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Infinite in Theory, Finite in Practice

Infinite in Theory, Finite in Practice

When we start to count on our fingers and continue by heart, we only have a small part of all the possible "natural integers": these are all the possible numbers we may count with.

We know that we may count until very very big numbers, the numbers we may imagine, and even further… These very big numbers may be the size of a Galaxy, the size of the known universe, the number of cells in a human body, in all the human bodies, and so on.

The mathematicians construct the set of all "natural integers" with the Peano lemmae:

  • The set begins with 0
  • Any natural integer has a "follower"
  • 0 is not a follower

The numbers contructed that way have no end. This is because if there were a natural integer that is the biggest one, the follower of that natural integer would be bigger that that, so the biggest one is not the biggest one.

We say that the natural integers "go to infinity", and this is a very useful property even in practice, because this means that there are no limits to construct numbers.

However, this is only a virtual property, because we can only write down finite numbers. And our computers can only deal with finite numbers.

Even more, as the number of digits we may use to represent numbers is finite, the exponential notation for big numbers are only approximations. For instance, a number of the order or the billions, that is 10 to the power 9, is very often represented as an approximation up to the millions, with only 3 digits after the decimal point in the mantissa.

Of course, powerful computer may increase the precision of the representation of big numbers. But there is always a limit, because the machines, and our mind as well, are limited.

Thus the moto "It goes up to infinity" is very useful to tell that there is no limit, but  it is not completely true in practice.

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