# Complex numbers

Categories: complex numbers imaginary numbers

The unit imaginary number, *i*, is defined as the square root of -1. It is called *imaginary* because the square of any real number is always positive or zero, so no real number can be the square root of a negative number.

A complex number is composed of two parts, a real part and an imaginary part. It takes the form *a + bi* where *a* and *b* ae real numbers and *i* is the unit imaginary number.

Many of the operators and functions we apply to real numbers can also be applied to complex numbers, of ten with interesting and useful results. For example:

- We can add, subtract, multiply and divide complex numbers.
- We can form powers and roots of complex numbers, even using complex exponents.
- We can create complex polynomials.
- Many analytic functions, such as the exponential function and the sine function, have complex number equivalents.
- We can perform calculus on many complex number functions.

## See also

- Imaginary and complex numbers
- Complex number arithmetic
- Argand diagrams
- Why does complex number multiplication cause rotation?
- Modulus-argument form of complex numbers
- De Moivre's theorem
- i to the power i
- Euler's formula - proof
- Complex powers and roots of complex numbers
- Semiprocal numbers - z to the power i
- Complex polynomials
- Complex number trigonometry functions

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