How do you determine whether this decimal expansion is a rational or irrational number: 0.7039927949564695286700690483... Thank you for help.
The ... means it goes on forever, but it doesn't have a pattern, and therefore cannot be written as a fraction. It is irrational
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.