Proof that 0.3333... = 0.3 recurring can be expressed as p/q

To prove that 0.3333... is equal to 0.3 recurring and can be expressed as the fraction p/q, where p and q are integers and q is not equal to 0, we need to follow a few steps:

Step 1: Let x = 0.3333...

We start by assuming that the repeating decimal 0.3333... can be represented as x.

Step 2: Multiply both sides of the equation by 10

To remove the repeating decimal, we can multiply both sides of the equation by 10. This will shift the decimal point to the right, allowing us to subtract the original equation from the multiplied equation to eliminate the repeating part.

10x = 3.3333...

Step 3: Subtract the original equation from the multiplied equation

By subtracting the original equation (x) from the multiplied equation (10x), we can eliminate the repeating decimal.

10x - x = 3.3333... - 0.3333...

This simplifies to:

9x = 3

Step 4: Solve for x

To solve for x, we divide both sides of the equation by 9.

9x/9 = 3/9

This simplifies to:

x = 1/3

Step 5: Rewrite x as 0.3 recurring

Since we assumed x = 0.3333..., we can conclude that 0.3333... is equal to 1/3.

Therefore, 0.3333... = 0.3 recurring = 1/3.

Conclusion

By following the steps above, we have shown that 0.3333... can be expressed as 0.3 recurring and as the fraction 1/3. This demonstrates that the decimal representation of 0.3333... is equivalent to the fraction 1/3.