Problem 63 - Precalculus Mathematics for Calculus

Question

Express the repeating decimal as a fraction. $$0.777 \ldots$$

Step-by-Step Solution

Answer

The repeating decimal 0.777... is equal to the fraction \(\frac{7}{9}\).

Step 1: Let the Repeating Decimal Be a Variable

Start by setting the repeating decimal to a variable. Let \( x = 0.777\ldots \). This means \( x \) represents the repeating decimal 0.777... which continues indefinitely.

Step 2: Multiply by Power of 10 to Shift the Decimal

Multiply both sides of the equation by 10 to shift the decimal point one place to the right. This gives us: \( 10x = 7.777\ldots \). The decimal part still repeats as before.

Step 3: Subtract to Eliminate Repeating Decimals

Now subtract the original equation (\( x = 0.777\ldots \)) from the equation we obtained after multiplying by 10 (\( 10x = 7.777\ldots \)). This looks like this: \( 10x - x = 7.777\ldots - 0.777\ldots \) Simplifying, we get \( 9x = 7 \).

Step 4: Solve for x

Solve \( 9x = 7 \) for \( x \) by dividing both sides by 9: \[ x = \frac{7}{9} \]Hence, the repeating decimal 0.777... can be expressed as the fraction \( \frac{7}{9} \).

Key Concepts

Fraction Conversion

Converting repeating decimals to fractions is a crucial skill that often appears in math. It involves expressing an endlessly recurring decimal in the form of a fraction. The key to solving such problems is recognizing patterns and using algebraic manipulation to eliminate the repeating part. We'll illustrate this concept by considering the example of converting the repeating decimal \(0.777\ldots\) to a fraction.

To start, we set the repeating decimal to a variable. In this case, \(x = 0.777\ldots\). By doing so, we identify the endless pattern of the decimal. Next, we multiply this equation by a power of ten—here it's 10—to shift the decimal point to the right and bring similar digits together. This step results in \(10x = 7.777\ldots\). This multiplication allows us to align the numbers for the next important step.

Subtracting the original decimal equation from the new one helps eliminate the repeating part. In our example, \(10x - x = 7.777\ldots - 0.777\ldots\) simplifies to \(9x = 7\). This step is crucial as it provides an equation free of repeating decimals. Finally, solving for \(x\) by dividing both sides by 9 gives us \(x = \frac{7}{9}\). Thus, the repeating decimal \(0.777\ldots\) is equivalent to the fraction \(\frac{7}{9}\). Keeping this process in mind is essential for mastering fraction conversions for any repeating decimal.

Precalculus

Precalculus plays a significant role in bridging the gap between basic algebra and more complex calculus concepts. Among various skills essential in precalculus, handling repeating decimals through fractions is fundamental. This skill not only assists in simplifying calculations but also prepares students for advanced mathematical problem solving.

The foundations of precalculus include understanding the nature and behavior of different types of numbers, such as rational numbers, which can be expressed as fractions. Repeating decimals often represent rational numbers, and converting them helps in recognizing this.

• In precalculus, repeating decimals like \(0.777\ldots\) illustrate how patterns extend up to infinity.
• The concept of a variable (like \(x\) in our example) serves as a reminder of algebraic principles, as it helps create equations from patterns.

Viewing repeating decimals through the lens of precalculus unlocks a deeper comprehension of number systems and their applications in mathematics. This understanding is pivotal in forming the groundwork for calculus, where complex number manipulations become necessary. Converting repeating decimals into fractions is a small yet essential piece of this precalculus framework, enhancing both numerical intuition and algebraic fluency.

Algebraic Manipulation

Algebraic manipulation involves using algebraic techniques to rearrange and simplify expressions or equations. It is a fundamental skill that helps solve various mathematical problems including the conversion of repeating decimals to fractions. This process presents an excellent opportunity to practice and apply these skills.

In our example, the variable \(x\) represents the repeating decimal \(0.777\ldots\). Through algebraic thought, we manipulate expressions to remove the repeating decimal part. The step of multiplying the whole equation by 10 (\(10x = 7.777\ldots\)) demonstrates how we can strategize to align terms, setting the stage for successful elimination of the repeating sequence.

• This multiplication is a clear example of using algebraic principles to isolate and deal with infinite patterns.
• Subtracting the original equation from the multiplied one simplifies the problem into a solvable linear equation \(9x = 7\).

Finally, solving this equation by dividing both sides by 9 shows how algebraic manipulation can lead directly to the solution, transforming a complex repeating decimal into the neat fraction \(\frac{7}{9}\). Mastering these algebraic skills is not only critical for handling repeating decimals but also instrumental across various fields of math where similar techniques apply.