0.33333...(Repeating) as a Fraction
The repeating decimal 0.33333... is a fascinating mathematical concept that has puzzled many mathematicians and students alike. But what if we told you that this decimal can be expressed as a fraction? In this article, we'll explore how to convert 0.33333... into a fraction and understand the mathematical principles behind it.
What is 0.33333...?
0.33333... is a decimal that goes on indefinitely in a repeating pattern. It's a non-terminating, non-repeating decimal, also known as a non-terminating decimal. This type of decimal has an infinite number of digits, and the pattern of digits never ends.
Converting 0.33333... to a Fraction
To convert 0.33333... to a fraction, we can use a mathematical technique called the "repeating decimal to fraction" method. Here's how it works:
Let x = 0.33333...
Multiply both sides by 10 to get:
10x = 3.33333...
Now, subtract the first equation from the second equation to eliminate the decimal part:
10x - x = 3.33333... - 0.33333... 9x = 3
Divide both sides by 9 to solve for x:
x = 3/9 x = 1/3
And there you have it! 0.33333... is equal to the fraction 1/3.
Understanding the Math Behind It
So why does this method work? To understand the math behind it, let's dive deeper into the concept of repeating decimals.
A repeating decimal is a decimal that has a pattern of digits that repeats indefinitely. In the case of 0.33333..., the pattern is 3, which repeats forever. When we multiply 0.33333... by 10, we're essentially shifting the decimal point one place to the right, so the pattern remains the same.
By subtracting the original equation from the new equation, we're able to eliminate the decimal part, leaving us with a whole number. This is because the decimal part of 0.33333... is the same as the decimal part of 3.33333... - 0.33333..., which is 0.
The resulting equation, 9x = 3, is a linear equation that can be easily solved to find the value of x. In this case, x = 1/3, which is the fraction equivalent of 0.33333....
Conclusion
In conclusion, 0.33333... can be expressed as a fraction, specifically 1/3. This is a fascinating example of how a repeating decimal can be converted into a simple fraction using mathematical techniques. By understanding the principles behind this conversion, we can gain a deeper appreciation for the beauty and complexity of mathematics.
