Problem 90 - Intermediate Algebra

Question

Explain why every integer is a rational number, but not every rational number is an integer.

Step-by-Step Solution

Answer

Every integer is a rational number because it can be expressed as a fraction with 1 as the denominator. Not every rational number is an integer because rational numbers can have fractional parts.

Step 1: Understanding Integers

Integers are whole numbers that can be positive, negative, or zero. Examples of integers include -3, 0, and 47. They do not contain fractions or decimals.

Step 2: Defining Rational Numbers

A rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. This means it can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).

Step 3: Showing Integers are Rational Numbers

Every integer \( n \) can be written as a rational number by expressing it as \( \frac{n}{1} \). For instance, 5 is an integer and can be written as \( \frac{5}{1} \), which is a rational number because it satisfies the condition \( \frac{a}{b} \) where \( b eq 0 \).

Step 4: Exploring Rational Numbers That Are Not Integers

A rational number is not always an integer. Consider the rational number \( \frac{3}{2} \). This cannot be expressed as a whole number, thus it is not an integer. Rational numbers can have non-integer values when their numerator is not a multiple of their denominator.

Key Concepts

Integers

Integers are the simple numbers we use in everyday counting without involving any fractions or parts of a number. They include:

• Positive numbers like 1, 2, 3
• Negative numbers like -1, -2, -3
• Zero, which stands alone

Integers are straightforward because they do not include any decimals or fractional parts. They are simply whole numbers that help us count and order things around us. Understanding integers is crucial because they form the foundation of more complex number systems, such as rational numbers. They are easy to work with in basic arithmetic operations like addition, subtraction, multiplication, and division (except division by zero, which is undefined). By using integers, you can easily express quantities, measurements, or even position in counting.

Fractions

Fractions are numbers that express a part of a whole. They are written in the form of a numerator over a denominator, like this: \[ \frac{a}{b} \] The numerator, \(a\), represents how many parts we have, while the denominator, \(b\), tells us into how many parts the whole is divided. For example, \( \frac{3}{4} \) means that 3 parts out of 4 are being considered. Fractions allow for a more detailed representation of quantities than integers. You can use fractions to represent values between two whole numbers. Some key points to remember about fractions include:

• A fraction is a rational number if both the numerator and denominator are integers and the denominator is not zero.
• Fractions can be simplified if the numerator and denominator have a common factor.
• Fractions can sometimes represent whole numbers, such as \( \frac{6}{3} = 2 \).

By understanding fractions, you gain the ability to express quantities that are not whole, allowing you to grasp a broader range of mathematical concepts.

Quotient of Integers

The concept of a quotient of integers is fundamental to understanding rational numbers. A quotient refers to the result of division between two integers. When one integer is divided by another, and the denominator (the integer you are dividing by) is not zero, then the result is a rational number. It can be expressed as: \[ \frac{a}{b} \] What makes a number a rational number is its ability to be expressed as this quotient of two integers. The properties of the quotient include:

• If \(b = 1\), then \( \frac{a}{1} = a\), showing that every integer is a rational number.
• If \(a\) is not a multiple of \(b\), then \( \frac{a}{b} \) will not be an integer, hence not every rational number is an integer.

This concept is central to understanding why integers fit comfortably within rational numbers, yet not all rational numbers are integers. By learning about the quotient of integers, you develop a clearer view of how numbers can be expressed and manipulated.