Problem 10 - Elementary Algebra

Question

Determine whether the following real numbers are integers, rational, or irrational. $$ -5 $$

Step-by-Step Solution

Answer

-5 is an integer and a rational number.

Step 1: Understand Integers

Integers are whole numbers that can be positive, negative, or zero, but they do not include fractions or decimals.

Step 2: Identify the Given Number

The number given is -5.

Step 3: Determine if the Number is an Integer

Since -5 is a whole number and does not involve a fraction or decimal point, it is an integer.

Step 4: Understand Rational Numbers

Rational numbers are any numbers that can be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \). This includes integers, as they can be written with a denominator of 1.

Step 5: Determine if the Number is Rational

The number -5 can be written as \( \frac{-5}{1} \), which is a quotient of two integers, satisfying the definition of a rational number.

Step 6: Understand Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as \( \frac{a}{b} \) with integers \( a \) and \( b \). Examples include \( \pi \) and the square root of 2.

Step 7: Determine if the Number is Irrational

Since -5 can be written as \( \frac{-5}{1} \), it does not satisfy the criteria for being irrational.

Key Concepts

Integers

Integers are the building blocks of real numbers – they are whole numbers without any fractional or decimal parts. These numbers can be either positive, negative, or zero. Think of integers as the numbers you count on your fingers, including the direction! For example:

• Positive integers: 1, 2, 3, 4, ...
• Negative integers: -1, -2, -3, -4, ...
• Zero: 0

An integer like -5 fits into this category because it is a whole number without any fractional or decimal component. This makes integers unique in that they are discrete and sequential, forming an endless line stretching from negative infinity through zero all the way to positive infinity.
These qualities of integers help them play vital roles in various mathematical operations, including addition, subtraction, and understanding patterns.

Rational Numbers

Rational numbers are like versatile artisans in the world of numbers. They include any number that can be expressed as the quotient of two integers. More explicitly, a rational number can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and importantly, \( b eq 0 \). This means almost any number you can think of is a rational number!Even integers themselves qualify as rational numbers because they can be expressed as fractions with a denominator of 1. For example:

• The number 7 is a rational number since it can be written as \( \frac{7}{1} \).
• Similarly, -5 is a rational number because it can be represented as \( \frac{-5}{1} \).

This property of being part of a fraction broadens the scope of rational numbers to include whole and fractional values alike. They are helpful in everyday arithmetic where both whole numbers and parts of whole numbers are needed.

Irrational Numbers

Irrational numbers add a touch of mystery to the realm of numbers. Unlike rational numbers, they cannot be neatly written as a fraction of two integers. This means they cannot be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, with \( b eq 0 \). In fact, their decimal representation goes on forever without repeating, showing no predictable pattern.One of the famous examples of an irrational number is \( \pi \), often represented as approximately 3.14159, and the square root of 2, written as \( \sqrt{2} \). These numbers cannot be expressed as a precise fraction of integers, making them unique.The number -5 is, however, not irrational because it can be expressed as \( \frac{-5}{1} \), categorizing it as a rational number. The allure of irrational numbers lies in their endless decimals and appearances in places such as geometry and on the number line as bridging gaps where no rational numbers precisely lie.