The decimal 0.737373... (where the "73" repeats infinitely) might seem confusing, but converting it to a fraction is surprisingly straightforward. This guide will walk you through the process, explaining the underlying math and answering common questions. Understanding this process unlocks a broader understanding of how repeating decimals and fractions relate.
What is a Repeating Decimal?
A repeating decimal is a decimal number that has a digit or a group of digits that repeat infinitely. We often denote repeating decimals using a bar over the repeating part, like this: 0.7̅3̅. This clearly indicates that the sequence "73" continues forever. Unlike terminating decimals (like 0.25 or 0.75), which have a finite number of digits, repeating decimals continue without end.
Converting 0.7̅3̅ to a Fraction: The Step-by-Step Process
Here's how to convert the repeating decimal 0.7̅3̅ into its fractional equivalent:
Step 1: Assign a Variable
Let's represent the repeating decimal with a variable, say 'x':
x = 0.737373...
Step 2: Multiply to Shift the Decimal
Multiply both sides of the equation by 100. We use 100 because the repeating block has two digits ("73"):
100x = 73.737373...
Step 3: Subtract the Original Equation
Now, subtract the original equation (x = 0.737373...) from the equation we just created:
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 0.7̅3̅ is 73/99.
Can this fraction be simplified?
No, the fraction 73/99 cannot be simplified further. 73 is a prime number, and it does not divide evenly into 99.
How to Convert Other Repeating Decimals to Fractions
The method above works for any repeating decimal. The key is to multiply by a power of 10 that shifts the repeating block to the left of the decimal point. For example:
- 0.333... (0.3̅): Multiply by 10, then subtract the original equation (resulting in 9x = 3, or x = 1/3).
- 0.121212... (0.1̅2̅): Multiply by 100, then subtract the original equation.
- 0.142857142857... (0.1̅4̅2̅8̅5̅7̅): Multiply by 1,000,000, then subtract the original equation.
Why Does This Method Work?
This method works because of the nature of repeating decimals and the manipulation of equations. By multiplying by powers of 10, we're essentially aligning the repeating parts of the decimal, allowing us to subtract them and isolate the whole number portion. The result is always a fraction where the denominator is a power of 10 minus 1.
Frequently Asked Questions (FAQ)
What is the simplest form of 0.73 repeating?
The simplest form of 0.73 repeating (0.7̅3̅) is 73/99. It cannot be reduced further.
How do you convert a repeating decimal to a fraction without a calculator?
The method described above (using variables and multiplication/subtraction) is entirely calculator-free.
Are all repeating decimals rational numbers?
Yes, all repeating decimals are rational numbers. A rational number is a number that can be expressed as a fraction (a/b) where 'a' and 'b' are integers, and 'b' is not zero.
Can all fractions be expressed as terminating or repeating decimals?
Yes, all fractions can be expressed as either terminating or repeating decimals. The key determining factor is whether the denominator of the fraction, when expressed in its simplest form, contains only prime factors of 2 and/or 5 (for terminating decimals) or any other prime factors (for repeating decimals).
This comprehensive guide provides a clear and concise understanding of how to convert the repeating decimal 0.7̅3̅ into its fractional form, as well as the broader context of repeating decimals and their relation to fractions. Hopefully, this will alleviate any confusion surrounding this mathematical concept.
