ΒΙΒΛΙΟ: Complex Numbers from A to ... Z (pdf)

Solving algebraic equations has been historically one of the favorite topics of mathematicians. While linear equations are always solvable in real numbers, not all quadratic equations have this property. The simplest such equation is $x^2 + 1 = 0$. 

 

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Until the 18th century, mathematicians avoided quadratic equations that were not solvable over R. Leonhard Euler broke the ice introducing the “number”  $\sqrt{-1}$  in his famous book Elements of Algebra as “ ... neither nothing, nor greater than nothing, nor less than nothing ... ” and observed “ ... notwithstanding this, these numbers present themselves to the mind; they exist in our imagination and we still have a sufficient idea of them; ... nothing prevents us from making use of these imaginary numbers, and employing them in calculation”. 

Euler denoted the number $\sqrt{-1}$ by $i$ and called it the imaginary unit.