Problem 10 - Precalculus: Mathematics for Calculus

Question

Real Numbers List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers $$\left\\{1.3,1.3333 \ldots, \sqrt{5}, 5.34,-500,1 \frac{2}{3}, \sqrt{16}, \frac{246}{579},-\frac{20}{5}\right\\}$$

Step-by-Step Solution

Answer

Natural numbers: \( \sqrt{16} \); Integers: \( \sqrt{16}, -500, -\frac{20}{5} \); Rational numbers: \(1.3, 1.3333\ldots, 5.34, \sqrt{16}, \frac{246}{579}, -500, -\frac{20}{5}\); Irrational numbers: \( \sqrt{5} \).

Step 1: Identify Natural Numbers

Natural numbers are the set of positive integers starting from 1 without any fractional or decimal parts. Out of the given set, natural numbers are: \( \sqrt{16} \), which simplifies to 4.

Step 2: Identify Integers

Integers include all whole numbers, both positive and negative, including zero. The elements that are integers in the given set are \( \sqrt{16} \) (which is 4) and \(-500\), \(-\frac{20}{5}\) (which is -4).

Step 3: Identify Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). The rational numbers in the set include: \(1.3\), \(1.3333\ldots\), \( 5.34\), \( \sqrt{16} \) (which is 4), \( \frac{246}{579} \), \(-500\), and \( -\frac{20}{5} \).

Step 4: Identify Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction; they have non-terminating, non-repeating decimal parts. In the given set, the irrational number is \( \sqrt{5} \).

Key Concepts

Natural Numbers

Natural numbers are one of the first number sets we learn about. These numbers are the foundation of basic counting and include all positive whole numbers starting from 1. Here’s a useful way to remember them:

• They do not include zero.
• They do not have fractional or decimal parts.
• They are always positive.

In the set provided in the exercise, the natural number we find is obtained by simplifying \( \sqrt{16} \), which is 4. This represents a typical natural number because it is a positive whole number and fits all the criteria mentioned above.

Integers

Integers are a broader concept than natural numbers because they include more types of numbers. They cover both positive and negative whole numbers, as well as zero, and are commonly used in everyday calculations.

• Integers include positive numbers like natural numbers.
• They also cover negative numbers.
• Zero is also considered an integer.

In the exercise set, we identify integers by transforming or directly using some numbers:

• \( \sqrt{16} \), which simplifies to 4.
• \(-500\), a straightforward negative integer.
• \(-\frac{20}{5}\), which simplifies to -4.

These numbers are integral because they meet the criteria of having no fractional or decimal parts, and include negatives and positives, as well as zero.

Rational Numbers

Rational numbers are more diverse compared to natural numbers and integers. These are numbers that can be expressed as fractions, where both the numerator and the denominator are integers, with the denominator not being zero.

• They can be either integers or non-integers.
• They include positive and negative numbers.
• They can end in a repeating or terminating decimal.

From the exercise, the rational numbers are easily picked by checking if they can be written as a fraction or having repeating or terminating decimals:

• \(1.3\) is a terminating decimal.
• \(1.3333\ldots\) is a repeating decimal.
• \(5.34\) is also a terminating decimal.
• \(\sqrt{16}\) simplifies to 4, another rational number.
• \(\frac{246}{579}\) is already in fraction form.
• \(-500\) is a negative integer, hence rational.
• \(-\frac{20}{5}\) simplifies to -4, which is an integer.

Rational numbers are thus straightforward to identify, provided they fit in a fraction format or end in repeating patterns.

Irrational Numbers

Irrational numbers add an interesting twist to our understanding of numbers. Unlike rational numbers, they cannot be expressed as simple fractions because their decimal parts go on forever without repeating a pattern.

• They have non-terminating and non-repeating decimals.
• They include certain roots which do not simplify to a whole number.
• They do not include numbers that terminate or repeat.

In the exercise set, the irrational number that stands out is \( \sqrt{5} \). This number is irrational because when calculated, its decimal expansion continues infinitely without repeating.