46 Repeating As A Fraction

Decoding the Mystery: 46 Repeating as a Fraction

The seemingly simple decimal 0.464646... (often written as 0.46̅) presents a fascinating challenge: how do we express this repeating decimal as a fraction? Understanding this process not only unveils the beauty of mathematical conversions but also provides a solid foundation for tackling more complex repeating decimals. This article will guide you through the steps, explain the underlying mathematics, and answer frequently asked questions, ensuring a comprehensive understanding of this topic.

Understanding Repeating Decimals

Before diving into the conversion process, it's crucial to understand what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digits "46" repeat endlessly. These repeating decimals are rational numbers, meaning they can be expressed as a fraction of two integers. This is in contrast to irrational numbers, such as π (pi) or √2 (the square root of 2), which have non-repeating, non-terminating decimal expansions.

Converting 0.46̅ to a Fraction: A Step-by-Step Guide

The conversion of a repeating decimal to a fraction involves a clever algebraic manipulation. Here's the breakdown:

Step 1: Assign a Variable

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

x = 0.464646...

Step 2: Multiply to Shift the Decimal

We need to manipulate the equation so that we can subtract the original equation and eliminate the repeating part. The key is to multiply the equation by a power of 10 that shifts the repeating block to the left of the decimal point. Since the repeating block has two digits ("46"), we multiply by 100:

100x = 46.464646...

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 0.464646...) from the multiplied equation (100x = 46.464646...):

100x - x = 46.464646... - 0.464646...

This simplifies to:

99x = 46

Step 4: Solve for x

Finally, solve for 'x' by dividing both sides of the equation by 99:

x = 46/99

Therefore, the fraction equivalent of the repeating decimal 0.46̅ is 46/99.

Simplifying the Fraction (If Possible)

In this case, the fraction 46/99 is already in its simplest form because the greatest common divisor (GCD) of 46 and 99 is 1. However, if you had obtained a fraction like 12/18, you would need to simplify it by finding the GCD (which is 6 in this example) and dividing both the numerator and denominator by it, resulting in the simplified fraction 2/3.

The Mathematical Rationale Behind the Method

The method we employed relies on the properties of infinite geometric series. The repeating decimal 0.46̅ can be written as:

0.46 + 0.0046 + 0.000046 + ...

This is an infinite geometric series with the first term (a) = 0.46 and the common ratio (r) = 0.01. The sum of an infinite geometric series is given by the formula:

Sum = a / (1 - r)

Substituting our values:

Sum = 0.46 / (1 - 0.01) = 0.46 / 0.99 = 46/99

This confirms our earlier result obtained through algebraic manipulation.

Converting Other Repeating Decimals

The same method can be applied to other repeating decimals. The only difference lies in the multiplier used in Step 2. If the repeating block has one digit, multiply by 10. If it has three digits, multiply by 1000, and so on. For example:

• 0.333... (0.3̅): Let x = 0.3̅. Then 10x = 3.3̅. Subtracting x from 10x gives 9x = 3, so x = 3/9 = 1/3.
• 0.123123... (0.123̅): Let x = 0.123̅. Then 1000x = 123.123̅. Subtracting x from 1000x gives 999x = 123, so x = 123/999 = 41/333.

Handling Repeating Decimals with Non-Repeating Parts

Things become slightly more complex when dealing with repeating decimals that have a non-repeating part before the repeating block. For instance, consider 0.246̅. You would handle the non-repeating part separately.

Step 1: Separate the non-repeating and repeating parts. In this example: 0.2 + 0.0464646...

Step 2: Convert the repeating part to a fraction using the method described above. 0.046̅ = 46/990

Step 3: Convert the non-repeating part to a fraction. 0.2 = 2/10 = 1/5

Step 4: Add the two fractions together: (1/5) + (46/990) = (198/990) + (46/990) = 244/990. This can be simplified to 122/495

Frequently Asked Questions (FAQ)

• Q: Why does this method work? A: The method works because it leverages the properties of infinite geometric series and allows us to eliminate the infinitely repeating part of the decimal by clever subtraction.

• Q: Can all repeating decimals be expressed as fractions? A: Yes, all repeating decimals are rational numbers and can be expressed as a fraction of two integers.

• Q: What if the repeating block is very long? A: The process remains the same, but the numbers will be larger. You might need a calculator to simplify the resulting fraction.

• Q: What if the decimal has a non-repeating part after the repeating part? A: This is not a standard repeating decimal. Such a number is still rational if its decimal representation terminates (ends) after a finite number of decimal places or is a repeating decimal. However, it might represent an irrational number if the non-repeating part goes on forever and there is no pattern in the non-repeating part.

Conclusion

Converting repeating decimals to fractions might seem daunting at first, but with the systematic approach outlined in this article, it becomes a manageable and even enjoyable process. The underlying mathematical principles are elegant and powerful, revealing the interconnectedness between different representations of numbers. Mastering this technique not only strengthens your understanding of decimals and fractions but also equips you with a valuable skill applicable to various mathematical problems and further mathematical exploration. Remember, practice is key! The more you work through different examples, the more confident and proficient you'll become in converting repeating decimals into their equivalent fractional forms. You will gain a deeper appreciation for the beauty and logic inherent within the realm of mathematics.