The Number That Turned Sideways
5 days ago
- #Euler-formula
- #mathematical-history
- #complex-numbers
- Complex numbers are pairs of real numbers with a real part (a) and an imaginary part (bi), where i is the square root of -1.
- The magnitude of a complex number is given by |z| = √(a² + b²).
- Complex numbers can be represented in polar form as z = |z|(cosθ + i sinθ), where θ is the angle.
- Multiplying a complex number by a real scalar stretches or shrinks it, while multiplying by i rotates it by π/2.
- Rafael Bombelli in 1572 defined arithmetic rules for complex numbers, treating them as valid entities despite initial skepticism.
- Leonhard Euler in 1748 connected complex numbers to exponential functions, showing e^(ix) = cos(x) + i sin(x).
- Between 1799 and 1831, Caspar Wessel, Jean-Robert Argand, and Carl Friedrich Gauss gave geometric meaning to complex numbers, placing them on a plane.
- Bombelli's rules, Euler's formula, and the geometric interpretation together explain why imaginary exponents rotate numbers.
- Euler's formula also revealed the roots of unity, showing how complex numbers divide the circle into equal parts.
- The development of complex numbers spanned three centuries, evolving from arithmetic rules to analytic connections and finally geometric meaning.