Problem 13 - Intermediate Algebra

Question

From the list \(0,14, \frac{2}{3}, \pi, \sqrt{7},-\frac{11}{14}\), \(2.34,-19, \frac{55}{8},-\sqrt{17}, 3.2 \overline{1}\), and \(-2.6\), identify each of the following. The rational numbers

Step-by-Step Solution

Answer

Rational numbers: 0, 14, \(\frac{2}{3}\), \(-\frac{11}{14}\), 2.34, -19, \(\frac{55}{8}\), 3.2\overline{1}, -2.6.

Step 1: Understand Rational Numbers

A rational number is any number that can be expressed as a fraction or ratio of two integers, where the denominator is not zero. Examples include whole numbers, integers, proper fractions, improper fractions, and terminating or repeating decimals.

Step 2: Identify Rational Numbers in the List

To find the rational numbers, we'll examine each number: 1. **0** is rational because it can be expressed as \(\frac{0}{1}.\)2. **14** is rational because it can be written as \(\frac{14}{1}.\)3. **\(\frac{2}{3}\)** is already in the form of a fraction of two integers.4. **\(\pi\)** is not rational because it cannot be expressed as a fraction. It is an irrational number.5. **\(\sqrt{7}\)** is not rational because it cannot be expressed as a fraction of two integers.6. **\(-\frac{11}{14}\)** is a fraction of two integers and is rational.7. **2.34** is rational because it is a terminating decimal and can be written as \(\frac{234}{100}.\)8. **-19** is rational because it can be written as \(-\frac{19}{1}.\)9. **\(\frac{55}{8}\)** is a fraction and rational.10. **\(-\sqrt{17}\)** is not rational for the same reason as \(\sqrt{7}.\)11. **\(3.2\overline{1}\)** is rational because it is a repeating decimal and can be expressed as a fraction.12. **\(-2.6\)** is rational because it is a terminating decimal and can be expressed as \(-\frac{26}{10}.\)

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole, using two integers. The number above the line is the numerator, and the number below the line is the denominator.
For instance, in the fraction \( \frac{2}{3} \), 2 is the numerator and 3 is the denominator.
Fractions allow us to describe parts of a whole and are essential for understanding other mathematical concepts.

• Proper Fractions: Numerator is smaller than the denominator, such as \( \frac{3}{4} \).
• Improper Fractions: Numerator is larger or equal to the denominator, such as \( \frac{5}{3} \).
• Mixed Numbers: A combination of an integer and a proper fraction, like 1\( \frac{1}{2} \).

Fractions are foundational to understanding rational numbers, as every fraction represents a rational number when the denominator is not zero.

Terminating Decimals

Terminating decimals are decimals that have a finite number of digits after the decimal point. They come to an end.
For example, 2.34 is a terminating decimal because it has two digits after the decimal. It can be converted into a fraction, like \( \frac{234}{100} \), which simplifies to \( \frac{117}{50} \).

• Easy to convert: They can be simply converted to fractions, making them rational.
• Clear endpoint: Terminating decimals stop at a certain point, ensuring there is no continuation in digits.

Remember, every terminating decimal is a rational number, as it can be expressed in fraction form.

Repeating Decimals

Repeating decimals have digits or groups of digits that repeat infinitely after the decimal point. Not all decimal numbers are terminating; some go on forever in a repeating pattern.
For example, the number \( 3.2\overline{1} \) means that 1 is the repeating digit. It's written with a bar over the repeating part.

• Full repetition: The same digit or digits repeat endlessly, such as \( 0.\overline{3} \) (repeating 3).
• Rational representation: Just like fractions, repeating decimals can be expressed as a fraction. For instance, \( 0.\overline{9} \) equals 1.

All repeating decimals are rational because they can be expressed as fractions of two integers.

Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.
Examples are -19, 0, 1, 14. These numbers are complete, without any fractional or decimal part.

• Positive Integers: Greater than zero, like 1, 2, 3.
• Negative Integers: Less than zero, like -1, -5, -19.
• Zero: Zero is also an integer and is neither positive nor negative.

Integers are always considered rational numbers because they can be expressed as a fraction by using 1 as the denominator, such as \( \frac{14}{1} \). This makes them straightforward to work with in rational number contexts.