Convert The Decimal Fraction 0.1(6) To An Irreducible Ordinary Fraction.

In the realm of mathematics, the ability to convert between different forms of numbers is a fundamental skill. One common conversion involves transforming decimal fractions into ordinary fractions, also known as common fractions. This article delves into the process of converting the decimal fraction 0.1(6) into an irreducible ordinary fraction. We will explore the underlying principles and demonstrate the step-by-step method to arrive at the solution. Understanding this conversion is crucial for various mathematical operations and problem-solving scenarios.

The decimal fraction 0.1(6) is a repeating decimal, where the digit 6 repeats infinitely. To convert this into an ordinary fraction, we need to employ a specific technique that involves algebraic manipulation. This technique allows us to eliminate the repeating decimal part and express the number as a ratio of two integers. The resulting fraction can then be simplified to its irreducible form, which is the simplest representation of the fraction.

This article aims to provide a comprehensive guide to this conversion process, making it accessible to students and anyone interested in enhancing their mathematical skills. We will begin by explaining the concept of repeating decimals and their representation. Then, we will proceed with the conversion steps, providing clear explanations and examples. Finally, we will discuss the significance of irreducible fractions and their role in mathematical calculations. By the end of this article, you will have a solid understanding of how to convert repeating decimals into ordinary fractions and appreciate the importance of this skill in mathematics.

Before diving into the conversion process, it's essential to understand the concept of repeating decimals. A repeating decimal, also known as a recurring decimal, is a decimal number in which one or more digits repeat infinitely. These repeating digits are called the repetend. Repeating decimals are a common occurrence when converting fractions whose denominators have prime factors other than 2 and 5.

Repeating decimals can be represented using a bar over the repeating digits or by enclosing the repeating digits in parentheses. For example, the decimal 0.333... can be written as 0.3 or 0.(3), and the decimal 0.1666... can be written as 0.16 or 0.1(6). In the latter case, the digit 6 is the repetend, indicating that it repeats infinitely.

To grasp the concept of repeating decimals, it's helpful to consider their fractional equivalents. For instance, the fraction 1/3 is equal to the repeating decimal 0.333..., and the fraction 1/6 is equal to the repeating decimal 0.1666.... These examples illustrate how certain fractions result in repeating decimals when converted to decimal form.

The ability to recognize and work with repeating decimals is crucial in various mathematical contexts. Understanding their representation and conversion to ordinary fractions allows for accurate calculations and problem-solving. In the case of 0.1(6), recognizing the repeating digit is the first step towards converting it into an irreducible fraction. The repeating 6 is what differentiates this decimal from a terminating decimal, and it requires a specific method to eliminate the repeating part during conversion.

In the next section, we will delve into the step-by-step process of converting the repeating decimal 0.1(6) into an ordinary fraction. This process involves algebraic manipulation and a clear understanding of the properties of repeating decimals. By mastering this process, you can confidently convert any repeating decimal into its fractional equivalent.

To convert the repeating decimal 0.1(6) into an ordinary fraction, we will follow a systematic approach that involves algebraic manipulation. This method allows us to eliminate the repeating decimal part and express the number as a ratio of two integers. Here are the steps involved:

1. Assign a variable: Let x = 0.1(6). This means x = 0.1666..., where the digit 6 repeats infinitely.
2. Multiply by 10: Since one digit repeats, we multiply both sides of the equation by 10 to shift the decimal point one place to the right: 10x = 1.666...
3. Multiply by 10 again: To isolate the repeating part, we multiply both sides of the original equation by 100 to shift the decimal two places to the right: 100x = 16.666...
4. Subtract the equations: Subtract the equation from step 2 (10x = 1.666...) from the equation in step 3 (100x = 16.666...). This will eliminate the repeating decimal part: 100x - 10x = 16.666... - 1.666... This simplifies to 90x = 15.
5. Solve for x: Divide both sides of the equation by 90 to solve for x: x = 15/90.
6. Simplify the fraction: The fraction 15/90 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 15. So, 15/90 = (15 ÷ 15) / (90 ÷ 15) = 1/6.

Therefore, the ordinary fraction equivalent of the repeating decimal 0.1(6) is 1/6. This fraction is in its irreducible form, meaning that the numerator and denominator have no common factors other than 1.

This step-by-step process demonstrates how algebraic manipulation can be used to convert repeating decimals into ordinary fractions. By carefully multiplying and subtracting equations, we can eliminate the repeating decimal part and express the number as a ratio of two integers. The resulting fraction can then be simplified to its irreducible form, providing the simplest representation of the number. In the next section, we will delve into the concept of irreducible fractions and their significance in mathematical calculations.

In mathematics, an irreducible fraction, also known as a simplified fraction or a fraction in lowest terms, is a fraction in which the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be simplified any further. Irreducible fractions are the preferred way to represent fractional values because they provide the simplest and most concise form.

The process of reducing a fraction to its irreducible form involves dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest positive integer that divides both numbers without leaving a remainder. For example, the GCD of 15 and 90 is 15, as we saw in the previous section.

Irreducible fractions are significant for several reasons. Firstly, they provide a clear and unambiguous representation of a fractional value. When a fraction is in its irreducible form, it is easy to compare it with other fractions and perform mathematical operations. Secondly, irreducible fractions are often required in mathematical problems and solutions. Expressing the answer in its simplest form is a common expectation in mathematics.

Furthermore, irreducible fractions play a crucial role in simplifying calculations. When working with fractions, it is often easier to perform operations with smaller numbers. Reducing fractions to their irreducible form before performing calculations can save time and effort. For instance, adding 15/90 and 1/3 would be more cumbersome than adding 1/6 and 1/3, as the latter involves smaller numbers and a simpler common denominator.

In the context of the problem at hand, converting the repeating decimal 0.1(6) to the irreducible fraction 1/6 is essential. The fraction 1/6 provides the simplest and most accurate representation of the decimal value. It allows for easy comparison with other fractions and simplifies any subsequent calculations involving this number.

In conclusion, irreducible fractions are a fundamental concept in mathematics. They provide the simplest and most concise representation of fractional values, facilitate calculations, and are often required in mathematical problems and solutions. The ability to reduce fractions to their irreducible form is a crucial skill for any mathematics student or professional.

In this article, we have explored the process of converting the repeating decimal 0.1(6) into an irreducible ordinary fraction. We began by understanding the concept of repeating decimals and their representation. Then, we delved into the step-by-step conversion process, which involved algebraic manipulation to eliminate the repeating decimal part. Finally, we discussed the significance of irreducible fractions and their role in mathematical calculations.

The conversion of 0.1(6) to the irreducible fraction 1/6 demonstrates the power of algebraic techniques in simplifying mathematical expressions. By carefully multiplying and subtracting equations, we were able to eliminate the repeating decimal part and express the number as a ratio of two integers. The resulting fraction, 1/6, is in its simplest form, providing a clear and concise representation of the decimal value.

Understanding the significance of irreducible fractions is crucial for various mathematical applications. Irreducible fractions simplify calculations, facilitate comparisons, and are often required in mathematical problems and solutions. The ability to reduce fractions to their irreducible form is a fundamental skill that enhances mathematical proficiency.

This article has provided a comprehensive guide to converting repeating decimals into ordinary fractions. By mastering this process, you can confidently convert any repeating decimal into its fractional equivalent. This skill is not only valuable in academic settings but also in real-world scenarios where fractions and decimals are encountered. Whether you are a student learning mathematics or a professional working with numbers, the ability to convert between decimal and fractional forms is an essential tool in your mathematical arsenal.

In summary, the conversion of 0.1(6) to 1/6 is a testament to the elegance and power of mathematics. It showcases how seemingly complex numbers can be simplified and expressed in their most fundamental form. By understanding the principles and techniques involved, you can unlock a deeper appreciation for the beauty and utility of mathematics.