Question

Label the following numbers as natural, negative, or rational. a. \(1.333333\) b. \(-1 / 3\) C. 1066 d. \(2 / 5\) e. \(6.2\) f. \(\pi(\mathrm{pi})\)

Short answer

a: Rational, b: Negative/Rational, c: Natural/Rational, d: Rational, e: Rational, f: Neither

Step-by-step solution

Understanding Types of Numbers

First, let's understand the definitions: - **Natural Numbers**: Positive integers starting from 1, 2, 3, and so on. - **Negative Numbers**: Numbers less than zero. - **Rational Numbers**: Numbers that can be expressed as a fraction or ratio, where both the numerator and the denominator are integers.

Classifying 1.333333

The number 1.333333 is a repeating decimal. It can be written as the fraction \( \frac{4}{3} \). Because it can be expressed as a fraction of two integers, it is a **rational number**.

Classifying -1 / 3

The number \(-1 / 3\) is a fraction, and it's less than zero. Therefore, it is both a **negative number** and a **rational number** (since it can be expressed as a fraction).

Classifying 1066

The number 1066 is a whole number greater than zero, which makes it a **natural number**. It's also considered a **rational number** since it can be expressed as \(\frac{1066}{1}\).

Classifying 2 / 5

The number \(2 / 5\) is a fraction, where both the numerator and the denominator are positive integers. It is a **rational number** because it can be expressed as a ratio of two integers.

Classifying 6.2

The number 6.2 is a decimal number, which can be expressed as the fraction \(\frac{62}{10}\), reducing to \(\frac{31}{5}\). Therefore, it is a **rational number** as it can be expressed as a fraction.

Classifying π (Pi)

The number \(\pi\) (pi) is an irrational number because it cannot be exactly expressed as a fraction of two integers. It is a non-repeating, non-terminating decimal. Therefore, it is **not a rational number**, **not a natural number**, and **not a negative number**.

Key concepts

Natural Numbers

Natural numbers are the easiest type of numbers to understand. They are the numbers you count with every day, like 1, 2, 3, and so on. These numbers are always positive, and they don't include fractions, decimals, or negative numbers.
Natural numbers are the building blocks you first learned to count with in elementary school. They do not include 0 because they start from 1.

• Examples include: 1, 2, 3, 4, 5,...
• They are used for counting objects or ordering things.
• Natural numbers can go on infinitely.

It's important to remember that all natural numbers are also rational numbers, but not all rational numbers are natural.

Rational Numbers

Rational numbers can be a bit trickier than natural numbers. Any number that can be expressed as the quotient or fraction of two integers is called a rational number. This means rational numbers include natural numbers, whole numbers, fractions, and repeating or terminating decimals.
The key idea is that if a number can be expressed in the form \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\), then it is a rational number.

• Examples of rational numbers: \( \frac{2}{3} \), 4, \(-\frac{1}{5} \), 0.75
• Repeating or terminating decimals like 1.333... or 0.5 are also rational because they can be written as fractions.
• Every integer can be considered a rational number because it can be expressed with a denominator of 1, such as \( \frac{6}{1} \).

Understanding rational numbers expands your ability to work with a wide range of mathematical problems.

Negative Numbers

Negative numbers are numbers that are less than zero. They are found to the left of zero on the number line and are useful in representing values below a baseline, like sea level or temperatures below freezing.
Negative numbers are critical in mathematics for showing losses, decreases, or quantities less than a certain reference point.

• Examples include: -1, -3.5, -100
• Negative numbers can also be rational. For example, \(-\frac{4}{5}\) is both a negative and a rational number because it can be expressed as a fraction.
• They play a vital role in numerous real-world applications, such as finance and physics.

Grasping negative numbers gives you the skills to understand more complex scenarios that involve deficits or losses.

Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction of two integers. These numbers have infinite, non-repeating decimal expansions. One of the most famous irrational numbers is \( \pi \). It represents the ratio of a circle's circumference to its diameter, and its decimal form goes on forever without repeating.
Another well-known irrational number is the square root of 2, often used in geometry.

• Examples include: \( \pi \), \(\sqrt{2}\), the golden ratio \( \phi \)
• They differ from rational numbers, which have either terminating or repeating decimal patterns.
• Irrational numbers are essential in higher mathematics, including calculus and advanced geometry.

Always remember that while irrational numbers might seem "unfriendly" due to their infinite decimal form, they are fundamental to many areas of mathematics and science.