# Complex Numbers #

Math.js supports the creation, manipulation, and calculations with complex numbers. Support of complex numbers is powered by the library complex.js.

In mathematics, a complex number is an expression of the form `a + bi`, where `a` and `b` are real numbers and `i` represents the imaginary number defined as `i^2 = -1`. (In other words, `i` is the square root of `-1`.) The real number `a` is called the real part of the complex number, and the real number `b` is the imaginary part. For example, `3 + 2i` is a complex number, having real part `3` and imaginary part `2`. Complex numbers are often used in applied mathematics, control theory, signal analysis, fluid dynamics and other fields.

## Usage #

A complex number is created using the function `math.complex`. This function accepts:

• two numbers representing the real and imaginary part of the value,
• a single string containing a complex value in the form `a + bi` where `a` and `b` respectively represent the real and imaginary part of the complex number.
• an object with either properties `re` and `im` for the real and imaginary part of the value, or two properties `r` and `phi` containing the polar coordinates of a complex value. The function returns a `Complex` object.

Syntax:

``````math.complex(re: number) : Complex
math.complex(re: number, im: number) : Complex
math.complex(complex: Complex) : Complex
math.complex({re: Number, im: Number}) : Complex
math.complex({r: number, phi: number}) : Complex
math.complex({abs: number, arg: number}) : Complex
math.complex(str: string) : Complex
``````

Examples:

``````const a = math.complex(2, 3)     // Complex 2 + 3i
a.re                             // Number 2
a.im                             // Number 3

const b = math.complex('4 - 2i') // Complex 4 - 2i
b.re = 5                         // Number 5
b                                // Complex 5 - 2i
``````

## Calculations #

Most functions of math.js support complex numbers. Complex and real numbers can be used together.

``````const a = math.complex(2, 3)     // Complex 2 + 3i
const b = math.complex('4 - 2i') // Complex 4 - 2i

math.re(a)                       // Number 2
math.im(a)                       // Number 3
math.conj(a)                     // Complex 2 - 3i

math.add(a, b)                   // Complex 6 + i
math.multiply(a, 2)              // Complex 4 + 6i
math.sqrt(-4)                    // Complex 2i
``````

## API #

A `Complex` object created by `math.complex` contains the following properties and functions:

### complex.re #

A number containing the real part of the complex number. Can be read and replaced.

### complex.im #

A number containing the imaginary part of the complex number. Can be read and replaced.

### complex.clone() #

Create a clone of the complex number.

### complex.equals(other) #

Test whether a complex number equals another complex value.

Two complex numbers are equal when both their real and imaginary parts are equal.

### complex.neg() #

Returns a complex number with a real part and an imaginary part equal in magnitude but opposite in sign to the current complex number.

### complex.conjugate() #

Returns a complex number with an equal real part and an imaginary part equal in magnitude but opposite in sign to the current complex number.

### complex.inverse() #

Returns a complex number that is inverse of the current complex number.

### complex.toVector() #

Get the vector representation of the current complex number. Returns an array of size 2.

### complex.toJSON() #

Returns a JSON representation of the complex number, with signature `{mathjs: 'Complex', re: number, im: number}`. Used when serializing a complex number, see Serialization.

### complex.toPolar() #

Get the polar coordinates of the complex number, returns an object with properties `r` and `phi`.

### complex.toString() #

Returns a string representation of the complex number, formatted as `a + bi` where `a` is the real part and `b` the imaginary part.

### complex.format([precision: number]) #

Get a string representation of the complex number, formatted as `a + bi` where `a` is the real part and `b` the imaginary part. If precision is defined, the units value will be rounded to the provided number of digits.

## Static methods #

The following static methods can be accessed using `math.Complex`

### Complex.fromJSON(json) #

Revive a complex number from a JSON object. Accepts An object `{mathjs: 'Complex', re: number, im: number}`, where the property `mathjs` is optional. Used when deserializing a complex number, see Serialization.

### Complex.fromPolar(r: number, phi: number) #

Create a complex number from polar coordinates.

### Complex.compare(a: Complex, b: Complex) #

Returns the comparision result of two complex number:

• Returns 1 when the real part of `a` is larger than the real part of `b`
• Returns -1 when the real part of `a` is smaller than the real part of `b`
• Returns 1 when the real parts are equal and the imaginary part of `a` is larger than the imaginary part of `b`
• Returns -1 when the real parts are equal and the imaginary part of `a` is smaller than the imaginary part of `b`
• Returns 0 when both real and imaginary parts are equal.

Example:

``````const a = math.complex(2, 3)   // Complex 2 + 3i
const b = math.complex(2, 1)   // Complex 2 + 1i
math.Complex.compare(a,b) // returns 1

//create from json
const c = math.Complex.fromJSON({mathjs: 'Complex', re: 4, im: 3})  // Complex 4 + 3i
``````