In mathematics, a non-terminating decimal is a decimal that does not have an end digit and has an infinite number of terms. These decimals can be converted into fractions using a specific process. This article will guide you through the steps of converting non-terminating decimals to fractions, using examples to illustrate the process.

### Key Facts

- Identify the repeating pattern: Non-terminating decimals have a repeating pattern of digits. Identify the repeating pattern in the decimal number.
- Set up an equation: Let x be the non-terminating decimal number.
- Multiply by a power of 10: Multiply both sides of the equation by a power of 10 that eliminates the decimal places in the repeating pattern. This will move the repeating pattern to the left of the decimal point.
- Subtract the original equation: Subtract the equation obtained in step 2 from the original equation. This will eliminate the repeating pattern.
- Simplify the equation: Simplify the resulting equation to obtain the fraction form.

Here is an example to illustrate the process:

Convert 0.77777… to a fraction.

Step 1: The repeating pattern is 7.

Step 2: Let x = 0.7777…

Step 3: Multiply both sides by 10 to move the repeating pattern to the left: 10x = 7.7777…

Step 4: Subtract the original equation: 10x – x = 7.7777… – 0.7777…

Step 5: Simplify the equation: 9x = 7

Step 6: Solve for x: x = 7/9

Therefore, 0.77777… can be expressed as the fraction 7/9.

## Identifying the Repeating Pattern

The first step in converting a non-terminating decimal to a fraction is to identify the repeating pattern of digits. Non-terminating decimals typically have a repeating pattern of digits that occurs indefinitely. Once the repeating pattern is identified, the conversion process can begin.

## Setting up the Equation

To convert a non-terminating decimal to a fraction, we introduce a variable, typically x, to represent the decimal. This variable is then equated to the non-terminating decimal. This step sets up the equation that will be used to solve for the fraction.

## Multiplying by a Power of 10

The next step involves multiplying both sides of the equation obtained in the previous step by a suitable power of 10. The power of 10 chosen should be sufficient to eliminate the decimal places in the repeating pattern. This operation effectively moves the repeating pattern to the left of the decimal point, making it easier to work with.

## Subtracting the Original Equation

Once the equation has been multiplied by the appropriate power of 10, the original equation is subtracted from the modified equation. This subtraction eliminates the repeating pattern, leaving an equation that can be simplified to obtain the fraction form.

## Simplifying the Equation

The resulting equation from the subtraction step is typically simplified to obtain the fraction form. This may involve algebraic operations such as combining like terms, factoring, and canceling common factors. The goal is to simplify the equation until it is in the form of a fraction, with the numerator and denominator being integers.

## Example: Converting 0.77777… to a Fraction

To illustrate the process, let’s convert the non-terminating decimal 0.77777… to a fraction.

**Step 1: Identifying the Repeating Pattern**

The repeating pattern in this decimal is 7.

**Step 2: Setting up the Equation**

Let x = 0.7777…

**Step 3: Multiplying by a Power of 10**

Multiplying both sides of the equation by 10 gives: 10x = 7.7777…

**Step 4: Subtracting the Original Equation**

Subtracting the original equation from the modified equation: 10x – x = 7.7777… – 0.7777…

**Step 5: Simplifying the Equation**

Simplifying the equation: 9x = 7

**Step 6: Solving for x**

Solving for x: x = 7/9

**Therefore, 0.77777… can be expressed as the fraction 7/9.**

### Conclusion

Converting non-terminating decimals to fractions involves identifying the repeating pattern, setting up an equation, multiplying by a power of 10, subtracting the original equation, and simplifying the resulting equation. This process allows us to express non-terminating decimals as fractions, which are often more convenient for calculations and mathematical operations.

**References:**

- “Non-terminating Decimal to Fraction – YouTube.” YouTube, YouTube, 2024, www.youtube.com/watch?v=ONJo8jNWaCM.
- “Repeating Decimal to Fraction.” Cuemath, 2023, www.cuemath.com/numbers/repeating-decimal-to-fraction/.
- “Repeating Decimal to Fraction (Conversion Method with Solved Examples).” Byju’s, 2023, byjus.com/maths/repeating-decimal-to-fraction/.

## FAQs

**1. What is a non-terminating decimal?**

Answer: A non-terminating decimal is a decimal that does not have an end digit and has an infinite number of terms. It is also known as an infinite decimal.

**2. Can all non-terminating decimals be converted to fractions?**

Answer: Yes, all non-terminating decimals can be converted to fractions. However, some non-terminating decimals may have repeating patterns, while others may not.

**3. How do you convert a non-terminating decimal to a fraction?**

Answer: To convert a non-terminating decimal to a fraction, you can follow these steps:

- Identify the repeating pattern (if there is one).
- Set up an equation using a variable to represent the decimal.
- Multiply both sides of the equation by a suitable power of 10 to eliminate the decimal places in the repeating pattern.
- Subtract the original equation from the modified equation to eliminate the repeating pattern.
- Simplify the resulting equation to obtain the fraction form.

**4. What if the non-terminating decimal does not have a repeating pattern?**

Answer: If the non-terminating decimal does not have a repeating pattern, it is known as an irrational number. Irrational numbers cannot be expressed as a fraction of two integers.

**5. Can non-terminating decimals be expressed as fractions with repeating digits?**

Answer: Yes, non-terminating decimals with repeating patterns can be expressed as fractions with repeating digits. The repeating digits are represented using a bar over the repeating part of the fraction.

**6. Are all fractions with repeating digits non-terminating decimals?**

Answer: No, not all fractions with repeating digits are non-terminating decimals. Some fractions with repeating digits may terminate after a certain number of digits.

**7. What are some examples of non-terminating decimals that can be converted to fractions?**

Answer: Some examples of non-terminating decimals that can be converted to fractions include:

- 0.3333… = 1/3
- 0.6666… = 2/3
- 0.123456789… = 123456789/999999999

**8. Why is it useful to convert non-terminating decimals to fractions?**

Answer: Converting non-terminating decimals to fractions can be useful for various reasons, such as:

- Simplifying calculations and mathematical operations.
- Comparing and ordering numbers more easily.
- Expressing numbers in a more compact and readable form.