In prehistoric times. Quite possibly be a primitive humanoid trying to figure out if his "tribe" had more members than the enemy.
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They were known to the ancient civilisations. Irrational numbers are supposed to have been discovered by Hippassus of Metapontum, a member of the Pythagorean school in ancient Greece (5th Century BC).
Hippasus, However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned.
This is not known.It is believed that the ancient Egyptians knew of them, but it is known that the ancient Greeks did.
It has not yet been discovered because numbers are infinite
They were discovered when Cardano solved the third degree equation. In the formulas that arose to solve the third degree equation, Cardano needed to take the square root of negative numbers and add them up in a certain way. The strange thing that happened was that the formulas used these complex numbers, even if the solutions to the equation where all real. This baffled the mathematicians of the time, because how could these strange numbers turn out to be "real"? Later this was considered totally correct, when the field of complex numbers was better undestood.