Problem 9 - Elementary Algebra

Question

Determine whether the following real numbers are integers, rational, or irrational. $$ 4.5 $$

Step-by-Step Solution

Answer

4.5 is a rational number.

Step 1: Identify the Type of Number

The number given is 4.5, which is expressed with a decimal point.

Step 2: Evaluate if it's an Integer

Integers are whole numbers without fractions or decimals. Since 4.5 has a decimal part, it is not an integer.

Step 3: Determine if it's Rational

A rational number can be expressed as a fraction where both the numerator and denominator are integers. Since 4.5 can be written as \( \frac{9}{2} \), it is a rational number.

Step 4: Check if it's Irrational

Irrational numbers cannot be expressed as a simple fraction. Since 4.5 can be written as \( \frac{9}{2} \), it is not irrational.

Key Concepts

Integers

An integer is a number that can be written without a fractional or decimal component. Simply put, these are the numbers that you can count on your fingers. They include positive whole numbers, negative whole numbers, and zero. Some examples of integers are:

• -3
• 0
• 4

These numbers do not involve any fraction or decimal part, such as 4.5. That's why 4.5 is not an integer, because it has a decimal point. Remember, if a number has any fractional or decimal part, it cannot be classified as an integer.

Rational Numbers

Rational numbers are fascinating! They are numbers that can be expressed as a fraction of two integers. This means a rational number can be written in the form of \( \frac{a}{b} \) where \( a \) and \( b \) are both integers and \( b eq 0 \). Here are some quick examples:

• \( \frac{1}{2} \)
• 5 (which is the same as \( \frac{5}{1} \))
• -3 (which can be written as \( \frac{-3}{1} \))

In the case of 4.5, we can rewrite this number as a fraction: \( \frac{9}{2} \). Since both the numerator (9) and the denominator (2) are integers, 4.5 is indeed a rational number. So, if you can write a number as a simple fraction, it is rational.

Irrational Numbers

Irrational numbers are unique because they cannot be written as a simple fraction. They are real numbers with non-repeating and non-terminating decimal expansions. Famous examples of irrational numbers include:

• \( \pi \)
• \( \sqrt{2} \)
• \( e \)

These numbers cannot be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are integers. For 4.5, since it can be written as \( \frac{9}{2} \), it does not meet the criteria for irrational numbers. Remember, all non-repeating and non-terminating decimals are irrational.