Determine if 0.292992999299992999992 is a rational or irrational and give a reason for your answer.
In order to determine if the number 0.292992999299992999992 is rational or irrational, we need to examine its decimal representation.
Let's break down the number into parts:
0.292992999299992999992 = 0.2929 + 0.000099299 + 0.000000099299 + ...
We can see a pattern emerging with the digits. The digit sequence "299" repeats indefinitely after the initial "29". Therefore, we can express this number as a fraction by assigning a variable to the repeating part:
x = 0.299299299...
Next, we multiply both sides of the equation by 1000 to shift the decimal point three places:
1000x = 299.299299...
Now, we subtract the original equation from this new equation to eliminate the repeating part:
1000x - x = 299.299299... - 0.299299299...
This simplifies to:
999x = 299
Finally, dividing both sides of the equation by 999:
x = 299/999
Since we were able to express the decimal number as a fraction (299/999), we can conclude that 0.292992999299992999992 is rational.