Problem 38 - Intermediate Algebra

Question

List the elements of $$ \left\\{-3,-\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75,916 . \overline{6}\right\\} $$ that belong to the following sets. Rational numbers

Step-by-Step Solution

Answer

Rational numbers: \(-3, -\frac{8}{5}, 0, \frac{2}{3}, 1, 2, 4.75, 916.\overline{6}\).

Step 1: Identify Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This includes integers, finite decimals, and repeating decimals.

Step 2: Assess Each Element

Evaluate whether each element can be expressed as a fraction:- \(-3\): can be written as \(-\frac{3}{1}\), so it is rational.- \(-\frac{8}{5}\): expressed as a fraction, so it is rational.- \(0\): can be written as \(\frac{0}{1}\), so it is rational.- \(\frac{2}{3}\): expressed as a fraction, so it is rational.- \(1\): can be written as \(\frac{1}{1}\), so it is rational.- \(\sqrt{3}\): cannot be expressed as a fraction of two integers, so it is not rational.- \(2\): can be written as \(\frac{2}{1}\), so it is rational.- \(\pi\): cannot be expressed as a fraction, so it is not rational.- \(4.75\): can be written as \(\frac{475}{100}\), so it is rational.- \(916.\overline{6}\): can be expressed as a fraction (e.g., using equation for repeating decimals), so it is rational.

Step 3: List Rational Elements

From the evaluations, the elements that are rational are: \(-3, -\frac{8}{5}, 0, \frac{2}{3}, 1, 2, 4.75, 916.\overline{6}\).

Key Concepts

Integers

An important group of numbers you'll often come across in math are integers. Integers include whole numbers that can be positive, negative, or zero. They don't have fractions or decimals in them, making them quite straightforward. Examples of integers from the set given are:

• -3
• 0
• 1
• 2

These integers can also be represented as fractions, where the denominator is 1, like \(-\frac{3}{1}\) for -3, \(\frac{0}{1}\) for 0, and so on. Since integers are whole numbers with no fractional parts, they naturally fit into the category of rational numbers, which we will discuss more in the fractions section.

Fractions

Fractions are a way to express numbers that aren't whole. They show parts of a whole, with a numerator on top and a denominator on the bottom. A rational number is any number that can be written as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. Here are some examples of fractions from our set:

• \(-\frac{8}{5}\)
• \(\frac{2}{3}\)
• 4.75, which can be written as \(\frac{475}{100}\)

A number can also be a fraction even if it looks like a simple number, such as an integer. That's because any integer can be expressed with a denominator of 1. Rational numbers are broad and include both simple and repeating fractions. Recognizing numbers as fractions helps to identify if they are rational.

Repeating Decimals

Repeating decimals are fascinating because they bridge fractions and decimals. A repeating decimal is a decimal number that has one or a group of digits after the decimal point that repeat infinitely. These decimals can always be converted into fractions, which qualifies them as rational numbers.
For instance, the number \(916.\overline{6}\) from our set is a repeating decimal. The bar over 6 indicates that 6 repeats indefinitely. Such numbers can be expressed as fractions through some calculations. The magic of repeating decimals is that despite their never-ending nature, they can be neatly written as a ratio of two integers.This concept highlights the close connection between decimals and fractions, showing that even numbers with endless repeating patterns can be rational.