Problem 23 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$-\sqrt{100}$$

Step-by-Step Solution

Answer

\(-\sqrt{100}\) belongs to the sets of integers (\(\mathbf{Z}\)) and rational numbers (\(\mathbf{Q}\)).

Step 1: Simplify the Expression

First, simplify the expression \(-\sqrt{100}\). Calculate \(\sqrt{100}\) which equals \(10\), hence \(-\sqrt{100} = -10\).

Step 2: Classify as a Natural or Whole Number

Natural numbers (\(\mathbf{N}\)) are positive integers starting from 1. Whole numbers (\(\mathbf{W}\)) include all natural numbers as well as zero. Since \(-10\) is neither positive nor zero, it does not belong to \(\mathbf{N}\) or \(\mathbf{W}\).

Step 3: Identify as an Integer

Integers (\(\mathbf{Z}\)) include negative numbers, zero, and positive numbers without fractions or decimals. Since \(-10\) is a whole negative number, \(-10\) is an integer. Therefore, \(-\sqrt{100}\) belongs to \(\mathbf{Z}\).

Step 4: Determine if Rational

Rational numbers (\(\mathbf{Q}\)) can be expressed as the quotient of two integers, i.e., \(\frac{a}{b}\) where \(b eq 0\). The number \(-10\) can be expressed as \(-\frac{10}{1}\), so \(-\sqrt{100}\) is a rational number. Therefore, \(-\sqrt{100}\) belongs to \(\mathbf{Q}\).

Step 5: Confirm it's Not Irrational

Irrational numbers (\(\mathbf{I}\)) cannot be expressed as a simple fraction. Since \(-10\) is a rational number, it cannot be irrational. Thus, \(-\sqrt{100}\) does not belong to \(\mathbf{I}\).

Key Concepts

Real Numbers

Real numbers are numbers that can be found on the number line. This includes every number that you use in everyday life. It incorporates both rational and irrational numbers, making the set of real numbers quite vast. Real numbers can be either positive, negative, or zero.

They can be categorized as:

• Rational numbers: numbers that can be expressed as fractions.
• Irrational numbers: numbers that cannot be expressed as fractions.

Real numbers encompass the integers as well, given that they include all whole numbers, like -10. Any number, such as \(-\sqrt{100}\), that you can point to on the number line is a real number.

Integer Numbers

Integer numbers are whole numbers that do not include any fractional or decimal components. They form a straightforward set of numbers including strictly positive numbers, strictly negative numbers, and zero. Examples of integer numbers are -3, 0, and 7.

Zero is a special integer, as it's neither positive nor negative. Integer numbers are denoted by \(\mathbf{Z}\).

Since -10 is a whole number without any fractional part, it is considered an integer. Hence, \(-\sqrt{100}\) is an integer as its simplified form is -10.

Rational Numbers

Rational numbers are a broad set of numbers that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(b\) is not zero, and both \(a\) and \(b\) are integers. This includes many types of numbers such as integers, finite decimals, and repeating decimals.

These numbers are denoted by \(\mathbf{Q}\) and embrace everything from simple numbers like 1/2 to integers like -3 or 5.

Since \(-10\) can be easily written as \(-\frac{10}{1}\), it fits into the category of rational numbers. Therefore, the number -\(\sqrt{100}\) is a rational number.

Number Sets

Number sets categorize numbers into different groups based on their characteristics and properties. Understanding these sets is crucial for mathematical classification and operations. Here is a brief rundown:

• Natural Numbers (\(\mathbf{N}\)): This set includes all the positive numbers starting from 1, like 1, 2, 3, ...
• Whole Numbers (\(\mathbf{W}\)): This extends natural numbers by including 0 as well, resulting in numbers like 0, 1, 2, 3, ...
• Integers (\(\mathbf{Z}\)): Comprising whole numbers and their negative counterparts, they look like ..., -2, -1, 0, 1, 2, ...
• Rational Numbers (\(\mathbf{Q}\)): These are represented as fractions, including integers and fractions that terminate or repeat in decimal form.
• Irrational Numbers (\(\mathbf{I}\)): These cannot be expressed as fractions and include numbers like \(\pi\) and the square root of 2.

Every real number can be grouped into one or more of these sets, providing a structured understanding of numbers.