Problem 18 - 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. $$-\sqrt{64}$$

Step-by-Step Solution

Answer

The number \(-\sqrt{64} = -8\) belongs to the sets of integers (\(\mathbf{Z}\)) and rational numbers (\(\mathbf{Q}\)).

Step 1: Evaluate the Square Root

First, calculate \( \sqrt{64} \). Since 64 is a perfect square, \( \sqrt{64} = 8 \). Hence, \( -\sqrt{64} = -8 \).

Step 2: Classify as an Integer

Since \( -8 \) is a whole number on the negative side of the number line, it is an integer. Therefore, \( -8 \in \mathbf{Z} \).

Step 3: Classify as a Rational Number

Rational numbers are those that can be expressed as a fraction of two integers. Since \( -8 \) can be written as \( \frac{-8}{1} \), it is a rational number. Thus, \( -8 \in \mathbf{Q} \).

Key Concepts

Integers

Integers are a set of numbers that include all whole numbers along with their negative counterparts. This means any number that does not have a fractional or decimal part can be classified as an integer. For example:

• Positive numbers like 1, 2, 3, and so on.
• Negative numbers such as -1, -2, -3, etc.
• Zero is also considered an integer.

It's important to note that integers do not include fractions or decimals. Understanding integers helps in situations like calculating debts or measuring levels below zero. In the exercise, we evaluated \(-\sqrt{64}\) to obtain \(-8\), which is an integer. Since \(-8\) fits into the negative whole number category, it can be identified as part of the integer set. This is a crucial set in mathematics, used to perform operations across various real-life scenarios.

Rational Numbers

Rational numbers are a broad category that comprises numbers that can be expressed as a fraction or ratio of two integers, where the denominator is not zero. This means rational numbers include:

• Positive and negative integers (since they can be written with a denominator of 1)
• Fractions like \(\frac{3}{4}\) or \(\frac{-2}{5}\)
• Decimals that either terminate (like 0.5, 0.75) or repeat (like 0.333...)

In the context of the exercise, after evaluating \(-\sqrt{64}\), we found that \(-8\) is a rational number. That's because it can be expressed as a fraction, \(\frac{-8}{1}\). Rational numbers are incredibly important because they allow us to represent quantities precisely, which is essential in fields like engineering and science.

Square Roots

The concept of square roots is a fundamental mathematical operation where you determine what number, when multiplied by itself, gives you the original number. This operation is particularly relevant for perfect squares, which are integers that have integer square roots, such as:

• \(4\), with a square root of \(2\)
• \(9\), with a square root of \(3\)
• \(64\), with a square root of \(8\)

Understanding square roots is key in many areas of mathematics and applied disciplines, as they help to solve equations and facilitate geometric calculations. In our example, we took the square root of \(64\) to get \(8\), and then added the negative sign, due to the initial expression \(-\sqrt{64}\). Thus, knowing how to handle square roots, including their overall properties and operations, aids significantly in solving complex mathematical problems.