Problem 13 - 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 rational numbers

Step-by-Step Solution

Answer

Rational numbers are: 0, 14, \( \frac{2}{3} \), \( -\frac{11}{14} \), 2.34, \( 3.2\overline{1} \), \( \frac{55}{8} \), and \( -2.6 \).

Step 1: Definition of Rational Numbers

A rational number is any number that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). This includes integers, fractions, and decimals that terminate or repeat.

Step 2: Identify Fractions and Integers

From the list, directly identify any integers and fractions. The numbers \( 0 \), \( 14 \), \( \frac{2}{3} \), and \( -\frac{11}{14} \) are rational because they fulfill the definition.

Step 3: Check Decimal Numbers for Rationality

Examine the decimal numbers: \( 2.34 \) is terminating, so it's rational. \( 3.2\overline{1} \) repeats, so it is also rational. \( \frac{55}{8} \) is already a fraction, hence rational.

Step 4: Identify Non-Rational Numbers

The numbers \( \pi \), \( \sqrt{7} \), \( -\sqrt{17} \), and \( -19 \) are not rational. \( \pi \) is an irrational constant, and square roots of non-perfect squares are irrational unless simplified to an integer.

Step 5: Check Negative Decimal

The number \( -2.6 \) is a terminating decimal, hence also a rational number.

Key Concepts

Fractions

Fractions are numbers expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \). They represent parts of a whole, divided into equal parts.
Fractions can have positive or negative signs, depending on the values of the numerator \( p \) or the denominator \( q \). Negative fractions, such as \(-\frac{11}{14}\), represent values below zero. When the numerator is smaller than the denominator, as in \(\frac{2}{3}\), the fraction is less than one. If the numerator is larger, it results in a value greater than one, such as \(\frac{55}{8}\).
Not all numbers can be expressed as fractions; this is what distinguishes rational numbers from irrational ones. When expressed as a decimal, fractions might end (also known as terminating, like 0.5) or repeat (like 0.333...).

Decimals

Decimals are another way to express fractions. They use base-10 system and a decimal point to indicate parts of numbers. Decimal numbers can be categorized based on their behavior:

• Terminating decimals: These decimals have a finite number of digits after the decimal point, such as 0.25 and 2.34. These are always rational because they can be rewritten as fractions.
• Repeating decimals: These decimals have one or more repeating digits after the point, like \(3.2\overline{1}\), which indicates a repeating 1. Repeating decimals are also rational as they represent a fraction.
• Non-terminating and non-repeating decimals: Such decimals, like \(\pi\) and \(\sqrt{2}\), are characteristic of irrational numbers.

Understanding decimals helps in identifying whether numbers are rational by checking for termination or repetition.

Integers

Integers are whole numbers that can be positive, negative, or zero. This includes numbers like \(0, 14,\) and \(-19\).
Key properties of integers include:

• They do not have fractional or decimal parts.
• Every integer is a rational number as it can be written as a fraction with a denominator of 1, like \(-19 = \frac{-19}{1}\).
• The operations of addition, subtraction, multiplication, and division (except by zero) on integers result in integers.

Understanding integers helps in recognizing their role within rational numbers and distinguishes them from decimal or fractional forms.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction. Their decimal forms are non-terminating and non-repeating. Examples include \(\pi\), \(\sqrt{7}\), and \(-\sqrt{17}\).
These numbers emerge when dealing with:

• Certain constants: Like \(\pi\), which represents the ratio of the circumference to the diameter of a circle.
• Square roots of non-perfect squares: Numbers whose roots are not whole numbers, like \(\sqrt{7}\), exemplify irrationality. These roots cannot be exactly expressed as a fraction.

Irrational numbers play a significant role in advanced mathematics, revealing structures and patterns beyond simple arithmetic.