1.83 Repeating As A Fraction

Decoding the Mystery: 1.83 Repeating as a Fraction

Understanding how to convert repeating decimals, like 1.83 repeating, into fractions might seem daunting at first. But with a systematic approach, it becomes a manageable and even fascinating mathematical puzzle. This article will guide you through the process, exploring not just the mechanics of the conversion but also the underlying mathematical principles. We’ll delve into the concept of repeating decimals, explain the conversion method step-by-step, and even tackle some frequently asked questions. By the end, you’ll be equipped to tackle any repeating decimal conversion with confidence!

Understanding Repeating Decimals

A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or a group of digits that repeat infinitely. The repeating part is usually indicated by a bar placed over the repeating digits. For instance, 1.83 repeating is written as 1.83̅. This notation signifies that the digits "83" repeat endlessly: 1.83838383...

Understanding the nature of repeating decimals is crucial before we attempt to convert them into fractions. The repeating part implies an infinite series, and converting it involves manipulating this series to obtain a finite fraction.

Converting 1.83 Repeating to a Fraction: A Step-by-Step Guide

Let's break down the conversion of 1.83̅ into a fraction. We'll employ a method that utilizes algebraic manipulation:

Step 1: Assign a Variable

Let's represent the repeating decimal 1.83̅ with the variable x:

x = 1.83̅

Step 2: Multiply to Shift the Repeating Part

We need to manipulate the equation to isolate the repeating part. Since the repeating part has two digits ("83"), we multiply both sides of the equation by 100:

100x = 183.83̅

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 1.83̅) from the equation we obtained in Step 2:

100x - x = 183.83̅ - 1.83̅

This subtraction cleverly eliminates the repeating part:

99x = 182

Step 4: Solve for x

Finally, solve for x by dividing both sides by 99:

x = 182/99

Therefore, 1.83̅ as a fraction is 182/99.

This fraction is in its simplest form because 182 and 99 don't share any common factors other than 1.

Simplifying Fractions: A Quick Recap

While 182/99 is the simplest form in this case, it's useful to review simplifying fractions. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, and divide both by that GCD. For example, to simplify 12/18:

1. Find the GCD of 12 and 18 (it's 6).
2. Divide both numerator and denominator by 6: 12/6 = 2 and 18/6 = 3.
3. The simplified fraction is 2/3.

In our case, the GCD of 182 and 99 is 1, so 182/99 is already in its simplest form.

The Mathematical Underpinnings: Infinite Geometric Series

The method we used relies on the concept of an infinite geometric series. A geometric series is a series where each term is obtained by multiplying the previous term by a constant value (the common ratio). A repeating decimal can be expressed as an infinite geometric series.

Let's analyze 1.83̅:

1.83̅ = 1 + 0.83 + 0.0083 + 0.000083 + ...

This is a geometric series with:

• First term (a) = 0.83
• Common ratio (r) = 0.01

The sum of an infinite geometric series is given by the formula: S = a / (1 - r), provided that |r| < 1 (the absolute value of the common ratio is less than 1).

In our case:

S = 0.83 / (1 - 0.01) = 0.83 / 0.99 = 83/99

Adding the whole number part (1), we get:

1 + 83/99 = (99 + 83) / 99 = 182/99

This confirms our result from the algebraic method.

Converting Other Repeating Decimals

The method demonstrated above can be applied to any repeating decimal. The key is to:

1. Identify the repeating part: Determine the digits that repeat.
2. Multiply by the appropriate power of 10: Multiply the equation by 10 raised to the power of the number of digits in the repeating part.
3. Subtract the original equation: This eliminates the repeating part.
4. Solve for x: Solve the resulting equation for the variable representing the repeating decimal.
5. Simplify the fraction: Reduce the fraction to its simplest form by finding the GCD of the numerator and denominator.

For example, let's convert 0.6̅ to a fraction:

x = 0.6̅ 10x = 6.6̅ 10x - x = 6.6̅ - 0.6̅ 9x = 6 x = 6/9 = 2/3

Frequently Asked Questions (FAQ)

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

For example, consider 2.16̅. First, separate the non-repeating part: 2 + 0.16̅. Then, convert 0.16̅ to a fraction using the method described above. Finally, add the whole number part.

x = 0.16̅ 100x = 16.16̅ 100x - x = 15 x = 15/99 = 5/33

Therefore, 2.16̅ = 2 + 5/33 = (66 + 5) / 33 = 71/33

Q2: Can I use a calculator to convert repeating decimals to fractions?

Some advanced calculators have built-in functions to handle this conversion. However, understanding the manual method is valuable for comprehending the underlying mathematical principles.

Q3: Are all repeating decimals rational numbers?

Yes, all repeating decimals are rational numbers. A rational number is a number that can be expressed as a fraction of two integers (where the denominator is not zero). The fact that we can convert a repeating decimal to a fraction proves its rationality.

Conclusion

Converting a repeating decimal like 1.83̅ to a fraction might seem challenging initially, but with a methodical approach and a grasp of the underlying mathematical concepts, it becomes an achievable and even enjoyable task. This article has provided a step-by-step guide, explored the concept of infinite geometric series, and answered some frequently asked questions. Now you possess the knowledge and skills to confidently convert any repeating decimal into its fractional equivalent, solidifying your understanding of rational numbers and decimal representation. Remember to practice, and you'll soon master this valuable mathematical skill!