Problem 82 - Precalculus: Mathematics for Calculus

Question

Express the repeating decimal as a fraction. $$0.123123123 \dots$$

Step-by-Step Solution

Answer

The repeating decimal \( 0.123123123 \ldots \) is equal to \( \frac{41}{333} \) as a fraction.

Step 1: Define the repeating decimal

Let \( x = 0.123123123 \ldots \), which shows the repeating pattern \(123\). Our goal is to express \( x \) as a fraction.

Step 2: Multiply to remove the repeating part

Since the repeating part is three digits long, multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right:\[1000x = 123.123123123 \ldots\]Now, subtract the original \( x \) from this result to eliminate the repeating decimal.

Step 3: Set up the equation to eliminate the repeating decimal

Subtract the original \( x \) from \( 1000x \):\[1000x - x = 123.123123123 \ldots - 0.123123123 \ldots\]This simplifies to:\[999x = 123\]

Step 4: Solve for \( x \)

Divide both sides by 999 to solve for \( x \):\[x = \frac{123}{999}\]

Step 5: Simplify the fraction

Simplify \( \frac{123}{999} \). Both numbers can be divided by 3. Simplifying gives:\[x = \frac{41}{333}\]Therefore, the repeating decimal \( 0.123123123 \ldots \) as a fraction is \( \frac{41}{333} \).

Key Concepts

Fraction Simplification

Simplifying a fraction means reducing it to its simplest form. This is done by dividing the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).
To simplify, find the largest number that can evenly divide both numbers. This makes the fraction easier to understand and compare with other numbers.
For example, in the fraction \(\frac{123}{999}\), you can divide both 123 and 999 by 3 since they are divisible by 3.

• Divide 123 by 3 to get 41.
• Divide 999 by 3 to get 333.

Therefore, \(\frac{123}{999}\) simplifies to \(\frac{41}{333}\). This fraction is the simplest form as 41 and 333 have no common divisors other than 1. Knowing how to simplify fractions allows you to express numbers more briefly and clearly.

Decimal Representation

Decimals are a way to represent fractions or parts of a whole using the base 10 numeral system.
A repeating decimal, like 0.123123123..., has digits that infinitely repeat. They are often noted with a bar over the repeating part, for example, 0.\(\overline{123}\). This indicates that the sequence '123' will keep going.
Understanding decimal representation helps in converting these repeating sequences into a fraction.

• Start by assigning a variable, say \(x\), to the repeating decimal.
• The purpose is to identify how the repeating part operates, in this case, as every 3 digits.

This step provides a way to convert the decimal back into a fraction by removing the repetitive part from the equation, which leads us to algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation involves using algebraic techniques to transform and simplify equations.
In our case, we have the equation \(x = 0.123123123\ldots\). To convert this repeating decimal into a fraction, algebraic manipulation becomes crucial.
Here's how it works:

• Multiply both sides of the equation by 1000 (the power of 10 equal to the number of repeating digits) to eliminate the decimal. This changes the equation to \(1000x = 123.123123123\ldots\).
• Next, subtract the original equation \(x = 0.123123123\ldots\) from \(1000x = 123.123123123\ldots\) to eliminate the repeating decimal.
• This manipulation gives you \(999x = 123\), simplifying the decimal to a straightforward algebraic equation.
• Solve for \(x\) by dividing both sides by 999 to find \(x = \frac{123}{999}\).

Mastering algebraic manipulation facilitates converting complex repeating decimals into neat fractions.