Problem 6 - Precalculus

Question

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form \(\sqrt{n},\) where \(n\) is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: \(\sqrt{6}\), \(\sqrt{10}-2,3 \sqrt{15},\) and \(-5 \sqrt{3} / 2,\) (a) \(\sqrt{99}\) (b) \(\sqrt{99}+1\)

Step-by-Step Solution

Answer

(a) Irrational, (b) Irrational.

Step 1: Understand the Categories

Natural numbers are positive integers starting from 1 (e.g., 1, 2, 3,...). Integers include all whole numbers, both positive and negative (e.g., -3, 0, 2). Rational numbers can be expressed as fractions of integers. Irrational numbers cannot be expressed as fractions of integers.

Step 2: Analyze √99

To determine whether √99 is rational or irrational, first check if 99 is a perfect square. The perfect squares close to 99 are 81 (9²) and 100 (10²), hence 99 is not a perfect square. Since it is not a perfect square, √99 is irrational.

Step 3: Analyze √99 + 1

Next, consider the number (√99 + 1). Since √99 is irrational, let's analyze √99 + 1. The sum of an irrational number and a rational number (1) is always irrational. Therefore, √99 + 1 is also irrational.

Step 4: Classify Each Number

Based on our analysis: - (a) √99 is an irrational number. - (b) √99 + 1 is also an irrational number.

Key Concepts

Natural Numbers

Natural numbers are the simplest form of numbers and are primarily used for counting. They begin at 1 and continue indefinitely like 1, 2, 3, and so on. Natural numbers are

• Positive integers, which means they don't include zero or any negative numbers.
• Used commonly in daily life for counting objects.

For example, when you count apples in a basket and say there are 5 apples, you're using a natural number. Additionally, natural numbers are always whole numbers; they cannot be fractions or decimals. It's important to understand that the set of natural numbers is a fundamental building block in the number classification system, serving as a foundation on which other numbers are defined.

Integers

Integers extend the concept of natural numbers by including zero and the negative counterparts of natural numbers. They cover a wide range of whole numbers, making them a more inclusive set. In mathematical terms, integers include:

• All positive whole numbers (just like natural numbers).
• Zero, which is neither positive nor negative.
• All negative whole numbers.

For example, -3, 0, and 2 are all integers. Whenever you are working with quantities that can decrease or increase, or represent a neutral point like zero, integers come into play. Consider temperatures on a thermometer, which can go below zero or above zero, illustrating the concept of integers in a practical context.
Understanding integers is vital because they help us perform more complex operations such as subtraction, which isn't fully possible with natural numbers alone.

Rational Numbers

Rational numbers broaden our understanding of what numbers can be, allowing for fractions and decimals. A number is considered rational if it can be expressed as a fraction \[ \frac{a}{b} \]where \(a\) and \(b\) are integers, and \(b eq 0\). Key features of rational numbers include:

• They can be both positive and negative.
• They include simple fractions and mixed numbers like \( \frac{3}{4} \) or -2.5 (which equals \[ -\frac{5}{2} \]).
• Even whole numbers are rational numbers because they can be expressed as a fraction with a denominator of 1.

Rational numbers are crucial because they allow us to make precise measurements and comparisons, and they can be added, subtracted, multiplied, and divided (as long as division by zero is avoided). Consider measuring ingredients in a recipe, where precision is crucial; rational numbers help achieve that precision.

Irrational Numbers

Irrational numbers are a fascinating category of numbers that cannot be expressed as fractions of two integers. They often arise from mathematical operations that don't result in neat, repeating fractions or whole numbers. Common characteristics of irrational numbers include:

• They are non-repeating and non-terminating when written as decimals (such as the number \(\pi\)).
• They include numbers such as roots of non-perfect squares, like \( \sqrt{99} \), which can't be precisely expressed as a fraction.
• The sum of an irrational number and a rational number remains irrational. For example, \( \sqrt{99} + 1 \) is still irrational.

Irrational numbers are important because they extend the number system into areas that rational numbers cannot cover, playing a crucial role in areas like geometry, where precise calculations often need to rely on irrational numbers. Understanding irrational numbers aids in comprehending more complex mathematical concepts and helps in knowing that not all numbers can be neatly ordered or exactly calculated.