What if you think a number is irrational, when it actually repeats or terminates past what you can see? Can this happen?
+23254 A number written in decimal form can terminate (or repeat) long after the last digit that you know.
So, knowing a few digits (even a hundred digits!) may not be enough.
If the decimal represents a root, such as the square root or cube root or fourth root (etc), then you can tell if the number is rational or irrational depending upon whether the decimal represents a perfect square, cube, fourth (etc.).
For instance, if the decimal is 0.10251663... and if it turns out to be 0.1025166375025 000000... it is a perfect square and has a square root of 0.1012505.
If the number is like this: 0.10110111011110111110... it has a pattern that doesn't repeat, so it's an irrational number.
If you look at some rational number such as: 7/125003 in decimal form, which starts thus:
0.000055998656032255225874579010103757509819764325656184251...
From just looking at this string, it is impossible to tell whether it is rational or irrational..... In fact, this decimal fraction goes on for 62,501 digits before it starts repeating again!!!!!. The test of a "perfect square" is not going to help you determine fractions such as this one, whether it is rational or irrational. However, sometimes there is a simple test to determine the rationality or irrationality of a decimal. It is called "continued fraction" test that you can do, even on hand-held calculator, to determine what kind of a number it is. It goes like this: You take the reciprocal of the above decimal fraction and you immediately see a pattern of convergence, thus:17,857.571428571428571428571428571.......... Then you subtract the integer part and keep the fractional part thus:0.57142857142857142857142857142857.....Then you take the reciprocal of this last decimal fraction and you get this:1.75. Subtract 1 and keep .75. Take its reciprocal and you get:1.3333333333......Subtract 1 again and take the reciprocal of .3333333333333333.... and you get 1. And that is it. It tells you that your decimal fraction is actually a rational number. You can list the continued fraction thus: {0., 17,857, 1, 1, 3}.
