0.777... as a Fraction
Introduction
The repeating decimal 0.777... is a fascinating mathematical concept that has puzzled many students and mathematicians alike. While it may seem simple at first glance, converting this decimal to a fraction can be a challenging task. In this article, we will explore how to convert 0.777... to a fraction and delve into the underlying mathematical concepts.
Repeating Decimals
A repeating decimal is a decimal that has a sequence of digits that repeats indefinitely. In the case of 0.777..., the sequence of digits is "7" repeating indefinitely. This type of decimal is also known as a non-terminating, repeating decimal.
Converting 0.777... to a Fraction
To convert 0.777... to a fraction, we can use the following steps:
Let x = 0.777...
Multiply both sides by 10 to get: 10x = 7.777...
Subtract x from both sides to get: 9x = 7
Divide both sides by 9 to get: x = 7/9
Therefore, 0.777... can be written as a fraction as:
7/9
Proof
To prove that 0.777... is equal to 7/9, we can use the following mathematical identities:
- 0.777... = 0.7 + 0.07 + 0.007 + ...
- 0.7 = 7/10
- 0.07 = 7/100
- 0.007 = 7/1000
Using these identities, we can rewrite 0.777... as:
0.777... = 7/10 + 7/100 + 7/1000 + ...
Factoring out 7 from each term, we get:
0.777... = 7(1/10 + 1/100 + 1/1000 + ...)
Simplifying the expression, we get:
0.777... = 7(1/9)
Therefore, 0.777... is equal to 7/9.
Conclusion
In conclusion, 0.777... can be converted to a fraction as 7/9. This conversion involves using mathematical identities and clever manipulations to arrive at the final answer. Understanding how to convert repeating decimals to fractions is an important concept in mathematics, and it has many real-world applications in fields such as engineering, physics, and economics.
