Problem 11 - 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 whole numbers

Step-by-Step Solution

Answer

The whole numbers are 0 and 14.

Step 1: Understanding Whole Numbers

Whole numbers are non-negative numbers without any fractional or decimal part. This includes 0, 1, 2, 3, etc.

Step 2: Identifying Whole Numbers

Go through each element in the list: - 0: It is a whole number.- 14: It is a whole number.- \( \frac{2}{3} \): It is a fraction and not a whole number.- \( \pi \): It is a mathematical constant and not a whole number.- \( \sqrt{7} \): It is an irrational number and not a whole number.- \( -\frac{11}{14} \): It is a negative fraction and not a whole number.- 2.34: It is a decimal and not a whole number.- -19: It is a negative number and not a whole number.- \( \frac{55}{8} \): It is a fraction and not a whole number.- \(-\sqrt{17}\): It is an irrational negative number and not a whole number.- \(3.2 \overline{1}\): It is a repeating decimal and not a whole number.- -2.6: It is a negative decimal and not a whole number.

Step 3: Listing the Whole Numbers

The whole numbers from the list are 0 and 14.

Key Concepts

Integers

Integers are a part of the number classification system. They are whole numbers that can be either positive, negative, or zero. Unlike decimals or fractions, integers are not expressed with a fractional or decimal part. When we say integers, we talk about numbers like - ...-3, -2, -1, 0, 1, 2, 3...This includes zero, meaning zero is an integer even though it's neither positive nor negative. When dealing with whole numbers specifically, we only look at non-negative integers, which exclude any numbers that have a fraction or decimal part. For example:

• Numbers like -5 and 100 are integers.
• Decimals such as 4.5 are not integers because they are not whole numbers.
• Fractions like \( \frac{1}{2} \) are also not considered integers.

Rational Numbers

Rational numbers are another important class in the realm of numbers. These are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \) where both \( a \) and \( b \) are integers and \( b eq 0 \). This includes fractions and any number that can be written as a repeating or terminating decimal.Some examples of rational numbers include:

• The number \( \frac{3}{4} \) because it can be written as a fraction.
• 0.75 is rational as it can be represented as \( \frac{75}{100} \) and simplifies to \( \frac{3}{4} \).
• Whole numbers like 5 are rational because they can be expressed as \( \frac{5}{1} \).

It is crucial to note that every integer is a rational number since it can be written with a denominator of 1.

Number Classification

Number classification helps us to understand and categorize the different types of numbers we encounter in mathematical contexts. At the basic level, numbers are broadly divided into:- **Natural Numbers**: These are the numbers starting from 1, 2, 3, and so on, basically counting numbers without fractions or decimals.- **Whole Numbers**: This category includes all natural numbers along with 0. It does not include negative numbers or fractions.- **Integers**: Comprising of whole numbers and their negative counterparts, this category excludes any fractional or decimal numbers.- **Rational Numbers**: As mentioned earlier, these numbers can be expressed as fractions where both the numerator and the denominator are integers.- **Irrational Numbers**: These cannot be expressed as fractions, such as \( \pi \) and \( \sqrt{2} \). They have non-repeating, non-terminating decimal representations.Understanding these categories helps in solving various mathematical problems and in identifying where a number fits in the grand scheme of things. For instance, distinguishing between rational and irrational numbers can be crucial for solving algebraic equations or understanding certain properties in geometry.