How do you determine whether this decimal expansion is a rational or irrational number: 0.7039927949564695286700690483... Thank you for help.
+357 Ummm...
The ... means it goes on forever, but it doesn't have a pattern, and therefore cannot be written as a fraction. It is irrational
Yay!
+1009 Actually...
it doesn't SEEM to have a pattern, but who knows if it does after the..., ...
If it doesn't, and confirmed, as you said, it isn't rational.
So lack of information tells us probably inrational.
You can determine it for certain if you take its "continued fraction", which is as follows:
[0; 1, 2, 2, 1, 1, 1, 4, 8, 1, 2], which terminates in 2. Therefore, it tells you, with absolute certainty, that it is a rational number. If you work out the fractions backwards, starting with:2, 1, 8.......etc. you will actually get the original rational fraction. In fact, I worked it out and as it happens the fraction is: 2345 / 3331. However, the decimal expansion doesn't repeat for 3330 digits!! Finding the "continued fraction" will always tell you whether a decimal expansion is rational or irrational.
