Problem 44 - Beginning and Intermediate Algebra

Question

Tell which set or sets each number belongs to: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. See Example 5. $$ -\frac{1}{2} $$

Step-by-Step Solution

Answer

The number \(-\frac{1}{2}\) is a rational and real number.

Step 1: Identifying Rational Numbers

The given number is \(-\frac{1}{2}\), which is expressed as a fraction \(\frac{a}{b}\) where both \(a\) and \(b\) are integers, and \(b eq 0\). Thus, it belongs to the set of rational numbers.

Step 2: Checking Other Sets

The number \(-\frac{1}{2}\) is negative, which means it does not belong to the set of natural numbers (positive integers starting from 1) or whole numbers (natural numbers including zero). Also, it is not an integer since it has a fractional part.

Step 3: Recognizing Real Numbers

All rational numbers are part of the real numbers. Since \(-\frac{1}{2}\) is a rational number, it therefore is also a real number.

Step 4: Excluding Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction. Since \(-\frac{1}{2}\) is a fraction, it is not an irrational number.

Key Concepts

Rational Numbers

Rational numbers are an important part of the mathematical world. They're numbers that can be expressed as the ratio of two integers, i.e., in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \). This means that fractions, whole numbers, and integers can all be included in the set of rational numbers when written appropriately.
For example,

• The number \( -\frac{1}{2} \), as shown in our problem, is a rational number. It consists of the integer \(-1\), over the integer \(2\).
• The number \(3\) can be written as \( \frac{3}{1} \), illustrating that it's also a rational number.

So, rational numbers give us a great way to express and manage both simple and complex values in mathematics.

Real Numbers

Real numbers constitute the broadest range of numbers we commonly use and encounter in mathematics. They include all rational numbers and irrational numbers. This means every number that can be located on a number line is a real number.
Real numbers are like the ultimate collection of both fractions and decimals, whether they are finite or non-repeating continuous decimals.

• Rational numbers are all part of the real numbers, so \(-\frac{1}{2}\) is also a real number.
• The same goes for numbers like \( \pi \) and the square root of 2, which can't be expressed as fractions. These are irrational but still real numbers.

Understanding real numbers helps us grasp the concept of a continuous spectrum of numbers.

Natural Numbers

Natural numbers are the most basic set of numbers used in everyday counting. They start from 1 and go upwards, like 1, 2, 3, and so on. These are all positive numbers and do not include zero or any fractions or negative numbers.

• 0 is not included in natural numbers but is included when we extend them to whole numbers.
• Natural numbers form the basis as they represent actual counts of items.

In the context of our problem, \(-\frac{1}{2}\) is clearly not a natural number because it is neither positive nor whole.

Integers

Integers are a set of numbers that include all whole numbers as well as their negatives. The set includes numbers like ..., -3, -2, -1, 0, 1, 2, 3, ... and so on. Integers do not include fractions or decimals. This means numbers with decimal or fractional parts are excluded from the integer set.

• 0 is an integer, as are all its positive and negative counterparts.
• Numbers like \(-\frac{1}{2}\) don't fit in the integer category because of their fractional component.

Understanding integers is crucial because they are foundational to operations involving whole quantities.