Converting a Decimal Number into a Fraction
Recurring decimal numbers, with their infinite repeating sequences, have always fascinated those exploring mathematics.
For instance, the decimal part of \(0.333...\) consists of an endless stream of digits.
What might seem like an unsolvable puzzle actually reveals a simple and elegant truth: every recurring decimal can be expressed as a fraction.
In other words, the number \(0.333...\), where the digit \(3\) repeats indefinitely, can be written as \(\frac{1}{3}\).
$$ 0.\overline{3} = 0.3333333... = \frac{1}{3} $$
It’s like a little "mathematical miracle." Let’s discover how it works. We can use two methods to convert a recurring decimal into a fraction.
Generating Fraction Method
Among the various techniques for this transformation, the method of generating fractions stands out for its speed and simplicity.
What is a generating fraction?
A generating fraction is the fractional form of a recurring decimal.
This method relies on a straightforward formula:
$$ \text{Fraction} = \frac{N - A}{D} $$
Where:
- \(N\): the complete number, obtained by removing the decimal point and including both the repeating and non-repeating parts.
- \(A\): the number without the repeating part, obtained by removing the decimal point and considering only the non-repeating digits.
- \(D\): a number made up of as many \(9\)'s as there are repeating digits, followed by as many \(0\)'s as there are non-repeating digits.
Let’s look at some practical examples.
Consider the decimal \(0.777...\), where the repeating part is \(7\).
Identify the components:
- \(N = 7\) (the complete number, without the decimal point)
- \(A = 0\) (there is no non-repeating part)
- \(D = 9\) (one digit in the repeating part corresponds to one \(9\))
Apply the formula:
$$ \frac{N - A}{D} = \frac{7 - 0}{9} = \frac{7}{9} $$
So, the decimal \(0.777...\) can be written as \(\frac{7}{9}\).
Example 2: A Number with Both Non-Repeating and Repeating Parts
Take \(2.134343...\), where \(2.13\) is the non-repeating part, and \(43\) is the repeating part.
Identify the components:
- \(N = 21343\) (the full number without the decimal point)
- \(A = 213\) (the non-repeating part only)
- \(D = 9900\) (two repeating digits correspond to \(99\), and two non-repeating digits correspond to \(00\)).
Apply the formula:
$$ \frac{N - A}{D} = \frac{21343 - 213}{9900} = \frac{21130}{9900} $$
Thus, the decimal \(2.134343...\) can be written as \(\frac{21130}{9900}\), which simplifies further to \( \frac{2113}{990} \).
Equation Method
For those who prefer a more systematic approach, the equation method provides another solution.
This method involves assigning the decimal to a variable, multiplying by powers of 10 to move the non-repeating and repeating parts to the integer portion, and subtracting the resulting equations.
Let’s break it down with a concrete example.
Take the decimal \(0.333...\).
Assign it to a variable:
$$ x = 0.333... $$
Multiply by \(10\):
$$ 10x = 3.333... $$
Subtract the first equation from the second:
$$ 10x - x = 3.333... - 0.333... $$
$$ 9x = 3 $$
Solve for \(x\):
$$ x = \frac{3}{9} = \frac{1}{3} $$
Thus, \(0.333...\) is equivalent to \(\frac{1}{3}\).
Example 2: A Number with Both Non-Repeating and Repeating Parts
Consider \(2.134343...\).
Assign it to a variable:
$$ x = 2.134343... $$
First, multiply by \(10\) to move the non-repeating part to the left of the decimal:
$$ 10x = 21.343434... $$
Next, multiply by \(100\) to move the repeating part as well:
$$ 1000x = 2134.343434... $$
Now subtract the earlier equation:
$$ 1000x - 10x = 2134.343434... - 21.343434... $$
$$ 990x = 2113 $$
This eliminates the repeating part.
Solve for \(x\):
$$ x = \frac{2113}{990} $$
So, the decimal \(2.134343...\) can be expressed as \(\frac{2113}{990}\).
Which Method Should You Use?
Both methods clearly demonstrate that recurring decimals are rational numbers.
The equation method is perfect for gaining a deeper understanding of the relationship between repeating decimals and fractions.
The generating fraction method, on the other hand, is faster and more practical.
Whether you opt for generating fractions or equations, the result is always the same.
What truly matters is appreciating the elegance and simplicity of mathematics. Every recurring decimal can be "captured" as a fraction, turning the seemingly infinite into something finite and comprehensible.
