Problem 30 - Thomas Calculus

Question

Express each of the numbers in Exercises \(23-30\) as the ratio of two integers. $$3 . \overline{142857}=3.142857142857 \ldots$$

Step-by-Step Solution

Answer

The number can be expressed as \(\frac{3142854}{999999}\).

Step 1: Assign a variable

Let's start by assigning a variable to the repeating decimal. Let \( x = 3.142857142857\ldots \), where the digits 142857 repeat indefinitely.

Step 2: Multiply to eliminate the decimal

To eliminate the repeating decimal, multiply both sides of the equation by a power of 10 that matches the length of the repeating sequence. Here, the repeating sequence has 6 digits, so multiply by \(1000000\). This gives us: \(1000000x = 3142857.142857142857\ldots\).

Step 3: Set up an equation to solve

You have two equations now: \(x = 3.142857142857\ldots\) and \(1000000x = 3142857.142857142857\ldots\). Subtract the first equation from the second one to eliminate the repeating part: \(1000000x - x = 3142857.142857142857 - 3.142857142857\).

Step 4: Simplify the subtraction

This subtraction simplifies to: \(999999x = 3142857 - 3 = 3142854\).

Step 5: Solve for x

Now, solve the equation \(999999x = 3142854\) by dividing both sides by 999999: \(x = \frac{3142854}{999999}\).

Step 6: Simplify the fraction

Simplify the fraction \(\frac{3142854}{999999}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 1, so the fraction is already in its simplest form.

Step 7: The final answer

Therefore, the decimal \(3.142857142857\ldots\) can be written as the fraction \(\frac{3142854}{999999}\).

Key Concepts

Fraction Conversion

Converting a repeating decimal into a fraction involves careful steps to accurately represent the decimal as a ratio of two integers. A repeating decimal is a decimal fraction that has a digit or a group of digits that repeat infinitely. This process starts by assigning a variable to the repeating decimal to make it easier to manipulate. For our exercise, we let \( x = 3.142857142857\ldots \) where the sequence 142857 repeats indefinitely.

To convert this decimal into a fraction, we need a way to eliminate the repeating part. This involves multiplying the variable by a power of 10 that matches the number of repeating digits. Here, since there are six repeating digits (142857), we use \( 10^6 \) or 1000000. This transformation gives a new equation of \( 1000000x = 3142857.142857\ldots \). The repeating part is now behind the decimal point, and by subtracting the original equation from this new equation, the repeat is eliminated.

The last step requires subtracting to create a clean equation without decimals. We do the subtraction: \( 1000000x - x = 3142857 - 3 \), leading to \( 999999x = 3142854 \). Finally, dividing both sides by 999999 results in \( x = \frac{3142854}{999999} \). This fraction is a truthful expression of the original repeating decimal.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Repeating decimals are a type of rational number because they can be converted into a fraction of two integers.

Understanding that repeating decimals can always be expressed as fractions helps to categorize them under rational numbers. Each rational number has a unique representation in this format, which makes it possible to identify and convert various decimal forms into fractions. In our case, the number \( 3.142857142857\ldots \) is rational because it can be expressed as \( \frac{3142854}{999999} \).

Rational numbers, like the fraction we have obtained, can be placed on the number line and used in various mathematical operations. They play a crucial role in both foundational mathematics and advanced mathematical studies, providing a bridge between simple integers and more complex numerical systems.

Simplification of Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1. This process makes calculations easier and results clearer.

After converting a decimal to a fraction, simplification is often necessary. You find the greatest common divisor (GCD) of the numerator and the denominator to simplify. However, in the exercise, the fraction \( \frac{3142854}{999999} \) is already in its simplest form since its GCD is 1.

Understanding simplification helps solidify the process of converting decimals to fractions by ensuring that the final answer is as concise and correct as possible. Simplified fractions are easier to work with, especially in arithmetic operations or comparisons. It's vital to always check whether a fraction can be reduced further to maintain mathematical elegance and precision.