Problem 67 - Algebra 2

Question

Name the sets of numbers to which each number belongs. \(-4 . \overline{2}\)

Step-by-Step Solution

Answer

\(-4.\overline{2}\) belongs to the rational and real number sets.

Step 1: Identify Rational Numbers

A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \). Since \(-4.\overline{2}\) represents a repeating decimal, it is a rational number because it can be converted into a fraction.

Step 2: Evaluate for Real Numbers

Real numbers include all rational and irrational numbers. Since \(-4.\overline{2}\) is a repeating decimal and can be expressed as a rational number, it is also a real number.

Step 3: Verify Non-Inclusion in Other Sets

Imaginary numbers involve the square root of negative numbers, which \(-4.\overline{2}\) is not. It is not a whole or natural number either because those are non-negative and \(-4.\overline{2}\) is negative. Thus, it only belongs to the rational and real number sets.

Key Concepts

Rational Numbers

Rational numbers are a fascinating category in the world of mathematics. They are numbers that can be precisely expressed as a fraction of two integers. This means you can write a rational number as \( \frac{a}{b} \), where \( a \) (the numerator) and \( b \) (the denominator) are integers, and \( b eq 0 \).
- Any whole number you can think of is automatically a rational number because it can be put over 1 (e.g., \( 5 = \frac{5}{1} \))
- Fractions like \( \frac{2}{3} \) and \( \frac{10}{5} \) are rational because they fit the form \( \frac{a}{b} \).
- Even decimals that repeat or end, such as \(-4.2\) or \(0.75\), are rational numbers.
Rational numbers also include negative fractions and decimals, because they are still expressed with two integers where the denominator isn't zero. This characteristic is fundamental when living in a world where measurements are rarely whole numbers. As such, rational numbers are incredibly useful in daily life, whether you are cooking, tailoring, or working with finances.

Real Numbers

The real number set is exceptionally broad and includes both rational and irrational numbers. Real numbers make up the entire number line and cover everything from simple counting numbers to complex decimal expansions.
- Rational numbers, as part of the real number set, include fractions, whole numbers, and repeating decimals.
- Irrational numbers complete the definition of real numbers. These include numbers like \( \pi \) or \( \sqrt{2} \), which cannot be exactly written as fractions.
Real numbers are the building blocks for mathematics as they include every possible number you can encounter in everyday scenarios. They are used in calculations involving length, area, time, and much more. Thus, understanding that rational numbers are a subset of real numbers is crucial to grasping the full spectrum of numerical value in our world.

Repeating Decimals

Repeating decimals are quite intriguing and occur when a decimal number carries on infinitely with a repeating pattern. This is something often seen when dividing numbers doesn’t provide a neatly ending decimal.
For example, the number \(-4.\overline{2}\) implies that the decimal goes on as \(-4.222...\) indefinitely. Despite this infinite nature, repeating decimals are classified as rational numbers because they can be transformed into fractions.
Here's how you can convert a repeating decimal like \(-4.\overline{2}\) into a fraction:

• Let \( x = -4.\overline{2} \).
• Write the repeating decimal equation: \( 10x = -42.\overline{2} \).
• Subtract the first equation from the second: \( 10x - x = -42.\overline{2} - (-4.\overline{2}) \).
• This simplifies to \( 9x = -38 \).
• Solve for \( x \) to get \( x = \frac{-38}{9} \), proving \(-4.\overline{2} = \frac{-38}{9} \).

Understanding repeating decimals is essential because they help convert complex calculations into simple fractions, providing an easier way to navigate through lengthy numerical figures.