Problem 18 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The nonpositive integers are 0 and -19.

Step 1: Understand the Concept of Nonpositive Integers

A nonpositive integer is an integer that is less than or equal to zero. This means it includes all negative integers as well as zero.

Step 2: Identify Integers from the List

Review the list: \(0, 14, \frac{2}{3}, \pi, \sqrt{7}, -\frac{11}{14}, 2.34, 3.2\overline{1}, \frac{55}{8}, -\sqrt{17}, -19, -2.6\). Identify which of these are integers. The integers from the list are \(0, 14,\) and \(-19\).

Step 3: Select Nonpositive Integers

From the integers identified in Step 2, select those that are nonpositive (less than or equal to zero). These are \(0\) and \(-19\).

Key Concepts

Nonpositive Integers

Nonpositive integers are a vital concept in math involving numbers that are equal to or less than zero. They include both negative integers and zero. Let's examine this more closely:

• Zero: This is the starting point of nonpositive integers and represents a neutral value.
• Negative Integers: These are numbers less than zero, like -1, -2, -3, etc. Each negative integer represents a value which is below zero.

Nonpositive integers do not include positive values. The easiest way to identify them is to look at their sign. Negative numbers have a minus (-) sign, while zero has no sign. This category is useful in circumstances like temperature scales or bank balances, where nonpositive values are commonplace.

Integer Identification

Determining whether a number is an integer involves checking if it’s a whole number and doesn’t include fractions or decimals. Here's a quick guide on how to identify them:

• Whole numbers: These are numbers that don't have any fractions or decimals. For example, 14 or -19 are integers.
• Not Whole Numbers: Numbers with fractions like \(\frac{2}{3}\), decimals such as 2.34, or even irrational numbers like \(\pi\), are not integers.

When looking at a list of numbers, inspecting each for whole number characteristics can help. Remember, integers can be positive, negative, or zero. They are part of the broader real numbers category but exclude fractions and decimals. This understanding is key in performing accurate mathematical operations and solving equations.

Negative Integers

Negative integers play an important role in many real-world scenarios. These are whole numbers less than zero, such as -1, -2, -19, etc. Here’s how they’re distinguished:

• Less than Zero: All negative integers are found on the number line to the left of zero.
• No Fractions or Decimals: Negative integers should not be confused with negative fractions or decimals.

Negative integers might appear daunting at first, but they show a deficit or decrease. Think about concepts like debts or temperatures below freezing, where these numbers naturally come into play. Understanding their role can simplify how you deal with negative balances and calculations.