1.3 Repeating As A Fraction

Decoding the Mystery of 1.3 Repeating: A Deep Dive into Fractions and Decimal Representation

Understanding the relationship between decimals and fractions is fundamental to grasping core mathematical concepts. Many find the concept of repeating decimals, like 1.3 repeating (often written as 1.3̅ or 1.$\overline{3}$), particularly challenging. This article will demystify this seemingly complex idea, guiding you through the process of converting 1.3 repeating into a fraction, explaining the underlying mathematical principles, and exploring related concepts. We'll also address frequently asked questions, ensuring a thorough understanding for readers of all levels.

Understanding Repeating Decimals

A repeating decimal is a decimal number that has a digit or a group of digits that repeat infinitely. The repeating part is indicated by a bar placed over the repeating digits. For instance, 0.333... is written as 0.$\overline{3}$, while 0.142857142857... is written as 0.$\overline{142857}$. These repeating decimals represent rational numbers – numbers that can be expressed as a fraction of two integers (a/b, where 'a' and 'b' are integers and b≠0). The challenge lies in determining the precise fraction that represents the repeating decimal.

Converting 1.3 Repeating to a Fraction: A Step-by-Step Guide

The conversion of 1.3̅ to a fraction involves a simple yet elegant algebraic method. Let's break down the process step-by-step:

Step 1: Represent the Repeating Decimal with a Variable

Let's represent the repeating decimal 1.3̅ with a variable, say 'x':

x = 1.3̅

Step 2: Multiply to Shift the Repeating Part

Multiply both sides of the equation by a power of 10 that shifts the repeating part to the left of the decimal point. Since only the digit '3' is repeating, we multiply by 10:

10x = 13.3̅

Step 3: Subtract the Original Equation

Subtract the original equation (x = 1.3̅) from the equation obtained in Step 2 (10x = 13.3̅):

10x - x = 13.3̅ - 1.3̅

This simplifies to:

9x = 12

Step 4: Solve for x

Solve for 'x' by dividing both sides by 9:

x = 12/9

Step 5: Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator (12) and the denominator (9). The GCD of 12 and 9 is 3. Dividing both the numerator and the denominator by 3 gives us the simplified fraction:

x = 4/3

Therefore, 1.3̅ is equal to 4/3.

A Deeper Look at the Mathematics Behind the Conversion

The method employed above leverages the properties of arithmetic and the unique nature of repeating decimals. By multiplying by a power of 10, we effectively shift the repeating decimal portion, allowing us to subtract the original equation and eliminate the infinitely repeating sequence. This leaves us with a simple algebraic equation that can be readily solved to find the equivalent fraction. The process essentially transforms an infinite series into a finite expression.

This method works because repeating decimals represent infinite geometric series. For example, 0.$\overline{3}$ can be written as:

0.3 + 0.03 + 0.003 + 0.0003 + ...

This is an infinite geometric series with the first term (a) = 0.3 and the common ratio (r) = 0.1. The sum of an infinite geometric series is given by the formula:

S = a / (1 - r)

In our case:

S = 0.3 / (1 - 0.1) = 0.3 / 0.9 = 1/3

This demonstrates the connection between the algebraic method and the concept of infinite geometric series. While the algebraic method is often more practical for calculations, understanding the underlying geometric series provides a more profound comprehension of the mathematical principles at play.

Exploring Other Repeating Decimals

The method described above can be applied to any repeating decimal. Let's consider another example: 0.$\overline{142857}$.

1. Let x = 0.$\overline{142857}$
2. Multiply by 1,000,000 (since there are six repeating digits): 1,000,000x = 142857.$\overline{142857}$
3. Subtract the original equation: 999,999x = 142857
4. Solve for x: x = 142857/999999
5. Simplify: x = 1/7

This illustrates the versatility of the method. The key is to identify the repeating block and multiply by the appropriate power of 10 to align the repeating sequence for subtraction.

Frequently Asked Questions (FAQ)

Q1: What if the repeating decimal starts after a non-repeating part?

A: For decimals with a non-repeating part followed by a repeating part (e.g., 1.2$\overline{3}$), you would modify the process slightly. You would first multiply by a power of 10 to move the non-repeating part to the left of the decimal. Then follow the steps outlined above for the repeating part. For example, for 1.2$\overline{3}$:

1. Let x = 1.2$\overline{3}$
2. Multiply by 10: 10x = 12.$\overline{3}$
3. Multiply by 100: 100x = 123.$\overline{3}$
4. Subtract 10x from 100x: 90x = 111
5. Solve for x: x = 111/90 = 37/30

Q2: Are all repeating decimals rational numbers?

A: Yes, all repeating decimals represent rational numbers. This is a fundamental property of repeating decimals, as they can always be expressed as a fraction of two integers using the methods described above.

Q3: Can non-repeating decimals be expressed as fractions?

A: Terminating decimals (decimals that end) can be expressed as fractions. For example, 0.75 = ¾. However, non-repeating, non-terminating decimals (like π or √2) are irrational numbers and cannot be expressed as a simple fraction.

Q4: Why is this conversion important?

A: This conversion is crucial for several reasons. It helps bridge the gap between decimal and fractional representation, vital for understanding mathematical operations and problem-solving in various fields like engineering, physics, and computer science. It also deepens the understanding of number systems and their relationships.

Conclusion

Converting 1.3 repeating (or any repeating decimal) into a fraction is a straightforward process when you understand the underlying mathematical principles. By employing a simple algebraic method, we can effectively transform an infinite repeating decimal into a concise and manageable fractional representation. This article provided a step-by-step guide, explained the mathematics behind the conversion, explored related concepts, and addressed common questions. Mastering this skill significantly enhances your understanding of number systems and lays a solid foundation for more advanced mathematical concepts. Remember, the key is to practice and apply the methods to various examples to solidify your comprehension. The seemingly complex world of repeating decimals becomes much more accessible once you break down the process and understand the underlying logic.