Problem 8 - 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. $$ \frac{3}{1},-\frac{5}{8}, \sqrt{7}, 0 . \overline{45}, 0,5.6 \times 10^{3} $$

Step-by-Step Solution

Answer

Natural numbers: 3, 5600. Integers: 3, 0, 5600. Rational numbers: 3, -\frac{5}{8}, 0.\overline{45}, 0, 5600. Irrational numbers: \sqrt{7}.

Step 1: Classify \( \frac{3}{1} \)

The fraction \( \frac{3}{1} \) simplifies to 3. This number is a natural number because it is a positive whole number, an integer since it can be written without a fractional component, and a rational number as it can be expressed as a fraction of two integers.

Step 2: Classify \( -\frac{5}{8} \)

The number \( -\frac{5}{8} \) is not a natural number because natural numbers are positive. However, it is a rational number because it can be expressed as a fraction of two integers. It is not an integer because integers cannot have fractional parts.

Step 3: Classify \( \sqrt{7} \)

The number \( \sqrt{7} \) is not a perfect square, so it can't be expressed as a simple fraction of two integers, making it an irrational number.

Step 4: Classify \( 0.\overline{45} \)

The number \( 0.\overline{45} \) is a repeating decimal. Repeating decimals are considered rational numbers because they can be expressed as a fraction of two integers.

Step 5: Classify 0

The number 0 is an integer since it is a whole number and it is also a rational number because it can be expressed as \( \frac{0}{1} \). However, it is not considered a natural number since natural numbers are strictly positive.

Step 6: Classify \( 5.6 \times 10^{3} \)

The expression \( 5.6 \times 10^{3} \) simplifies to 5600. This is a natural number because it is a positive integer, an integer as it can be written without a fractional component, and a rational number because it can be expressed as a fraction \( \frac{5600}{1} \).

Key Concepts

Natural Numbers

Natural numbers are the basic building blocks of mathematics.
They are the counting numbers you first encountered as a child: 1, 2, 3, and so on.

• They are always positive.
• They do not include zero or any negative numbers.
• They are whole numbers, meaning they do not have fractions or decimal parts.

If you're shopping and you have 5 apples, you're using natural numbers.
In our original exercise, the number 3 (simplified from \( \frac{3}{1} \)) and the number 5600 (from \( 5.6 \times 10^{3} \)) are natural numbers.
They are both positive, whole, and without fractions.

Integers

Integers extend the number line both ways, encompassing all whole numbers.
They include positive numbers (natural numbers), zero, and negative numbers.

• They can be positive, negative, or zero.
• They don't have any decimal or fractional parts.

For instance, you might owe someone 3 volleys in a game, which can be thought of as \(-3\).
From the exercise, the numbers 3 and 5600 are not only natural but also integers.
Additionally, 0 is an integer, even though it is neither positive nor negative.

Rational Numbers

Rational numbers are a broad category that includes integers and more.
They are numbers that can be written as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).

• They can have fractional or decimal forms.
• Repeating or terminating decimals are considered rational numbers.

Think about sharing a pizza, you might get half, or \( \frac{1}{2} \) of it.
In the exercise, the numbers \( \frac{3}{1} \), \(-\frac{5}{8}\), \(0.\overline{45}\), and even 0 fall under rational numbers.
They can all be expressed neatly as fractions of integers.

Irrational Numbers

Irrational numbers are a fascinating group, offering numbers that cannot be written as a simple fraction of two integers.
They have decimal expansions that are non-repeating and non-terminating.

• They cannot be precisely written as fractions.
• Examples include \( \pi \) and the square root of numbers that aren't perfect squares.

If you'd infinitely try to write down \( \sqrt{2} \), you'd never find a fraction equal to it exactly.
In the exercise, \( \sqrt{7} \) is an irrational number because it has a non-repeating, non-ending decimal form.
It escapes the grasp of being written as a fraction like \( \frac{a}{b} \).