Problem 21 - 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. $$-\frac{56}{8}$$

Step-by-Step Solution

Answer

\(-7\) belongs to the sets of integers (\(\mathbf{Z}\)) and rational numbers (\(\mathbf{Q}\)).

Step 1: Simplify the Fraction

First, simplify the given fraction \(-\frac{56}{8}\). Divide both the numerator and the denominator by 8, the greatest common divisor. This simplifies to \(-7\).

Step 2: Identify the Number Set

Now, determine which sets this simplified number \(-7\) belongs to. Since \(-7\) is a negative whole number, it can be classified as an integer. Therefore, \(-7\) belongs to the set of integers \(\mathbf{Z}\).

Step 3: Check for Rationality

A number is rational if it can be expressed as a fraction of two integers where the denominator is not zero. \(-7\) can be written as \(-\frac{7}{1}\), so it belongs to the rational numbers \(\mathbf{Q}\) as well.

Key Concepts

Integer

Integers are a set of numbers that include all of the whole numbers and their negative counterparts. This means they range from negative numbers to positive numbers, including zero. Unlike fractions and decimals, integers have no fractional part. They can be either negative or positive.

• Examples of integers include -3, 0, and 27.
• However, numbers like 3.5 or \(\frac{2}{3}\) are not integers because they include fractions or decimals.

Breaking down the integer set further, positive integers are numbers greater than zero, negative integers are numbers less than zero, and zero itself is considered part of the integers as well. The set of integers can be denoted by the symbol \(\mathbf{Z}\), which originates from the German word "Zahlen," meaning "numbers." This set is crucial as it forms the foundation for more complex number sets.

Rational Number

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Essentially, if you can write a number as a fraction, it is considered rational. Thus, these numbers can be either positive or negative, whole numbers, or even fractions themselves.

• An easy example of a rational number is 2, which can be expressed as \(\frac{2}{1}\).
• Another example is \(-3.5\), which can also be expressed as a fraction: \(-\frac{7}{2}\).

The concept of rational numbers is important as it allows operations such as addition, subtraction, multiplication, and division (excluding division by zero) to be performed more flexibly on a broader set compared to integers. The set of rational numbers is denoted by \(\mathbf{Q}\), stemming from "quotient," which highlights the idea of a number being the division result of one integer by another.

Number Sets

Number sets refer to different classifications of numbers, grouping them according to specific properties and characteristics. It's a way of organizing numbers to better understand their roles and how they relate to one another.There are several key sets of numbers:

• Natural Numbers (\(\mathbf{N}\)): These are the numbers used for counting, starting from 1, 2, 3, and so forth.
• Whole Numbers (\(\mathbf{W}\)): These include all natural numbers plus zero.
• Integers (\(\mathbf{Z}\)): Encompassing whole numbers and their negative counterparts.
• Rational Numbers (\(\mathbf{Q}\)): These numbers can be expressed as a fraction of two integers.
• Irrational Numbers (I): Numbers that cannot be expressed as a simple fraction. Examples include \(\pi\) and \(\sqrt{2}\).

Each set has its own unique features that make them an essential component of the number system. Understanding these sets is crucial for solving mathematical problems and understanding the numeric world.