09 repeating as a fraction
Understand the Problem
The question is asking how to express the repeating decimal 0.999... as a fraction. This involves using algebraic methods to convert a decimal into a fraction.
Answer
The repeating decimal \( 0.999... \) is equal to 1.
Answer for screen readers
The repeating decimal ( 0.999... ) can be expressed as the fraction ( \frac{1}{1} ) or simply 1.
Steps to Solve
- Assign a variable to the repeating decimal
Let ( x = 0.999...).
- Multiply both sides by 10
To eliminate the repeating part, multiply both sides by 10:
$$ 10x = 9.999... $$
- Set up the equation
Now, you have two equations:
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( x = 0.999... )
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( 10x = 9.999... )
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Subtract the first equation from the second
Now subtract the first equation from the second:
$$ 10x - x = 9.999... - 0.999... $$
- Simplify the equation
This simplifies to:
$$ 9x = 9 $$
- Solve for ( x )
Now, divide both sides by 9:
$$ x = 1 $$
The repeating decimal ( 0.999... ) can be expressed as the fraction ( \frac{1}{1} ) or simply 1.
More Information
Interestingly, this shows that the repeating decimal ( 0.999... ) is actually equal to 1. Many people find this surprising, but it is a well-established fact in mathematics.
Tips
- A common mistake is thinking that ( 0.999... ) and 1 are different. Remember that they are the same value.
- Another mistake is forgetting to subtract the original equation from the multiplied one, which may lead to incorrect solutions.
