For the purpose of keeping track of the growing number of sheep herded by man, he started to count. This simple act was the beginning of a remarkable journey, which has made every known advancement of the modern age possible. Just think where would mankind be if he did not know what numbers are. Numbers are the just that, numbers. They cannot be invented, created or modified. They have an independent existence of their own as if it does not matter whether there is some intelligent creature capable of using them; numbers will still be there even if there were no universe simply because one is one, two is two and so on.
Whenever one thinks of a number, it’s the set of positive integers, i.e. whole numbers that comes to mind like 1,2,3,4… Not only is there an infinity of such numbers but there is also an infinity of numbers between any two such numbers. For example, between 1 and 2, there are decimal numbers like 1.23, 1.57, 1.7986524 and so on. Just these make another infinity. But the point is that both these kinds of numbers are countable although there are an infinite number of them. This means that there is order among them and one can go through all of them if one is immortal! In contrast, there are the irrational numbers, which are also infinite but are also uncountable. That means there is no way one can go through all of them because they don’t make any sense at all and just keep going on forever. Some of the most common examples are:
p = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679.... (whoa! That’s as far as I have memorized it up to 100 decimal places.)
e = 2.71828182845904523536028747135266…
Ö2 = 1.4142135623730950488016887242097…
As you can see, there is no way of predicting what the next decimal place is going to be. There is just no pattern unlike the rational number 1/3 = 0.33333333333333333333333333333333… in which it is obvious that the next decimal place is always 3 although this number also goes on forever. Another example is 1/7 = 0.142857 142857 142857 142857 142857… in which the sequence is just as obvious as above though much more interesting. The point is that irrational numbers like p are uncountable while rational numbers are countable.
There is yet another category of numbers, which are neither rational nor irrational, lie somewhere between the rational and irrational and form an infinite set of their own. They are called the transcendental numbers.