0.4 Repeating and 0.1 Repeating: Understanding Repeating Decimals
What are Repeating Decimals?
Repeating decimals, also known as recurring decimals, are a type of decimal number that has a sequence of digits that repeats indefinitely. These decimals are a result of dividing one integer by another, but the result is not a finite decimal. Instead, the digits repeat in a cycle, often with a finite number of digits.
0.4 Repeating (0.444...)
0.4 repeating, also written as 0.444..., is a repeating decimal where the digit 4 repeats indefinitely. This decimal is equivalent to the fraction 4/9.
Conversion to Fraction
To convert 0.4 repeating to a fraction, we can use the following steps:
- Let x = 0.4 repeating
- Multiply both sides by 10 to get 10x = 4.4 repeating
- Subtract x from both sides to get 9x = 4
- Divide both sides by 9 to get x = 4/9
Properties
- 0.4 repeating is a rational number, which means it can be expressed as a finite decimal or a ratio of integers.
- It is a non-terminating decimal, meaning it cannot be written in finite decimal form.
- The decimal 0.4 repeating is an infinite, non-repeating sequence of digits.
0.1 Repeating (0.111...)
0.1 repeating, also written as 0.111..., is a repeating decimal where the digit 1 repeats indefinitely. This decimal is equivalent to the fraction 1/9.
Conversion to Fraction
To convert 0.1 repeating to a fraction, we can use the following steps:
- Let x = 0.1 repeating
- Multiply both sides by 10 to get 10x = 1.1 repeating
- Subtract x from both sides to get 9x = 1
- Divide both sides by 9 to get x = 1/9
Properties
- 0.1 repeating is a rational number, which means it can be expressed as a finite decimal or a ratio of integers.
- It is a non-terminating decimal, meaning it cannot be written in finite decimal form.
- The decimal 0.1 repeating is an infinite, non-repeating sequence of digits.
Conclusion
In conclusion, 0.4 repeating and 0.1 repeating are two examples of repeating decimals that can be converted to fractions. Understanding these decimals and their properties is essential in mathematics, particularly in algebra and number theory.
