Answer: A) Yes, because the number can be written as the quotient of two integers
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Explanation:
Any repeating decimal can be expressed as a fraction of two integers.
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Optional section:
Here's a proof showing how the given number can be expressed as a fraction
Let x = 0.020020200202002...
The block of digits "02002" repeats forever.
Multiply both sides by 10^5 = 100,000 to move the decimal over 5 spots to the right.
The equation
x = 0.020020200202002...
will turn into
100000x = 2002.0200202002...
We have this system of equations
[tex]\begin{cases} x = 0.020020200202002\ldots\\100000x = 2002.0200202002\ldots\end{cases}[/tex]
Subtracting said equations leads to
100,000x - x = 99,999x on the left
2002.0200202002... - 0.0200202002... = 2002 on the left
Note how the decimal portions line up perfectly to cancel out when we subtract
After those two subtractions, we end up with the equation
99,999x = 2002
Divide both sides by 99,999 and we isolate x
x = 2002/(99,999)
Use your calculator to find that,
2002/(99,999) = 0.02002020020202...
which helps confirm our answer.
