are not whole, thus not nonpositive integers. Recognizing these criteria can clarify many algebraic challenges.

Whole Numbers

Whole numbers are a subset of real numbers that do not involve fractions or decimals. They are always non-negative and include numbers starting from zero:

• 0
• 1
• 2
• 3
• ...

Whole numbers form the foundation of basic counting and arithmetic operations. They differ from integers as integers include both negative and positive numbers, whereas whole numbers start at zero and go upwards. It is essential to distinguish between the two correctly, especially when solving algebraic problems. Recognizing whether a number is a whole number helps in understanding its properties and limitations respectively. In the context of problem-solving in algebra, this understanding ensures that numbers like - \(\frac{2}{3}\)- and \(-2.6\) are not mistakenly treated as whole numbers.

Problem Solving in Algebra

In algebra, problem-solving often involves careful examination and categorization of numbers based on their properties. Understanding number types, such as nonpositive integers and whole numbers, is vital. Here’s how to approach such problems:

• First, identify what type of numbers you are working with. For instance, determine whether they are integers, fractions, decimals, or irrational.
• Secondly, decide how these numbers fit into the categories of whole numbers and nonpositive integers.
• Finally, verify if the numbers meet the specific criteria needed for your algebraic problem.

Applying these steps ensures you accurately identify usable numbers in a given set. For example, when tasked to find nonpositive integers in a list, we look for those numbers that satisfy both the integer condition and the nonpositive condition, such as 0 and -19 in the original exercise. Mastering these steps is crucial for effectively solving algebraic challenges.