page_title: Complex number
section_title: Polar complex plane
hierarchical_section_title: Complex number / Visualization / Polar complex plane
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:
caption: Argument φ and modulus r locate a point in the complex plane.