What Is 1.6 Repeating As A Fraction

What is 1.6 Repeating as a Fraction? A Comprehensive Guide

The question of how to represent the repeating decimal 1.666... as a fraction might seem simple at first glance. However, understanding the underlying process reveals a fundamental concept in mathematics concerning the conversion of repeating decimals to rational numbers (fractions). This article will delve into various methods for solving this problem, explaining the reasoning behind each step and providing a deeper understanding of the mathematical principles involved. We'll also explore related concepts and address common misconceptions.

Understanding Repeating Decimals

Before tackling the conversion, let's define what a repeating decimal is. 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. In our case, the number 1.666... has the digit "6" repeating infinitely. We can represent this repeating part using a vinculum (a horizontal bar) placed above the repeating digit(s): 1.$\overline{6}$. This notation clearly indicates the infinite repetition.

Method 1: Algebraic Manipulation

This method is a classic and elegant approach to converting repeating decimals to fractions. It involves using algebraic equations to isolate the repeating part.

Steps:

1. Let x equal the repeating decimal: Let x = 1.$\overline{6}$

2. Multiply by a power of 10: We need to shift the decimal point to align the repeating part. Since only one digit repeats, we multiply by 10: 10x = 16.$\overline{6}$

3. Subtract the original equation: Subtract the equation in step 1 from the equation in step 2:

   10x - x = 16.$\overline{6}$ - 1.$\overline{6}$

   This simplifies to:

   9x = 15

4. Solve for x: Divide both sides by 9:

   x = 15/9

5. Simplify the fraction: Both the numerator and the denominator are divisible by 3:

   x = 5/3

Therefore, 1.6 repeating is equal to 5/3.

Method 2: Using the Formula for Repeating Decimals

A more generalized formula can be used to directly convert repeating decimals to fractions. For a decimal of the form 0.a₁a₂...a_n$\overline{b₁b₂...b_m}$, where a_i are the non-repeating digits and b_i are the repeating digits, the formula is:

$\frac{N - A}{10^n (10^m - 1)}$

Where:

• N is the number formed by the entire decimal (including repeating part) up to a certain point.
• A is the number formed by the non-repeating digits.
• n is the number of non-repeating digits.
• m is the number of repeating digits.

In our case, 1.$\overline{6}$:

• N can be any number formed by the repeating part, such as 16.666... (we can take a portion of the repeating sequence)
• A = 1
• n = 1
• m = 1

Let's use N = 16 (This works because of the infinite nature of the repeating digits)

The formula becomes:

$\frac{16 - 1}{10^1 (10^1 - 1)} = \frac{15}{10(9)} = \frac{15}{90} = \frac{5}{30} = \frac{1}{6}$

Notice that this gives us an incorrect answer. However, it is not suitable here because we need to represent the initial integer value "1" before the repeating decimal part. For this reason this method must be used carefully, focusing on the repeating part only. Thus, to calculate the fraction for 0.$\overline{6}$ we can use the formula:

• N = 6
• A = 0
• n = 0
• m = 1

$\frac{6 - 0}{10^0(10^1-1)} = \frac{6}{9} = \frac{2}{3}$

We found the fraction for 0.$\overline{6}$, which is $\frac{2}{3}$. Now, we add the whole number part:

$1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}$

This confirms our previous result. The formula is generally useful but needs careful application, particularly when dealing with numbers containing a whole number component and a repeating decimal part.

Method 3: Geometric Series

This method leverages the concept of an infinite geometric series. The repeating decimal 1.$\overline{6}$ can be expressed as:

1 + 0.6 + 0.06 + 0.006 + ...

This is an infinite geometric series with the first term a = 0.6 and the common ratio r = 0.1. The sum of an infinite geometric series is given by the formula:

S = a / (1 - r) where |r| < 1

Substituting our values:

S = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 6/9 = 2/3

Adding the whole number part (1) back in, we get 1 + 2/3 = 5/3.

Understanding the Result: 5/3

The fraction 5/3 represents an improper fraction, meaning the numerator (5) is greater than the denominator (3). This is perfectly acceptable and accurately reflects the value of the repeating decimal 1.$\overline{6}$. It can also be expressed as a mixed number: 1 2/3.

Common Mistakes and Misconceptions

• Rounding: Rounding the repeating decimal to a finite number of decimal places (e.g., 1.666667) will lead to an approximation, not an exact fractional representation. The key is to acknowledge the infinite repetition.
• Incorrect Application of Formulas: As shown in Method 2, the general formula for converting repeating decimals to fractions needs careful consideration of the whole number component and the position of the repeating sequence. It's crucial to understand the structure of the formula and to apply it correctly based on the form of the given number.
• Ignoring the Infinite Repetition: Treating the repeating decimal as if it were a finite decimal will always lead to an inaccurate result. The infinite nature of the repetition is crucial to the mathematical manipulation needed for accurate conversion.

Further Exploration: Other Repeating Decimals

The methods outlined above can be applied to convert any repeating decimal into a fraction. The complexity might increase with more repeating digits or longer non-repeating sections, but the underlying principles remain the same. For example, try converting 0.$\overline{142857}$ or 2.3$\overline{18}$ using the algebraic manipulation method or the geometric series method.

Conclusion

Converting a repeating decimal like 1.666... to a fraction involves understanding the properties of repeating decimals and applying appropriate mathematical techniques. Whether using algebraic manipulation, the formula for repeating decimals, or the geometric series approach, the result remains consistent: 1.6 repeating is equivalent to the fraction 5/3. This article has provided a comprehensive guide, addressing common pitfalls and offering multiple methods to enhance your understanding of this fundamental concept in mathematics. The ability to convert repeating decimals to fractions is a valuable skill that is important in numerous mathematical applications. Remember to always focus on the infinite nature of the repeating part to achieve the precise fractional representation.