The seemingly simple decimal 0.737373... (or 0.73 with a bar over the 73 indicating repetition) might look innocuous, but converting repeating decimals to fractions requires a specific mathematical approach. This comprehensive guide will walk you through the process, revealing the underlying logic and providing you with the tools to tackle similar conversions. We'll explore the steps involved, address common questions, and even delve into the mathematical theory behind it.
Understanding Repeating Decimals
Before diving into the conversion, let's solidify our understanding of what a repeating decimal actually is. A repeating decimal, also known as a recurring decimal, is a decimal representation of a number whose digits eventually become periodic, repeating the same sequence infinitely. In our case, the sequence "73" repeats endlessly. This indicates a rational number—a number that can be expressed as a fraction of two integers.
Converting 0.737373... to a Fraction
Here's the step-by-step method to convert the repeating decimal 0.737373... into a fraction:
Step 1: Assign a Variable
Let's represent the repeating decimal with the variable x:
x = 0.737373...
Step 2: Multiply to Shift the Decimal
Multiply both sides of the equation by 100 (since the repeating block has two digits):
100x = 73.737373...
Step 3: Subtract the Original Equation
Subtract the original equation (Step 1) from the equation in Step 2:
100x - x = 73.737373... - 0.737373...
This simplifies to:
99x = 73
Step 4: Solve for x
Divide both sides by 99 to isolate x:
x = 73/99
Therefore, the fraction equivalent of the repeating decimal 0.737373... is 73/99.
Can This Fraction Be Simplified?
Yes, let's check to see if the fraction can be simplified. The numbers 73 and 99 do not share any common factors other than 1, indicating that the fraction 73/99 is already in its simplest form.
Why Does This Method Work?
The method works because multiplying by a power of 10 shifts the decimal point, allowing us to align the repeating parts and effectively subtract them, leaving only an integer value. This integer is then used to create the numerator of the fraction, while the number used to multiply (less 1) becomes the denominator. This process works for any repeating decimal, regardless of the length of the repeating block.
What if the Repeating Decimal Starts After Some Non-Repeating Digits?
If the repeating decimal has non-repeating digits before the repeating block, a slightly modified approach is necessary. You'll need to account for the non-repeating portion before applying the multiplication and subtraction steps described above. For instance, converting 0.123333... would require a different approach, first addressing the "12" before handling the repeating "3".
Other Common Questions About Repeating Decimals
How do I convert other repeating decimals to fractions?
Follow the steps outlined above. The key is to multiply by the appropriate power of 10 to shift the decimal point so that the repeating part aligns, then subtract the original equation. This leaves you with an integer that forms the numerator of the fraction, with the denominator being the difference between the power of 10 and 1.
Are all repeating decimals rational numbers?
Yes, all repeating decimals represent rational numbers. Rational numbers are those that can be expressed as a ratio of two integers (a fraction).
This comprehensive guide provides you with the knowledge and skills to convert repeating decimals like 0.737373... into fractions. Remember the key steps: assign a variable, multiply to shift the decimal, subtract, and solve. By understanding this process, you can confidently tackle similar conversions in the future.
