0.5416 Repeating as a Fraction
In mathematics, a repeating decimal is a decimal representation of a number that recurs indefinitely in a predictable pattern. One such example is the decimal 0.5416, which repeating indefinitely. But have you ever wondered what this decimal represents as a fraction?
What is 0.5416 as a Fraction?
To convert 0.5416 to a fraction, we can use a few different methods. One way is to use the fact that the decimal repeats every four digits. Let's call this repeating part "r". We can set up an equation:
r = 0.5416
Since the decimal repeats every four digits, we can multiply both sides of the equation by 10,000 (10^4) to get:
10,000r = 5,416.5416
Now, subtract the original equation from this new one:
9,999r = 5,416
Next, divide both sides by 9,999:
r = 5,416 / 9,999
Simplifying the Fraction
The fraction 5,416 / 9,999 can be simplified by finding the greatest common divisor (GCD) of the numerator and denominator. Using the Euclidean algorithm, we find that the GCD is 1, so the fraction is already in its simplest form.
Therefore, 0.5416 repeating as a fraction is equal to:
5416/9999
This fraction represents the same value as the repeating decimal 0.5416.
Conclusion
In this article, we've seen how to convert the repeating decimal 0.5416 to a fraction using simple algebra and the properties of repeating decimals. The resulting fraction, 5416/9999, is a simplified and exact representation of the original decimal. Understanding how to work with repeating decimals and convert them to fractions is an important skill in mathematics, with applications in many areas of math and science.
