Problem 8 - 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. \(-3\) (The annual percent change in the area of tropical rain forests)

Step-by-Step Solution

Answer

-3 is an integer, rational number, and real number.

Step 1: Identify Natural Numbers

Natural numbers are positive integers starting from 1 (1, 2, 3, ...). Since \(-3\) is negative, it is not a natural number.

Step 2: Identify Integers

Integers include the set of whole numbers and their negative counterparts (-3, -2, -1, 0, 1, 2, 3, ...). Since \(-3\) is a negative whole number, it is classified as an integer.

Step 3: Identify Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \) where \ 'a' \ and \ 'b' \ are integers and \ 'b' \ is not zero. \-3\ can be expressed as \(-\frac{3}{1}\), so it is a rational number.

Step 4: Identify Real Numbers

Real numbers include all the numbers that can be found on the number line, including both rational and irrational numbers. Since \(-3\) is a rational number, and all rational numbers are also real numbers, \(-3\) is classified as a real number.

Key Concepts

Natural Numbers

Natural numbers are some of the simplest numbers we encounter. These are the numbers we naturally begin to count with: 1, 2, 3, 4, and so on.
They are positive, whole numbers, which do not include zero. For example, when counting apples in a basket, we use natural numbers.
Natural numbers are a subset of a larger group called "integers".

• Positive: Natural numbers are always positive.
• No Fractions or Decimals: A natural number cannot be a fraction or include a decimal point.

However, it's important to note that negative numbers like \(-3\) fall outside the natural number category.

Integers

Integers extend the concept of whole numbers by including negative values and zero. They form a set that consists of...
... , -3, -2, -1, 0, 1, 2, 3, ... and continue infinitely in both directions.

• Whole Numbers: Unlike fractions, integers are whole, meaning they don’t have any decimal places or fractional parts.
• Include Zero: Zero is considered an integer, which often makes it a part of basic arithmetic operations.
• Negative and Positive: Integers can be positive, negative, or zero.

Therefore, a negative number like \(-3\) qualifies as an integer.

Rational Numbers

Rational numbers broaden our number classification to include fractions and decimals that can be written as a ratio. Any number that can be expressed as the quotient of two integers is a rational number.
For instance, the number \(-3\) can be rewritten as \(-\frac{3}{1}\), showing that it is a rational number.

• Expressed as a Fraction: Rational numbers can always be expressed as \(\frac{a}{b}\) where both 'a' and 'b' are integers, and 'b' is not zero.
• Include Simple Fractions: Numbers such as 1/2 or 3/4 are also rational numbers.
• Terminating or Repeating Decimals: Rational numbers can be decimals that terminate or repeat, like 0.75 or 0.333...

Thus \(-3\) clearly fits into the rational category, as it can be expressed through a simple fraction.

Real Numbers

The set of real numbers encompasses almost any number you can think of. Real numbers include both rational and irrational numbers. They can be found on the number line, connecting them to concepts across the mathematical spectrum.

• Rational and Irrational: Real numbers include both rational numbers, like 6.5, and irrational numbers, like \(\sqrt{2}\).
• Continuous on the Number Line: Real numbers cover every point on the number line without any gaps.

A number like \(-3\) is positioned on the number line and is rational, making it undoubtedly a part of the real number category.