Problem 46 - Algebra and Trigonometry

Question

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots $$

Step-by-Step Solution

Answer

The repeating decimal \(0.\overline{1}\) is represented by the fraction \(\frac{1}{9}\).

Step 1: Expression Setup

To start, let's assign the repeating decimal to a variable, like \(x\). \[x = 0.1 + 0.01 + 0.001 + 0.0001 + ...\]

Step 2: Multiply by Radix

Next, multiply \(x\) by 10, which will shift the decimals one place to the right. \[10x = 1 + 0.1 + 0.01 + 0.001 + ...\]

Step 3: Calculate Difference

Now, subtract \(x\) from \(10x\) to eliminate the repeating decimals. From the equations above, \[10x - x = 1 + 0.1 + 0.01 + 0.001 + ... - (0.1 + 0.01 + 0.001 + ...)\] This simplifies to \[9x = 1\]

Step 4: Equation Solution

Finally, solve the equation for \(x\) by dividing by 9. So, \(x = 1/9\). This is our answer.

Key Concepts

Understanding Fractions

Fractions are a way to express parts of a whole. They consist of two numbers: the numerator and the denominator. The numerator is the top number that tells how many parts we have. The denominator is the bottom number that tells how many of those parts make up a whole. For example, in the fraction \(\frac{1}{9}\), \(1\) is the numerator and \(9\) is the denominator.

• Fractions can represent division, where the numerator is divided by the denominator.
• They are used in various mathematical calculations involving measurements and proportions.
• It's important to express fractions in their simplest form or lowest terms by dividing both the numerator and the denominator by their greatest common divisor.

In the context of repeating decimals, converting them to fractions can help simplify the representation of a number. For example, rather than writing \(0.\overline{1}\), you can express it as \(\frac{1}{9}\) to immediately understand the number’s place in arithmetic.

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is created by adding a constant to the previous term. This is known as the common difference. When dealing with repeating decimals, sometimes it's useful to look at them as a form of arithmetic sequence.

• For example, the decimal sequence \(0.1, 0.01, 0.001, 0.0001, \ldots\) is created by dividing each term by 10 to get the next term.
• The common ratio in this case is \(\frac{1}{10}\), which is the constant factor between consecutive terms.
• Arithmetic sequences help in understanding how these decimals add up and eventually contribute to the whole number representation when expressed as fractions.

Recognizing that repeating decimals can form a sequence is crucial to transforming them into a fraction. Identifying the pattern leads to simplified calculations and understanding of the number’s behavior.

Understanding Decimal Notation

Decimal notation is a way to represent fractions or real numbers in standard base-10 format. It uses a decimal point to separate the whole number part from the fractional part. Every digit after the decimal point represents a power of ten.

• The decimal \(0.\overline{1}\) means that the digit \(1\) repeats indefinitely.
• In decimal notation, repeating decimals can become cumbersome, but they can be converted into more manageable fractions for easier calculations.
• Knowing how to move between decimal notation and fractions can make it easier to perform arithmetic, compare numbers, and solve equations.

When you recognize a repeating pattern, you can assign a variable and use algebra to find a corresponding fraction. This understanding simplifies the mathematical expression and aligns with how numbers are used in real-world applications.