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Why real numbers are generally not real

The numbers defined in mathematics are in the increasing complexity order:

- The natural integers, from 0 to infinity, that we use to count

- The relative integers, that are the natural integers plus the negative integers, that we use for instance for accountancy "credit" or "debit" notions

- The rational numbers, that we use to share a quantity: they are the result of divisions

- The real numbers, that are all the possible finite or infinite combination of digits, with or without a decimal point: the possibiliy of an infinite combination is only for the digits after the decimal point

Thus, from a naive point of view, the real numbers are "all the numbers". They include the natural integers, the relative integers, the rational numbers, as well as the decimal numbers, that are the real numbers with a finite number of decimal digits.

But note that amazing fact: only the decimal numbers can be represented on a calculator or a computer. Moreover, we only understand "how much" a number is when we have its decimal representation, that is to say that, if the number of the decimals is infinite or simply very big, we only understand its decimal approximation up to a certain number of digits after the decimal point.

For instance, 1/3 may be known as 0.333333, or simply 0.33, depending on the accuracy we wish to assign to our calculations. And the determination of the decimals of pi is a well known challenge for mathematicians, even if scholars tell that it is 3.14 (and in fact 3.14 is a sufficient accuracy for many simple calculations).

So, let me see. The real numbers are all the numbers. They are the results of many calculations and as such, they are very useful. But for many of them, we may know only approximations in the real world.

That is, real numbers are not real!

Or more precisely, what mathematicians call real numbers are not the numbers we use in everyday lifeā€¦

Amazing, isn't it?

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