It seems confusing that there are many, many more of irrational numbers as compared to rationals. There are, after all, an infinite number of rational numbers and an infinite number of irrational numbers. Without getting into the discussion that one of these infinities is infinitely greater than the other infinity, we can get an idea as to how rare are the rational numbers in a space of real numbers.
Consider all numbers, rational and irrational, to be made up of an infinite number of digits. For example,
3 would be 3.000…
5.0125 would be 5.0125000…
23.56565656… will remain 23.56565656…
1.41421356237… will remain 1.41421356237…
…and so on . So we need an infinite number of digits for every number. Also recall that, by definition, a rational number either has a terminating decimal expansion (like 37.0183) or an infinite but recurring expansion (like 2.141141141141…)
Now try creating a number by randomly generating each of the infinite digits of that number. Chances are (and overwhelmingly so), it’s irrational. Why so? In order to create a rational number, one way is to have a terminating decimal expansion; this means that you must have an infinite number of zeroes at the end, and probability of randomly getting infinite zeroes one after the other is zero. The only other way of getting a rational number is getting an infinite recurring pattern, but that too is very unlikely if all the numbers are being randomly generated.
Therefore, the probability of getting a rational number is almost zero. In other words, almost any number generated this way, will be an irrational number.
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