Problem 14 - 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 integers

Step-by-Step Solution

Answer

The integers are 0, 14, and -19.

Step 1: Understand Integers

Integers are whole numbers that can be positive, negative, or zero, and do not include any fractions or decimals. Examples include -3, 0, 1, 7, and so on.

Step 2: Analyze the List

Examine each number in the list: 1. \(0\)2. \(14\)3. \(\frac{2}{3}\)4. \(\pi\)5. \(\sqrt{7}\)6. \(-\frac{11}{14}\)7. \(2.34\)8. \(-19\)9. \(\frac{55}{8}\)10. \(-\sqrt{17}\)11. \(3.2\overline{1}\)12. \(-2.6\)

Step 3: Identify Integers in the List

Identify which of the numbers are integers from the list analyzed in Step 2:- \(0\): An integer- \(14\): An integer- \(-19\): An integerOther numbers either involve fractions, decimals, non-terminating decimals, or are irrational numbers, which do not qualify as integers.

Key Concepts

Whole Numbers

Whole numbers are a fundamental part of the number system. They include all the non-negative numbers without any fractional or decimal part. This means whole numbers can be 0, 1, 2, 3, and so on.
Whole numbers start from zero, which is unique because it represents an absence of quantity. Here are key aspects you should know about whole numbers:

• They are all non-negative integers.
• They do not include any decimal or fractional parts.
• Examples include numbers like 0, 4, and 25.
• They form the basis for counting and basic arithmetic operations.

Whole numbers are simple to understand and are used in everyday life for counting things like apples, days, or cars.

Number Systems

The concept of number systems is critical in mathematics. Number systems provide a framework to classify numbers into different categories.
Different types of numbers have different characteristics and uses in mathematics. One commonly known system is the set of integers, which includes whole numbers, their negative counterparts, as well as zero.
Various Number Systems:

• Natural Numbers: These are the numbers you naturally count with, starting from 1 onwards (1, 2, 3,...).
• Whole Numbers: Natural numbers including zero, hence starting from 0 (0, 1, 2,...).
• Integers: Both positive and negative whole numbers, and zero (…, -3, -2, -1, 0, 1, 2, 3,…).
• Rational Numbers: Numbers that can be expressed as a fraction or ratio of two integers ( frac{a}{b}$ where $b$ is not zero).
• Irrational Numbers: Numbers that cannot be written as a simple fraction, such as $ pi$ or $ sqrt{2}$.

Understanding number systems helps to perform arithmetic more efficiently and solve mathematical problems with clarity.

Positive and Negative Numbers

Positive and negative numbers are critical in understanding the full scope of the number line.
The number line extends infinitely in both the positive and negative directions, with each number having an opposite counterpart. Understanding Positive and Negative Numbers:

• Positive Numbers: These are the numbers greater than zero and are located on the right side of zero on the number line.
• Negative Numbers: Located on the left side of zero, these numbers are less than zero and are denoted with a minus sign (-).
• Zero: It is neither positive nor negative and acts as a dividing point between positive and negative numbers.

Positive and negative numbers are immensely useful when dealing with real-world situations such as temperature changes, financial gains and losses, or elevations above and below sea level. They allow for a better understanding and representation of quantities that can vary in both directions.