Problem 57 - Single Variable Calculus

Question

Express each number as a rational number. \(1 . \overline{213}=1.213213213 \ldots\)

Step-by-Step Solution

Answer

The number \(1 . \overline{213}\) can be expressed as a rational number by following these steps: 1. Let the number be represented as x: \(x = 1.213213213 \ldots\) 2. Multiply both sides by a power of 10 to move the repeating part to the left side of the decimal point: \(1000x = 1213.213213 \ldots\) 3. Subtract the original equation from the new equation: \(999x = 1212\) 4. Divide by the coefficient of x: \(x = \frac{1212}{999}\) 5. Simplify the fraction: \(x = \frac{404}{333}\) Hence, the given number \(1 . \overline{213}\) can be expressed as a rational number \(\frac{404}{333}\).

Step 1: Find the variable for the repeating decimal

Let the number be represented as x. So, \[x = 1.213213213 \ldots\]

Step 2: Multiply both sides by a power of 10 to move the repeating part to the left side of the decimal point

The repeating part has a length of 3 digits (213). Therefore, multiply both sides of the equation by 1000. \[1000x = 1213.213213 \ldots\]

Step 3: Subtract the original equation from the new equation

Subtract the original equation (1) from the new equation (2), which will help us remove the repeating decimals. \[1000x - x = 1213.213213 \ldots - 1.213213 \ldots\] \[999x = 1212\]

Step 4: Divide by the coefficient of x

Divide both sides by 999 to isolate x. \[x = \frac{1212}{999}\]

Step 5: Simplify the fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (gcd) which is 3. \[x = \frac{1212 \div 3}{999 \div 3} = \frac{404}{333}\] The given number \(1 . \overline{213}\) can be expressed as a rational number \(\frac{404}{333}\).

Key Concepts

Repeating Decimals

Repeating decimals occur when a certain digit or a sequence of digits in a decimal keeps repeating indefinitely. They are represented with a line or bar over the repeating part, like in the number \(1 . \overline{213}\). The overline signifies that the sequence "213" repeats endlessly after the decimal point. Understanding repeating decimals is essential because they can often be converted into a simpler form. Although they may seem complex at first, recognizing the repeating pattern helps transform these numbers into fractions, revealing their rational nature. This conversion is highly useful in various mathematical applications and real-world contexts.

Converting Decimals to Fractions

To convert repeating decimals to fractions, the first step involves setting an equation for the decimal. Assign a variable, like \(x\), to the repeating decimal. For instance, in our case with \(1.213213213...\), we express it as \(x = 1.213213213\ldots\). Next, we need to eliminate the repeating part by multiplying the equation by a power of 10 that matches the length of the repeating sequence. Here, the sequence "213" is three digits long, so we multiply by 1000, resulting in \(1000x = 1213.213213\ldots\). By strategically subtracting the original equation from this new one, the repeating decimals cancel out, simplifying the equation to \(999x = 1212\). Divide both sides by 999 to solve for \(x\), resulting in \(x = \frac{1212}{999}\). This process not only reveals the fraction form but also confirms that repeating decimals are indeed rational numbers, as they can be expressed by the ratio of two integers.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its lowest terms, making it easier to understand and work with. The fraction \(\frac{1212}{999}\) consists of both a numerator and a denominator that can be divided by their greatest common divisor (GCD) for simplification. In this case, the GCD of 1212 and 999 is 3. By dividing the numerator and the denominator by this number, the fraction simplifies to \(\frac{404}{333}\).Simplification helps in reducing computation complexity and gives a clearer, more elegant form of the rational number. It is important to note that a fraction is considered fully simplified only when the numerator and denominator no longer have a common factor apart from 1. This is a fundamental step in working with fractions, especially after converting from repeating decimals, as it ensures the most efficient form of a rational number.