Problem 12 - 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 natural numbers

Step-by-Step Solution

Answer

14 is the natural number on the list.

Step 1: Understand Natural Numbers

Natural numbers are the set of positive integers starting from 1 going forward, i.e., \ \(1, 2, 3, 4, \ldots\). They do not include zero, fractions, decimals, or negative numbers.

Step 2: Evaluate Each Number

From the list: - \(0\) is not a natural number as natural numbers start from 1.- \(14\) is a positive integer, so it is a natural number.- \(\frac{2}{3}\) is a fraction, not a natural number.- \(\pi\) (approximately 3.14159) is not an integer, hence not a natural number.- \(\sqrt{7}\) is an irrational number, hence not a natural number.- \(-\frac{11}{14}\) is a negative fraction, hence not a natural number.- \(2.34\) is a decimal number, not a natural number.- \(-19\) is a negative integer, not a natural number.- \(\frac{55}{8}\) is a fraction, hence not a natural number.- \(-\sqrt{17}\) is a negative irrational number, hence not a natural number.- \(3.2\overline{1}\) (recurring decimal) is not a natural number.- \(-2.6\) is a negative decimal, not a natural number.

Step 3: Conclusion

From the evaluation, the only number from the list that is a natural number is \(14\). It is a positive integer greater than zero.

Key Concepts

Integers

Integers are a collection of numbers that include all whole numbers and their negative counterparts. Think of them as the backbone of basic arithmetic, providing a straightforward numerical path without fractions or decimals. They include:

• Whole numbers: such as 0, 1, 2, 3, etc.
• Negative numbers: such as -1, -2, -3, etc.

It's important to note that while integers encompass zero and negative numbers, they do not include fractions or decimals. For instance, numbers like \(\frac{3}{4}\) or 2.5 are not integers. This distinction is crucial when solving mathematical problems that strictly require integer inputs or outputs.
Consider the list provided in the exercise. Among the numbers, \(-19\) is an example of a negative integer, while \(14\) is a positive integer. While both are integers, only positive integers are considered when identifying natural numbers.

Positive Integers

Positive integers are a subcategory of integers, specifically including only those greater than zero. They represent the counting numbers that come naturally to us, e.g., when enumerating objects or entities. Thus, the sequence is 1, 2, 3, 4, and so forth.
Here are some key points about positive integers:

• They do not include zero or negative numbers.
• They are used for counting and ordering.

The exclusion of zero from this group is what differentiates them from the broader integer category. For example, from the exercise, \(14\) is the sole candidate that fits the description of a positive integer, making it the only natural number in the given list.
When working with math problems involving natural numbers, ensure you only choose those greater than zero.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. These numbers have non-repeating, non-terminating decimal expansions. Classic examples include \(\pi\) and \(\sqrt{2}\). Their exact value cannot be completely expressed in decimal or fractional form, setting them apart from integers or rational numbers.
Some important characteristics of irrational numbers are:

• They cannot be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(beq0\).
• Their decimal expansions go on forever without repeating.

In the given exercise, numbers like \(\pi\), \(\sqrt{7}\), and \(-\sqrt{17}\) fall under this category because they cannot be simplified into a finite or repeating decimal form. Understanding these nuances is crucial when identifying number types and determining their properties in various mathematical contexts.