Euclid did not have real numbers. He could say that the length of one line segment goes into the length of another a certain number of times, or that the ratio of lengths of two segments was the same as that of two specified numbers ("numbers" were $2,3,4,5,...$ and in particular $1$ was not a "number") or that the ratio of the length of the diagonal of a square to the side was not the ratio of any two numbers.

There is a much celebrated essay by James Wilkinson titled _The Perfidious Polynomial_ that I have not brought myself to read beyond the first couple of pages because of the profound stupidity of what Wilkinson claims is the history of the development of number systems. He states with a straight face that negative numbers were introduced for the purpose of solving polynomial equations that had no positive solutions, that rational numbers were introduced after that for the purpose of solving yet more equations, that irrational numbers were then introduced for solving yet more, and finally complex numbers for the purpose of solving quadratic equations that lacked real solutions. Did Wilkinson never hear that Euclid proved the statement above about the ratio of diagonal to side of a square, but Euclid had never heard of negative numbers, and did not even have anything that we would consider a system of numbers that includes positive rational and irrational numbers? That imaginary numbers were used to find <b>real</b> solutions of <b>third</b>-degree polynomials? That negative numbers were introduced by accountants for their purposes? And that all of that obviously makes more sense and is more elegant and intellectually satisfying than the childish story he tells? In order to do what Wilkinson says was done, one would need certain definitions that were introduced only much later.