Problem 53 - Calculus: Early Transcendentals

Question

Express the number as a ratio of intergers. \( 2. \overline {516} = 2.516516516 . . . \)

Step-by-Step Solution

Answer

The number is expressed as the ratio \( \frac{838}{333} \).

Step 1: Define the repeating decimal

Let \( x \) be the repeating decimal, so \( x = 2.\overline{516} \). This means that \( x = 2.516516516... \).

Step 2: Remove the whole part

Separate the whole number from the decimal part: Let \( y = 0.\overline{516} \) so that \( x = 2 + y \).

Step 3: Equation for the decimal part

We have \( y = 0.\overline{516} \). To eliminate the repeating part, multiply both sides by 1000 (since the repeat length is 3 digits): \( 1000y = 516.516516... \).

Step 4: Create an equation to subtract

Now subtract the original \( y = 0.516516... \) from the multiplied equation: \( 1000y - y = 516.516516... - 0.516516... \).

Step 5: Simplify the subtraction

The subtraction gives \( 999y = 516 \), since the decimals cancel out. Thus, \( y = \frac{516}{999} \).

Step 6: Simplify the fraction

Simplify \( \frac{516}{999} \) by finding the greatest common divisor (GCD) of 516 and 999, which is 3. Divide both the numerator and the denominator by 3: \( \frac{516}{999} = \frac{172}{333} \).

Step 7: Formulate the final ratio

Recall \( x = 2 + y \), which becomes \( x = 2 + \frac{172}{333} \). Write this as a single fraction: \( x = \frac{2 \times 333 + 172}{333} = \frac{838}{333} \).

Step 8: Verify the solution

Confirm that \( \frac{838}{333} = 2.516516516... \) by performing division to ensure it matches the original decimal.

Key Concepts

Repeating Decimal

Repeating decimals, or recurring decimals, are numbers that have digits repeating in an indefinite sequence after the decimal point. In other words, after the decimal point, there's a block of one or more digits that continuously repeat. For example, in the number \( 2.\overline{516} \), the digits 516 are the repeating component. Repeating decimals are most often denoted using a bar above the repeating sequence.

Recognizing and converting repeating decimals is foundational in algebra and number theory. To convert a repeating decimal into a fraction, knowledge of its repeating portion and non-repeating portion is essential.

Why does a decimal repeat? When you divide any two integers, you either get a terminating or repeating decimal. This is because at some point, in the division of two integers, the remainders begin repeating, leading to the numerators and entire decimal sequence repeating.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, with both the numerator and the denominator being integers and the denominator not equal to zero. Every number that can be written as a fraction of two integers is a rational number, including repeating decimals. This is why \( 2.\overline{516} \) can be converted to the fraction \( \frac{838}{333} \).

Rational numbers include:

• Whole numbers, such as 1, 7, and 0.
• Fractions, like \( \frac{1}{2} \) or \( \frac{5}{8} \).
• Negative numbers, such as \(-3\).
• Repeating and terminating decimals, like \( 0.75 \) (\( \frac{3}{4} \)) and that pesky repeating decimal we’re discussing.

Recognizing a repeating decimal as a rational number allows us to better understand its properties and convert it into an equivalent fraction.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form, where the numerator and denominator share no common factors other than 1. This process makes fractions easier to work with and understand.

To simplify the fraction \( \frac{516}{999} \) in our example:

• Determine the greatest common divisor (GCD) of 516 and 999, which is 3.
• Divide both the numerator and the denominator by that GCD.
• Thus, \( \frac{516}{999} \) becomes \( \frac{172}{333} \).

This simplification makes the fraction more comprehensible and readily usable for further calculations.

Understanding how to simplify fractions is crucial since it helps in recognizing equivalent values and comparisons in mathematical problems.