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The real number system was not invented: it evolved.

People started off counting, using positive integers - the natural numbers. But they soon hit a problem when dealing with borrowing (or owing). So negative numbers were added, and the system became the set of integers.

This was still not up to sharing evenly and so rational numbers [fractions] were added to the number system.

But even that system was inadequate when it came to measuring the hypotenuse of a right angled triangle with sides of unit length. It was therefore necessary to add Irrational Numbers and we had the real number system.

Some people believe that the ancient Indians knew about the existence of irrational numbers but this is not well documented. There is some evidence that the Greeks, in the 5th Century BCE, were aware of irrational numbers.

Of course, the real number system is not good enough, as anyone who has come across the square root of a negative number will testify. So, the imaginary numbers were added to the system and so was born the Complex field.

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Q: Who invented real number system?

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