\( x^2 + 1 = 0 \) has no real solutions.
The magic number \( i = \sqrt{-1} \)
Cubic equations \( x^3 = px + q \)
has a solution given by Del Ferro‘s formula
\( x \ = \sqrt[3]{ \frac{q}{2} + \sqrt{\frac {q^2}{4} - \frac {p^3}{27}} } + {\sqrt[3]{ \frac{q}{2} - \sqrt{\frac {q^2}{4} - \frac {p^3}{27}} }} \)
Rafael Bombelli considered the cubic equation given by \( x^3 = 15x + 4 \) and found
\( x = \sqrt[3]{ 2 + \sqrt{-121} } + \sqrt[3]{ 2 - \sqrt{-121} } = \sqrt[3] { 2 + 11i } + \sqrt[3]{ 2 - 11i } \)
\( x = 2 + \sqrt{-1} + 2 - \sqrt{-1} = 2 + i + 2 - i = 4 \)
In fact, \( x^3 - 15x - 4 = 0 \)
\( (x-4)(x^2+4x+1) = 0 \) has three real solutions.
\(
\begin{align*}
\begin{cases}
x_1 = 4 \\
x_2 = \sqrt{3} - 2 \\
x_3 = -2 - \sqrt{3}
\end{cases}
\end{align*}
\)
Bombelli demonstrated how real numbers are engendered from complex ones, He proved how a combination of imaginary roots could lead to a real number.
Bombelli’s discovery is considered the “Birth of Complex Analysis”.
The imaginary unit \( i \) appears in the Schrödinger equation
\(
\displaystyle \huge i\hbar {\frac {d}{dt}}\vert \Psi (t)\rangle ={\hat {H}}\vert \Psi (t)\rangle
\)
- Week1Lecture1: History of complex numbers by Petra Bonfert-Taylor
- In Imagining numbers (particularly the square root of \( \sqrt{-15} \) ), Barry Mazur mentioned Bombelli’s work many times.
- Bombelli, however, was interested in pursuing the opposite, more obstreperous case - that is, when \( \frac {q^2}{4} - \frac {p^3}{27} \) is a negative real number.(Chapter 7 Bombelli’s Puzzle, Imagining numbers)
- But the equation \( x^3 = 15x + 4 \) clearly did have a solution — indeed, \( x = 4 \) is one — it was just that applying the cubic formula required computing \( \sqrt{-121} \). … It was bombelli, also a mathematician and engineer, who decided to bite the bullet and just see what happened. (The Princeton Companion to Mathematics - Part II The Origins of Modern Mathematics, II.1. From Numbers to Number Systems, 6 Real, False, Imaginary)
- Galois theory offers deep insight into what is happening with casus irreducibilis.
- In fact, complex numbers are endemic in the formalism of quantum theory. (Chapter 2 The light dawns - 8 Probalility amplitude, Quantum Theory: A Very Short Introduction, John Polkinghorne)
- It would, no doubt, have come as a great surprise to all those who had voiced their suspicion of complex numbers to find that, according to the physics of the latter three quarters of the 20th century, the laws governing the behaviour of the world, at its tiniest scales, is fundamentally governed by the complex number system. (Chapter 4 Magical complex numbers, The Road to Reality, Roger Penrose)