Probably all mathematicians, from very ancient times.
It has been known from very ancient times that the length of the diagonal of a unit square is not a rational number. There were no specific mathematicians who "discovered" real numbers. Furthermore, all mathematicians of any significance, contributed to our understanding of real numbers.
There are mathematical concepts that mathematicians call "imaginary numbers" these are a multiple of the square root of minus 1. Infinity is not a real number either.
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There is no real number whose square root can be negative so there is no real solution. So mathematicians invented the imaginary number i with the property that i*i = -1 i is fundamental to complex numbers.
In all likelihood, all the numbers that you will come across during school will belong to the Real number system.In the UK , it is only when you study "Further Pure" mathematics at A-levels, or at university that you will study complex numbers which belong to a set that is the next step beyond Real numbers.
In real life one can do very well with nothing but rational numbers. Real numbers that are not rational can be approximated with whatever degree of accuracy is needed, which is exactly what computers and calculators do . Real numbers are essential to mathematical theory but unless you are a mathematician or need to understand higher mathematics in a rigorous way, you can do very well with only a vague , intuitive understanding of real numbers. In fact, it was not until 1870 ,or so, that mathematicians devised a satisfactory definition of real numbers.
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier as Virahankanumbers in Indian mathematics.
Rene Descartes came up with the word imaginary in 1637 to describe them. It was a derogatory term. He (and many other mathematicians of that age) did not like imaginary numbers. Many people didn't believe in them, because they were not real.
In many countries, you will only come across real numbers up to the age of around 16. If you continue to study mathematics beyond that you will find that the number system extends beyond the real numbers: to imaginary and complex numbers, and further still to quaternions.
Mathematics is beautiful in itself. Back in the 1700s and later, mathematicians studied "imaginary" numbers (numbers that involve a factor of the square root of -1) knowing that they didn't describe anything "real", the way "real numbers" do. But when beauty can be melded to practicality, things get REALLY interesting. It turns out that you can use imaginary numbers and "complex numbers" (which have a "real" component and an "imaginary" component) to describe the way radiation and electromagnetic fields behave.
They both considered "identity elements". 0 is actually the identity element under addition for the real numbers, since if a is any real number, a + 0 = 0 + a = a. Mathematicians refers to 0 as the additive identity (or better said, the reflexive identity of addition). 1 is a separate and special entity called 'Unity' or 'Identity element'. 1 is actually the identity element under multiplication for the real numbers, since a x 1 = 1 x a = a. Mathematicians refers to 1 as the multiplicative identity (or better said, the reflex identity of multiplication).