The seemingly simple decimal .737373... (or 0.73 repeating) hides a fascinating connection to the world of fractions. Understanding how to convert repeating decimals into fractions is a key concept in mathematics, and this particular decimal provides a perfect illustration of the process. This article will explore the conversion, delve into the underlying principles, and answer some frequently asked questions about repeating decimals.
What is a Repeating Decimal?
Before we dive into converting 0.73 repeating, let's define what a repeating decimal is. A repeating decimal is a decimal number that has a digit or group of digits that repeat infinitely. These repeating digits are often indicated by a bar placed over them. For example, 0.73 repeating is written as 0.. This notation clearly shows that the digits "73" continue to repeat endlessly.
How to Convert 0.73 Repeating to a Fraction
Converting repeating decimals to fractions involves a straightforward algebraic method. Here's how to convert 0.:
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Let x = 0.737373... This assigns a variable to our repeating decimal.
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Multiply by 100: Since two digits repeat, we multiply both sides of the equation by 100: 100x = 73.737373...
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Subtract the original equation: Subtract the original equation (x = 0.737373...) from the equation in step 2:
100x - x = 73.737373... - 0.737373...
This simplifies to: 99x = 73
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Solve for x: Divide both sides by 99:
x = 73/99
Therefore, the fraction equivalent of 0.73 repeating is 73/99. This fraction is in its simplest form, as 73 is a prime number and doesn't share any common factors with 99.
What are some other examples of repeating decimals and their fractions?
Many repeating decimals can be converted to fractions using this method. For example:
- 0.: This is equivalent to 1/3.
- 0.: This is equivalent to 1/7.
- 0.: This is equivalent to 2/3.
The key is to multiply by a power of 10 that shifts the repeating block to the left of the decimal point. Then, subtract the original equation, leaving you with an equation you can easily solve for x.
How do I convert other repeating decimals into fractions?
The method described above works for any repeating decimal. The key is to identify the repeating block of digits and multiply by 10 raised to the power of the number of digits in the repeating block. If you have a decimal with a non-repeating part before the repeating part (e.g., 1.2), you will need to adjust the method slightly to account for this non-repeating part.
Can all decimals be converted into fractions?
No. Only terminating and repeating decimals can be expressed as fractions. Non-repeating, non-terminating decimals, like pi (π), are irrational numbers and cannot be expressed as a simple fraction.
Why is understanding this conversion important?
Understanding how to convert repeating decimals to fractions is crucial for various mathematical applications, including algebra, calculus, and even computer science. It helps build a stronger foundation in number systems and improves problem-solving skills.
In conclusion, the seemingly simple 0.73 repeating offers a clear and concise example of how to convert a repeating decimal into its fractional equivalent. This understanding opens the door to a deeper appreciation of the relationship between decimals and fractions, highlighting the interconnectedness of seemingly disparate mathematical concepts.
