Problem 107 - Precalculus with Limits

Question

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 0.\overline{36} \)

Step-by-Step Solution

Answer

The rational representation of \(0.\overline{36}\) is \( \frac{4}{11} \).

Step 1: Define the decimal as a variable

Let \( x = 0.\overline{36} \) . Here, we denote the given repeating decimal as \( x \) . It's important to consider that the repeating digit is 36.

Step 2: Construct a second equation

Form an equation with the repeating part shifted, so that it can be subtracted from the original equation to remove the repeating decimal. This can be done by multiplying \( x \) by 100 to give \( 100x = 36.\overline{36} \).

Step 3: Subtract the two equations

Subtract the first equation from the second to eliminate the decimal: \( 100x - x = 36.\overline{36} - 0.\overline{36} \) simplifies to \( 99x = 36 \).

Step 4: Solve for the variable

Solving for \( x \) gives \( x = 36/99 \).

Step 5: Simplify the fraction

By removing common factors, we simplify \( 36/99 \) to \(\frac{4}{11}\).

Key Concepts

Rational Numbers

Rational numbers are an essential concept in mathematics.
They are numbers that can be expressed as a quotient or fraction of two integers, where the numerator is a whole number and the denominator is a non-zero whole number.
This means any number of the form \( \frac{a}{b} \), where both `a` and `b` are integers, and `b` is not zero, is a rational number.
These numbers include:

• Fractions like \( \frac{2}{3} \)
• Whole numbers like 5 (which can be written as \( \frac{5}{1} \))
• Negative numbers like -3 (written as \( \frac{-3}{1} \))

Decimals can also be rational numbers if they are terminating or repeating.
For example, the repeating decimal \( 0.\overline{36} \) corresponds to a rational number because it can be written as a fraction, \( \frac{4}{11} \).
Recognizing decimals as rational numbers aids in understanding their structure and how they relate to fractions.

Fractions

Fractions are foundational in mathematics, representing parts of a whole.
A fraction is made up of two parts:

• The **numerator**, which is the number above the fraction bar. It represents how many parts are being considered.
• The **denominator**, the number below the fraction bar, which shows how many equal parts make up a whole.

For example, in the fraction \( \frac{3}{4} \), 3 is the numerator indicating 3 parts out of a possible 4.
This concept helps in transitioning repeating decimals into a fraction form.
When given a repeating decimal like \( 0.\overline{36} \), the conversion process involves:

• Assigning it an algebraic variable
• Formulating an equation that involves shifting the decimal points to establish a second equation
• Subtracting these equations to form a straightforward fraction

Thus, a repeating decimal is translated into a neat fraction, like \( \frac{36}{99} \), and further simplified when necessary.

Simplification

Simplifying fractions involves reducing them to their smallest possible terms.
This process makes fractions easier to understand and work with.
To simplify a fraction, you determine the greatest common divisor (GCD) of both the numerator and denominator.
You then divide both by this number.
For example, with the fraction \( \frac{36}{99} \), notice that both numbers share common factors.
By identifying that the greatest common factor is 9:

• Divide 36 by 9 to get 4
• Divide 99 by 9 to get 11

This results in the simplified fraction \( \frac{4}{11} \).
Simplifying fractions is vital because it presents the number in an uncomplicated form, making it easier to interpret and use in calculations.
This simplified form demonstrates the simplicity and elegance of rational numbers in mathematics.