Problem 53 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(N=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and $\mathbf{I}=\text { irrational numbers. $$\sqrt{63}$$

Step-by-Step Solution

Answer

\( \sqrt{63} \) belongs to the set of irrational numbers, \( \mathbf{I} \).

Step 1: Understand the Number

The number given is \( \sqrt{63} \). We need to determine its exact value to categorize it into different sets. Since \( \sqrt{63} \) is not a perfect square, it is approximately equal to 7.94, and it cannot be expressed as an exact fraction.

Step 2: Determine if it Belongs to Natural Numbers (N)

Natural numbers are counting numbers, starting from 1 onwards (1, 2, 3,...). Since \( \sqrt{63} \) is not a whole number and not even an integer, it does not belong to the natural numbers.

Step 3: Determine if it Belongs to Whole Numbers (W)

Whole numbers include all natural numbers and zero (0, 1, 2, 3,...). Since \( \sqrt{63} \) is not a whole number, it does not belong to whole numbers.

Step 4: Determine if it Belongs to Integers (Z)

Integers include all whole numbers and their negatives (-3, -2, -1, 0, 1, 2,...). Since \( \sqrt{63} \) is not an integer, it is not included in the set of integers.

Step 5: Determine if it Belongs to Rational Numbers (Q)

Rational numbers are numbers that can be expressed as a fraction of two integers. \( \sqrt{63} \) cannot be exactly expressed as a fraction and has a non-terminating, non-repeating decimal expansion, thus it is not a rational number.

Step 6: Determine if it Belongs to Irrational Numbers (I)

Irrational numbers cannot be expressed as a fraction, and they have non-repeating, non-terminating decimals. Since \( \sqrt{63} \) meets these criteria (it's not a perfect square and not expressible as a fraction), it is an irrational number.

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers. They start from 1 and continue upwards to infinity. Think of them as the numbers you first learned to count. So, when you think of natural numbers, you just imagine counting objects like 1 apple, 2 apples, and so on.

• They are also known as counting numbers.
• Natural numbers include numbers like 1, 2, 3, 4, etc.

However, it's important to note that natural numbers do not include zero or negative numbers. "Is a number a natural number?" is often a question asked when deciding if it belongs to this category. If the number is a decimal, fraction, or negative, it’s not natural.

Whole Numbers

Whole numbers make up an extended set including all natural numbers and the number 0.

• Whole numbers start from 0 and extend infinitely in the positive direction: 0, 1, 2, 3, and so on.
• They do not include fractions, decimals, or negative numbers.

Adding zero to the set of natural numbers allows whole numbers to represent quantities, including the absence of quantity, which is the idea behind zero. This makes them useful in a wide array of mathematical contexts, like counting objects even when there are none.

Integers

Integers broaden the concept further by including negative numbers. They encompass positive whole numbers, zero, and their negative counterparts.

• Examples include -3, -2, -1, 0, 1, 2, 3, etc.
• The set of integers is infinite in both the positive and negative directions.

Integers are crucial for calculations where subtraction and the concept of opposites are involved. They help us work with positive and negative changes, such as temperature fluctuations, banking, and more.

Rational Numbers

Rational numbers represent a broader set of numbers that includes fractions. A rational number can be expressed as the quotient or fraction of two integers (a/b), where 'a' and 'b' are both integers, and 'b' is not zero.

• They include numbers like 1/2, -4/3, and even numbers like 5 (which can be written as 5/1).
• Rational numbers have either terminating or repeating decimal expansions.

This property allows them to be written as exact decimal values, making them very practical for precise calculations in various fields including science, engineering, and finance.

Irrational Numbers

Irrational numbers are a unique set of numbers that cannot be expressed as a simple fraction of two integers. They have non-terminating, non-repeating decimal expansions. This means they go on forever without a repeating pattern.

• Common examples include \(\pi\) and \(\sqrt{2}\).
• Irrational numbers cannot be exactly written as a fraction.

They arise naturally in geometry, as the calculations involving circles and certain right triangles reveal these numbers. Despite their "irrational" name, they appear frequently in the natural world and in mathematical problems, such as in the case of \(\sqrt{63}\), which is not a perfect square and fits into this category.