page_title: Complex number
section_title: Multiplication and division in polar form
hierarchical_section_title: Complex number / Relations and operations / Multiplication and division in polar form
context_page_description: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i² = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted using the symbol. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
  Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation
  has no real solution, since the square of a real number cannot be negative. Complex numbers, however, provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i taken to satisfy the relation i² = −1, so that solutions to equations like the preceding one can be found. In this case, the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i² = −1:
context_section_description: Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z₁ = r₁(cos φ₁ + i sin φ₁) and z₂ = r₂(cos φ₂ + i sin φ₂), because of the trigonometric identities
  we may derive
  In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i² = −1. The picture at the right illustrates the multiplication of
  Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
  holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π.
  Similarly, division is given by
caption: Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.