Problem 14 - 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 integers

Step-by-Step Solution

Answer

The integers are 0, 14, and -19.

Step 1: Understand what an integer is

Integers are whole numbers that can be positive, negative, or zero. They do not have fractional or decimal parts. Examples include -3, 0, and 4.

Step 2: Analyze the given numbers

Look at each number in 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. Notice which numbers do not have decimal or fraction components.

Step 3: Identify the integers

From the list, select the numbers that meet the criteria of being whole, without fractions or decimals. These are: 0, 14, and -19.

Key Concepts

Properties of Integers

Integers are a fundamental part of mathematics, representing whole numbers without any fractional or decimal components. An integer can be either positive, zero, or negative. Each integer is a member of the set \( \mathbb{Z} \), which includes:

• Positive whole numbers (e.g., 1, 2, 3, ...)
• Negative whole numbers (e.g., -1, -2, -3, ...)
• The number zero (0), which is neither positive nor negative

One key property of integers is that they are closed under addition, subtraction, and multiplication. This means that adding, subtracting, or multiplying any two integers will always yield another integer. For example, adding two integers like 5 and -3 results in 2, which is also an integer.
However, division of integers does not always result in an integer, which is why integers are not closed under division. For instance, dividing 1 by 2 gives a fraction \( \frac{1}{2} \), which is not an integer. This simple distinction is crucial when working with integers in algebra and other mathematical fields.

Number Classification

Number classification is the system by which we categorize numbers according to their properties. Understanding these categories helps us identify what kinds of mathematical operations and transformations are possible with each type of number.Here is a brief overview of the primary categories:

• Natural Numbers: These are the counting numbers beginning from 1 upwards (1, 2, 3, ...). They do not include zero or negative numbers.
• Whole Numbers: This category includes all natural numbers plus zero (0, 1, 2, 3, ...).
• Integers: As mentioned, integers include negative, zero, and positive whole numbers (e.g., -2, 0, 3).
• Rational Numbers: These are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \(a\) and \(b\) are integers, and \(b\) is not zero. Rational numbers include integers and fractions.
• Irrational Numbers: These cannot be expressed as simple fractions, as their decimal form is non-repeating and non-terminating. Examples include \( \pi \) and \( \sqrt{2} \).

By recognizing the classification of a number, you can determine its characteristics and the operations that can be performed with it, making math problems more approachable.

Whole Numbers

Whole numbers are a subset of integers that start from zero and include all the positive integers without any fractional or decimal parts. Essentially, whole numbers are just like natural numbers but include the number zero. They are useful in counting and ordering objects around us.

• Examples of Whole Numbers: 0, 1, 2, 3, 4, ...
• Non-Examples: -1, 1.5, \( \frac{3}{4} \)

Whole numbers have closed properties under addition and multiplication. This means that when you add or multiply two whole numbers, the result is always another whole number. For example, adding 2 and 3 gives 5, a whole number. Similarly, multiplying 2 by 4 results in 8.
However, whole numbers are not closed under subtraction; for instance, subtracting 5 from 3 gives -2, which is not a whole number. Similarly, whole numbers are not closed under division as dividing 5 by 2 results in 2.5, which is not a whole number. Understanding the properties of whole numbers can aid in solving problems correctly and efficiently in various mathematical contexts.