Problem 10 - A Graphical Approach to College Algebra

Question

Classify each number as one or more of the following: natural number, integer, rational number, or real number. \(-3.5\) (The amount in billions of dollars that the Motion Picture Association of America estimates is lost annually due to piracy)

Step-by-Step Solution

Answer

The number \(-3.5\) is a rational number and a real number.

Step 1: Determine if the Number is a Natural Number

Natural numbers are the numbers used for counting, typically starting from 1, 2, 3, and so on. These numbers do not include fractions, decimals, or negative numbers. The number \(-3.5\) is a negative decimal, so it cannot be classified as a natural number.

Step 2: Determine if the Number is an Integer

Integers include all whole numbers, both positive and negative, as well as zero. An integer does not have any fractional or decimal parts. Since \(-3.5\) contains a decimal, it is not an integer.

Step 3: Determine if the Number is a Rational Number

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. The number \(-3.5\) can be written as \(-\frac{7}{2}\), which is a ratio of two integers. Therefore, \(-3.5\) is a rational number.

Step 4: Determine if the Number is a Real Number

Real numbers include all the rational numbers, integers, and all irrational numbers, which are numbers that cannot be expressed as a simple fraction. All numbers you encounter on a number line are real numbers, including \(-3.5\). Thus, \(-3.5\) is a real number.

Key Concepts

Natural Numbers

Natural numbers are the simplest form of numbers that we encounter. They begin from 1 and go on infinitely: 1, 2, 3, 4, and so on. Natural numbers are used for counting objects, steps, or anything that needs to be tallied. The key characteristics of natural numbers are:

• They are positive and start from 1, excluding zero.
• They do not include fractions, decimals, or negative values.
• They are also referred to as counting numbers.

This simplicity makes natural numbers the foundation of mathematics. However, because they lack decimals and negative values, numbers like -3.5 don't fit as natural numbers. It's good to remember that whenever you are dealing with decimals or negative signs, you're outside the realm of natural numbers.

Integers

Moving one step beyond natural numbers, we have integers. Integers include all the whole numbers and their negatives along with zero. They appear on both sides of zero on the number line: ..., -3, -2, -1, 0, 1, 2, 3, ... This makes them more versatile than natural numbers. The main features of integers are:

• They encompass all natural numbers, zero, and negative numbers.
• No fractional or decimal components are present in integers.

This is why -3.5 cannot be considered an integer. Integers are useful in situations where you need both positive and negative counting, such as temperatures or financial balances. But remember, if it has a decimal or fraction, it's not an integer.

Rational Numbers

Rational numbers broaden the spectrum by allowing us to express numbers as a fraction of two integers. A rational number can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). Some key points about rational numbers include:

• They can be positive or negative, and may include zero.
• Fractions like \( \frac{1}{2} \) or decimals like 0.75 fall under rational numbers.
• Every integer is also a rational number.

For the number -3.5, it can be expressed as \( -\frac{7}{2} \), clearly showing it as a rational number. This property of expressing decimals and fractions makes rational numbers incredibly useful for precise calculations and real-world applications.

Real Numbers

Real numbers encompass nearly all the numbers you encounter in daily life. They include all rational numbers and cover irrational numbers too. Real numbers can be depicted conveniently on the number line, capturing both finite and infinite decimals. The significant aspects of real numbers are:

• They include rational numbers (integers and fractions).
• They cover irrational numbers, which cannot be expressed as simple fractions, like \( \sqrt{2} \) or \( \pi \).
• Real numbers are the most comprehensive set of numbers, covering anything you can think of along the line, from negative infinity to positive infinity.

Given this extensive set, -3.5 is definitely a real number since it comfortably fits on the number line. Understanding real numbers is crucial since it means understanding the full extent of numbers you will deal with in life.