Problem 20 - Thomas Calculus

Question

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0 . \overline{234}=0.234234234 \ldots $$

Step-by-Step Solution

Answer

The number \( 0.\overline{234} \) can be expressed as \( \frac{26}{111} \).

Step 1: Set the repeating decimal as a variable

Let's assign the repeating decimal to a variable for better manipulation. Let \( x = 0.\overline{234} \). This means \( x = 0.234234234 \ldots \).

Step 2: Multiply to remove the repeating part

We must multiply \( x \) by a power of 10 such that the decimal after the comma aligns. This can be achieved by multiplying by 1000 since the repeating sequence has three digits. This gives us \( 1000x = 234.234234234 \ldots \).

Step 3: Subtract the original equation from the multiplied equation

To eliminate the repeating decimal, subtract the original variable equation from the new one obtained in Step 2: \( 1000x = 234.234234 \ldots \) and \( x = 0.234234 \ldots \). This results in: \[ 1000x - x = 234.234234 \ldots - 0.234234 \ldots \]Simplifying gives \( 999x = 234 \).

Step 4: Solve for x to find the integer ratio

Now, solve for \( x \) by dividing both sides by 999:\[ x = \frac{234}{999} \]

Step 5: Simplify the fraction

To ensure the fraction is in its simplest form, find the greatest common divisor (GCD) of 234 and 999. The GCD is 9. Thus, divide both the numerator and the denominator by 9:\[ \frac{234 \div 9}{999 \div 9} = \frac{26}{111} \]

Step 6: Express the final ratio

The repeating decimal \( 0.\overline{234} \) is expressed as the ratio of two integers in its simplest form: \( \frac{26}{111} \).

Key Concepts

Decimal to Fraction Conversion

Converting repeating decimals to fractions is a key mathematical exercise. When facing a repeating decimal, such as \( 0.\overline{234} \), the goal is to express it as a fraction, which is the ratio of two integers. This process involves a few crucial steps.
First, assign the repeating decimal to a variable, say \( x \). For our example, \( x = 0.234234234 \ldots \). We multiply \( x \) by 1000, a power of 10 that matches the length of the repeating block (three digits, in this case). Thus, \( 1000x = 234.234234 \ldots \). Adding 234 before the decimal aligns the decimals for subtraction.
By subtracting the initial equation from the newly multiplied one \( 1000x - x \), we are left with \( 999x = 234 \). This step successfully removes the repeating decimal, allowing us to solve for \( x \) as a fraction:

• \( x = \frac{234}{999} \)

After determining the fraction, it’s simplified by dividing both the numerator and the denominator by their greatest common divisor. This process reduces fractions to their simplest form, making them easier to interpret.

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. This set includes integers, fractions, and most importantly for this exercise, repeating decimals.
Repeating decimals are inherently rational because they can always be converted back into a fraction.
For instance, the repeating decimal \( 0.\overline{234} \) was initially represented as the fraction \( \frac{234}{999} \), and further reduced to \( \frac{26}{111} \).

• This conversion confirms its rational nature.

The capability to express a seemingly endless sequence in a precise and finite form underscores the nature of rational numbers. This feature distinguishes them from irrational numbers, which cannot be precisely represented by fractions.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest positive integer that divides two or more integers without leaving a remainder. It plays a crucial role in simplifying fractions and ensuring they are in their simplest form.
To find the GCD, you can list the factors of each number and identify the largest common one. For the fraction \( \frac{234}{999} \), the GCD is 9.
Here's how it works in practice:

• Divide \( 234 \) by 9, which gives 26.
• Divide \( 999 \) by 9, which gives 111.

Using the GCD reduces the fraction to \( \frac{26}{111} \), which is its simplest form.
This procedure confirms the importance of divisibility in arithmetic, ensuring clarity and precision in mathematical solutions.