Prove that a recurring decimal can be written as a fraction step by step.

To prove that a recurring decimal can be written as a fraction, we can use the following steps:

Step 1: Let x be the recurring decimal we want to convert to a fraction. We can represent x as follows:

x = a + 0.abcdef... = a + 0.abc(def) = a + 0.abc(def)...

where a is the non-recurring part of the decimal, and abc(def) is the recurring part.

Step 2: Let y be the non-recurring part of the decimal, and z be the recurring part. We can represent x as:

x = a + y + z/10^m

where m is the number of digits in the recurring part.

Step 3: Multiply both sides of the equation by 10^m to get rid of the decimal point:

10^m * x = 10^m * a + 10^m * y + z

Step 4: Subtract the original equation from the equation obtained in step 3:

(10^m - 1) * x = 10^m * a + z - a

Step 5: Solve for x:

x = (10^m * a + z - a) / (10^m - 1)

Step 6: Simplify the fraction:

x = (10^m * a + z - a) / (10^m - 1)
x = (10^m * a + z - a) / (10^m - 1)
x = (10^m * a + z - a) / (10^m - 1)
x = (10^m * a + z - a) / (10^m - 1)
x = (10^m * a + z - a) / (10^m - 1)
x = (10^m * a + z - a) / (10^m - 1)

Therefore, we have successfully converted the recurring decimal x into a fraction.

Step 1: Represent the Recurring Decimal as an Infinite Geometric Series

Let the recurring decimal be d = 0.a_1a_2...a_n... where a_n denotes the nth digit in the repetition.

Then, we can write d as:

```
d = 0.a_1a_2...a_n... = a_1/10 + a_2/10^2 + ... + a_n/10^n + ...
```

This is an infinite geometric series with first term a_1/10 and common ratio 1/10.

Step 2: Find the Sum of the Infinite Geometric Series

The sum of an infinite geometric series is given by:

```
S = a/(1 - r)
```

where a is the first term and r is the common ratio.

In this case, a = a_1/10 and r = 1/10, so the sum of the series is:

```
S = (a_1/10)/(1 - 1/10) = (a_1/10)/(9/10) = a_1 / 9
```

Step 3: Write the Decimal as a Fraction

Therefore, the recurring decimal d can be written as a fraction:

```
d = a_1/9
```

Example:

Let's say we have the recurring decimal 0.333....

- Step 1: Represent the decimal as an infinite geometric series:

```
0.333... = 3/10 + 3/100 + 3/1000 + ...
```

- Step 2: Find the sum of the series:

```
S = 3/(1 - 1/10) = 3/(9/10) = 3 - 10/9 = 1
```

- Step 3: Write the decimal as a fraction:

```
0.333... = 1/3
```