Problem 7 - College Algebra with Modeling and Visualization

Question

Classify each number as one or more of the following: natural number, integer, rational number, or irrational number. $$ \pi,-3, \frac{2}{9}, \sqrt{9}, 1 . \overline{3},-\sqrt{2} $$

Step-by-Step Solution

Answer

\(\pi\) and \(-\sqrt{2}\) are irrational; -3 is an integer and rational; \(\frac{2}{9}\) and 1.\overline{3} are rational; \(\sqrt{9}\) (3) is natural, integer, and rational.

Step 1: Classify Pi (π)

The number π (pi) is a well-known irrational number because it cannot be expressed as a fraction of two integers. It cannot be fully expressed in a decimal format as it is non-terminating and non-repeating.

Step 2: Classify -3

The number -3 is an integer because it is a whole number without a fractional or decimal component. However, since it is negative, it is not classified as a natural number. It is also a rational number because it can be expressed as -3/1, where both the numerator and denominator are integers.

Step 3: Classify \(\frac{2}{9}\)

\(\frac{2}{9}\) is a rational number because it is expressed as a fraction of two integers: 2 and 9. Rational numbers also include fractions that have repeating decimals. This fraction is not an integer or a natural number.

Step 4: Classify \(\sqrt{9}\)

The square root of 9 is 3, which is a natural number since it is a non-negative whole number. It is also an integer since all natural numbers are also integers. Additionally, 3 is a rational number as it can be expressed as 3/1.

Step 5: Classify 1.\overline{3}

1.\overline{3} represents the repeating decimal 1.333..., which can be expressed as the fraction \(\frac{4}{3}\). Therefore, it is classified as a rational number. It is not a natural number or an integer because it has a decimal part.

Step 6: Classify \(-\sqrt{2}\)

\(-\sqrt{2}\) is an irrational number, as it cannot be expressed exactly as a fraction of two integers. The square root of 2 is a non-repeating, non-terminating decimal and negating it does not change its irrational property.

Key Concepts

Natural Numbers

Natural numbers are the simplest form of numbers that we use in everyday counting. Think of them as the numbers you start learning when you start counting: 1, 2, 3, and so on. These numbers are positive and do not include zero or any fractions.

• The smallest natural number is 1.
• Natural numbers do not include negative numbers or decimals.
• They are sometimes referred to as "counting numbers" because they are used for counting objects.

In our original exercise, when we evaluated \( \sqrt{9} \), we determined it was a natural number because the result is 3, which fits the criteria above.

Integers

Integers are a broader category compared to natural numbers. They include all whole numbers, both positive and negative, plus zero. While natural numbers start from 1, integers continue beyond zero, including negative numbers like -1, -2, etc.

• Zero is an integer, but it is neither positive nor negative.
• Integers do not have fractional or decimal parts.
• Examples of integers include -3, 0, and 4.

In the exercise, -3 is a perfect example of an integer because it is a whole number but falls below zero, separating it from natural numbers.

Rational Numbers

Rational numbers might sound complicated, but they're quite simple. A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b eq 0 \). They also include integers, because any integer \( x \) can be written as \( \frac{x}{1} \).

• Decimals that repeat or terminate are also rational numbers.
• Examples include 0.5, which is \( \frac{1}{2} \), and 1.333..., which is \( \frac{4}{3} \).

In the problem, \( \frac{2}{9} \) and 1.\overline{3} are identified as rational numbers, as they can be exactly expressed as fractions.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. This means their decimal form is non-repeating and non-terminating, which makes them quite unique.

• Common examples of irrational numbers include \( \pi \) and \( \sqrt{2} \).
• Irrational numbers can't be precisely written in fraction form.
• Despite their complexity, these numbers are crucial in areas of mathematics dealing with geometry and calculus.

In the original exercise, both \( \pi \) and \(-\sqrt{2} \) fit this description perfectly, signifying they're irrational.