Problem 13 - Intermediate Algebra

Question

Determine whether each number is a repeating or a nonrepeating decimal, and whether it is a rational or an irrational number. $$ 5.41414141 \ldots $$

Step-by-Step Solution

Answer

5.41414141... is a repeating decimal and a rational number.

Step 1: Identify Pattern

The number given is 5.41414141.... Notice that after the decimal point, the digits '41' repeat indefinitely. This indicates a repeating decimal pattern with '41' as the repeating block.

Step 2: Determine Type of Decimal

Since the number has a repeating block ('41'), we classify this as a repeating decimal.

Step 3: Classify the Number as Rational or Irrational

A number with a repeating decimal is rational. Rational numbers can be expressed as the ratio of two integers. Hence, 5.41414141... is a rational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or a ratio of two integers, where the denominator is not zero. They include fractions like \( \frac{3}{4} \), integers like 5, and even decimals that terminate or repeat a pattern indefinitely. This means:

• If you can write the number as \( \frac{a}{b} \), where both \( a \) and \( b \) are integers and \( b eq 0 \), it's rational.
• Decimals that are repeating or terminating are also rational numbers because they can be converted into fraction form.

For instance, the repeating decimal 5.41414141... meets the criteria for a rational number because it can be rewritten as a fraction representing the repeating cycle after some algebraic manipulation. Rational numbers are incredibly useful because they make up a big portion of the numbers we use in everyday life.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction. This means they can't be expressed as the ratio of two integers. They arise in mathematics when we consider decimal numbers that neither terminate nor repeat a sequence.

Here are some key characteristics:

• They have a non-terminating and non-repeating decimal expansion. For example, the number \( \pi \) (pi) is approximately 3.14159 and goes on forever without a pattern.
• They include non-perfect square roots, like \( \sqrt{2} \), which cannot be written as a fraction of two integers.

Irrational numbers are crucial in mathematics, highlighting the continuous nature of numbers and proving useful in a variety of complex calculations and geometrical constructions.

Nonrepeating Decimals

Nonrepeating decimals are all about how decimal numbers can behave. When we describe a decimal as nonrepeating, we mean that after the decimal point, the digits don't cycle or form a recurring pattern.

Understanding nonrepeating decimals involves:

• Recognizing that they don't end in a repeating sequence of numbers, unlike 5.41414141..., which repeats '41'.
• Realizing that these types of decimals are often linked with irrational numbers since nonrepeating and non-terminating decimals can't be written as simple fractions.

This is why nonrepeating decimals are a clear indicator of irrational numbers. Examples include the decimal representations of the square root of 2 or pi, where the digits continue infinitely without a periodic pattern. They offer insights into numbers that can't be captured by a finite or repeating process.