Question

Find a rational number representing each of the following repeating decimals: (a) \(0.537537537537537537537537537 \ldots\) (b) \(31.25469696969696969696969 \ldots\)

Short answer

(a) \( \frac{179}{333} \), (b) \( \frac{618843}{19800} \).

Step-by-step solution

Convert just the repeating part to a fraction

The decimal in part (a) is repeating in the form 0.537. To convert this, let's express it as a fraction. Set \( x = 0.537537537\ldots \). Notice that 0.537 repeats every three digits. Thus, multiply \( x \) by 1000 to shift the decimal point over three places: \( 1000x = 537.537537\ldots \). Now, subtract the original \( x \) from this equation: \( 1000x - x = 537.537537\ldots - 0.537537\ldots \). This simplifies to \( 999x = 537 \). Finally, solve for \( x \): \( x = \frac{537}{999} \), which can be simplified to \( \frac{179}{333} \) after dividing by their greatest common divisor, 3.

Express the non-repeating part as a fraction and add

Part (b) contains a non-repeating part 31.254 and a repeating part 6969. To handle this, first express the whole decimal as \( x = 31.254696969\ldots \). Multiply \( x \) by 1000 to shift the repeating part: \( 1000x = 31254.696969\ldots \). Next, multiply by 100 again to account for the 2-digit repeating pattern: \( 100000x = 3125469.696969\ldots \). Now subtract the 1000x equation from it: \( 100000x - 1000x = 3125469.696969\ldots - 31254.696969\ldots \), which simplifies to \( 99000x = 3094215 \). Solving for \( x \), we get \( x = \frac{3094215}{99000} \). Simplify this fraction by dividing by their greatest common divisor, 5, to \( \frac{618843}{19800} \), and continue simplifying if possible.

Key concepts

Rational Numbers

A rational number is any number that can be expressed as the fraction of two integers, where the denominator is not zero. This includes not only the fractions we're accustomed to, like \( \frac{1}{2} \), but also integers like 7 (which can be represented as \( \frac{7}{1} \)).
Rational numbers also cover decimal numbers that either terminate (like 0.75) or repeat (like 0.333...). This is because they can be represented as the ratio of two integers. In our exercise, the goal is to convert repeating decimals, such as 0.537537..., back into their rational form so they can be expressed as fractions. Understanding this concept clarifies that the decimals we encounter in these exercises are indeed rational. They only need the right procedure to reveal their fraction form.

Fraction Conversion

Converting a repeating decimal into a fraction might seem daunting, but it's a systematic process. Let’s break it down:

• Identify the repeating part of the decimal.
• Assign a variable, say \( x \), to the repeating decimal.
• Multiply \( x \) by a power of 10 that matches the length of the repeating sequence to shift the decimal point.
• Subtract the original \( x \) from this new equation to eliminate the repeating part.
• Solve for \( x \) by dividing by the resulting coefficient.

This whole process effectively turns the repeating decimal into a fraction of two integers, illustrating that such decimals are indeed rational numbers. This is shown in the original exercise, where values such as 0.537537... are simplified to a fraction like \( \frac{179}{333} \).

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when simplifying fractions. It is the largest positive integer that evenly divides two or more numbers without leaving a remainder.
To simplify a fraction, find the GCD of the numerator and the denominator and divide both by this number. Doing so gives the fraction in its simplest form.
For example, in the solution, the fraction \( \frac{537}{999} \) simplifies to \( \frac{179}{333} \) when both 537 and 999 are divided by their GCD, which is 3. Utilizing the GCD ensures that the fraction you derive from a repeating decimal is in its simplest form, making it easy to understand and work with.

Decimal Representation

Understanding decimal representation, especially with repeating decimals, is crucial in recognizing the connection to rational numbers. A repeating decimal has one or more repeating digits after the decimal point. This can be expressed as a geometrically repeating pattern that continues indefinitely.
In our problem, the decimals such as 0.537 and 31.2546969... show repeating patterns. Identifying these repeating blocks allows us to write equivalent fractions effectively, demonstrating their rationality.
Through procedures like multiplying by powers of 10 and the subsequent subtraction, we can convert these repeating decimals into recognizable fractions of integers. This process reveals the inherent order in what might appear to be chaotic sequences of numbers. Therefore, understanding decimal representation not only helps in converting to fractions but in appreciating the structure and predictability within mathematics itself.