Problem 24 - Thomas Calculus

Question

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 1 . \overline{414}=1.414414414 \ldots $$

Step-by-Step Solution

Answer

\(1.\overline{414} = \frac{140027}{99000}\) as a fraction.

Step 1: Understanding the Problem

We need to express the repeating decimal \(1.\overline{414} = 1.414414414\ldots\) as a fraction i.e., the ratio of two integers.

Step 2: Identify the Repeating Part

The repeating part of the decimal is \(.414\), which repeats indefinitely.

Step 3: Set Up an Equation for the Decimal

Let \(x = 1.414414414\ldots\). This way we can manipulate this equation to eliminate the repeating part.

Step 4: Eliminate the Repeating Part

Multiply both sides of the equation \(x = 1.414414414\ldots\) by 1000 to shift the repeating part:\(1000x = 1414.414414\ldots\).

Step 5: Create a Second Equation

Now, multiply both sides of \(x = 1.414414414\ldots\) by 10:\(10x = 14.144144\ldots\).

Step 6: Subtract the Two Equations

Subtract the second equation from the first:\(1000x - 10x = 1414.414414\ldots - 14.144144\ldots\), which simplifies to \(990x = 1400.27\).

Step 7: Solve for x

Divide both sides by 990:\(x = \frac{1400.27}{990}\). Simplify this fraction by multiplying both the numerator and the denominator by 100, then factor out common terms:\(\frac{140027}{99000}\rightarrow\frac{140027}{99000} \text{ (simplified as much as possible)}\).

Step 8: Verify the Fraction

Verify by dividing the numerator by the denominator to ensure it returns the original decimal 1.414414414... This confirms our fraction is equivalent to the repeating decimal.

Key Concepts

Fractions

A fraction represents a part of a whole and is composed of two integers: a numerator and a denominator. The numerator is the top number, indicating how many parts we have. The denominator is the bottom number, showing the number of equal parts the whole is divided into.

Fractions can also express ratios or portions, which can be useful when working with repeating decimals. Converting repeating decimals into fractions makes complex numbers simpler to handle or compare.

• In mathematics, fractions help to express larger numbers in simpler forms.
• They are also crucial in communicating quantities that are not whole numbers, like probabilities or rates.

Using fractions, we translate the endless cycle of repeating decimals into a succinct and exact representation. Doing so requires understanding the repeating portion and fitting it into a logical numeric equality.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and the denominator are integers. Every number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \), is a rational number.

Repeating decimals are a quintessential example of rational numbers. They might seem complex, but because they can be expressed as fractions, they fit the definition of rational numbers.

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Performing operations like converting repeating decimals to a fraction showcases the concept of rationality within numbers, hence establishing their predictability and consistency:

• Every rational number has a repeating pattern, either finite or infinite, when expressed as a decimal.
• Working with rational numbers is foundational for algebra and higher-level math.

Understanding these conversions enhances students’ fluency in both rational and irrational arenas.

Ratio of Integers

A ratio of integers is another way to describe a fraction, emphasizing the relationship between two quantities. When expressing a decimal as a ratio of two integers, you convert it into a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are whole numbers.

This conversion is beneficial because it takes an infinitely repeating decimal and turns it into a manageable expression.

• We eliminate the infinity by representing the decimal in a finalized form.
• Using algebraic manipulation, we adjust the decimal to provide a clean ratio.

This is useful when needing to perform arithmetic operations or when precision is important. Rather than working with a non-terminating decimal, a ratio of integers maintains accuracy and is easier to work with in formulas or equations.