# Rational Irrational Numbers

I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a decimal, but it is predictable. This is observed in most if not all rational numbers, as far as I know.

So would it be possible for a number to be partially predictable?

Could a number have the exact same digits as pi, but instead have a periodic, predictable digit in it?

Is there a category of Rational irrational numbers (or irrational rational numbers)? For instance, could a number with digits like pi or the square root of 2 have rational parts?

• Yes, all rational numbers eventually have a repeating pattern when represented in decimal (or any integer base.) (Numbers that terminate, like $1/5$ can be thought of as having repeating $0$ digits.) – Thomas Andrews Apr 30 '15 at 12:41
• $\pi$ has predictable digits - you just compute $\pi$ to enough digits to predict it. The notion of what is "predictable" is a vague notion. – Thomas Andrews Apr 30 '15 at 12:42
• I'm not sure this makes any sense. What precisely does it mean to have the "exact same digits as $\pi$, but instead have a periodic, predictable digit in it"? What does it mean for an irrational number to have "rational parts"? – Travis Willse Apr 30 '15 at 12:44
• I think he's asking if there is a class of irrational numbers where particular digits in the decimal expansion appears predictably and patterned. Like for example, $$0.19528872964211126372...$$ where a 2 appears predictably every fourth decimal term... The first three decimal terms are random (or if not random, just unpatterned) – Eleven-Eleven Apr 30 '15 at 12:50
• @tennispro1213, it would be helpful, though, if you chimed in here and helped clarify what you are looking for. – Eleven-Eleven Apr 30 '15 at 12:54

## 2 Answers

It is possible for an irrational number to have a predictable pattern; consider $0.1101001000100001...$. It is also possible to have an irrational number that is another irrational number away from a rational; i.e. $x-y = r$, where $x,y$ irrational and $r$ rational; in fact the equivalence classes of such numbers are dense in the reals. So you can subtract some irrational number from $\pi$ and get a number with a repeating pattern... in fact any pattern that you want.

• Does this also mean that one could work backwards, too? Like if one starts with a number with a repeating pattern, would one be able to add and subtract numbers from it till pi or any other number is "created"? – tkhanna42 Apr 30 '15 at 15:02
• Yes, you can construct pi using, for example, a continued fraction. Not through addition and subtraction only, however; division is necessary. In any case, you need to clarify what you mean by "pattern". As I mentioned, we can concoct any number of patterns. That something is a pattern doesn't make it profound or interesting. – Emily Apr 30 '15 at 15:11
• By pattern I mean a recognizable recurrence of digits or a relationship between them. In pi, the digits start off as 3,1,4,1,5,9....there isn't much that we can predict using a simple expression. But for a number with the digits 3,1,2,4,1,2,5,9,2,.....We see that there is a simple pattern of repeating twos. I understand that everything may seem to have a pattern if we analyze it macroscopically, but I would like to first verify that it holds true for basic patterns and then extend it to more complex patterns. – tkhanna42 Apr 30 '15 at 15:21

A rational number is one that is the ratio of two integers. Rational numbers will have a predictable pattern in their decimal representations but that is not the definition of a rational number. As others have said, this pattern is that, eventually,a block of digits will start to repeat. In a very simple case, e.g. $1/3$ this will just be the single digit $3$ repeating. Slightly more complex is $1/7000 = 0.000 142857 142857 ...$ which has three $0$s followed by repeating blocks of $142857$. As others have said, this includes the terminating case by considering the repeating block to be $0$.

All sorts of other predictable and interesting patterns are possible but they will not be rational numbers. Read about Liouville numbers. These have a very predictable pattern to their digits but they are not rational, in fact they were the first known transcendental numbers.

So, no - you cannot say that a rational is one whose digits have a pattern. It is only rational if the digits follow the quite precise pattern of ending with a repeating block of digits.