There are always two real roots of a positive number. For example, if x2 is 25, x is ±5. However, if x2 is -25 real roots do not exist. The square root of any negative number is the square root of its absolute value multiplied by an imaginary unit j = √−1.
Hence √−25 = √25 𝑋−1 = √25 × √−1 = 5j
A Complex number consists of real and imaginary component. It is represented as x+yj. Both x and y are real numbers. Y multiplied by imaginary unit forms an imaginary part of complex number.
Examples: 3+2j, 10-5.5J, 9.55+2.3j, 5.11e-6+4j
Python has a built-in complex data type. A complex number object can be created by literal representation as follows −
>>> x = 2+3j >>> type(x)
The complex number object has two attributes real (returns the real component) and imag (returns imaginary component excluding imaginary unit j)
>>> x.real 2.0 >>> x.imag 3.0
It also has conjugate() method. Conjugate of a complex number has the same real component and imaginary component with the opposite sign. Hence conjugate of 2+3j is 2-3j
>>> x.conjugate() (2-3j)
Python also has built-in complex() function which returns a complex number object. The function takes two parameters, one each for real and imaginary component. They can be of any numeric type (int, float or complex)
>>> complex(9,5) (9+5j) >>> complex(-6, -2.5) (-6-2.5j) >>> complex(1.5j, 2.5j) (-2.5+1.5j)
If only one parameter is given it is treated as real component, imaginary component is assumed to be zero.
>>> complex(15) (15+0j)
The function can also take a string as argument provided it contains numeric representation.
>>> complex('51')
(51+0j)
>>> complex('1.5')
(1.5+0j)Addition and subtraction of complex numbers is similar to that of integers or floats. Real and imaginary parts are added / subtracted separately.
>>> a = 6+4j >>> b = 3+6j >>> a+b (9+10j) >>> a-b (3-2j)
For multiplication consider complex number as a binomial and multiply each term in the first number by each term in the second number.
a = 6+4j b = 3+2j c = a*b c = (6+4j)*(3+2j) c = (18+12j+12j+8*-1) c = 10+24j
In Python console the result verifies this −
>>> a = 6+4j >>> b = 3+2j >>> a*b (10+24j)
Division of complex numbers is done as follows −
Let the two numbers be
a = 2+4j
b = 1-2j
and we want to calculate a/b.
Obtain conjugate of denominator which is 1+2j
Multiply numerator and denominator by the conjugate of the denominator to get result of division
c = a/b c = (2+4j)*(1+2j)/(1-2j)(1+2j) c = (2+4j+4j+8*-1)/(1+2j-2j-4*-1) c = (-6+8j)/5 c = -1.2+1.6j
Following Python console session verifies above treatment.
>>> a = 2+4j >>> b = 1-2j >>> a/b (-1.2+1.6j)
The cmath module
Mathematical functions defined in math module of Python’s standard library process floating point numbers. For complex numbers, Python library contains cmath module.
Complex number z = x+yj is a Cartesian representation. It is internally represented in polar coordinates with its modulus r (as returned by built-in abs() function) and the phase angle Φ (pronounced as phi) which is counterclockwise angle in radians, between the x axis and line joining x with the origin. Following diagram illustrates polar representation of complex number −
Functions in cmath module allow conversion of Cartesian representation to polar representation and vice versa.
polar() − This function returns polar representation of a Cartesian notation of complex number. The return value is a tuple consisting of modulus and phase.
>>> import cmath >>> a = 2+4j >>> cmath.polar(a) (4.47213595499958, 1.1071487177940904)
Note that the modulus is returned by abs() function
>>> abs(a) 4.47213595499958
phase() − This function returns counterclockwise angle between x axis and segment joining a with origin. The angle is represented in radians and is between π and -π
>>> cmath.phase(a) 1.1071487177940904 z = x+yj Φ
rect() − This function returns Cartesian representation of complex number represented in polar form i.e. in modulus and phase
>>> cmath.rect(4.47213595499958, 1.1071487177940904) (2.0000000000000004+4j)
The cmath module contains alternatives for all mathematical functions defined in math module. There are trigonometric and logarithmic functions as explained below −
cmath.sin() − This function returns the sine trigonometric ratio for phase angle represented in radians.
>>> import cmath >>> a = 2+4j >>> p = cmath.phase(a) >>> cmath.sin(p) (0.8944271909999159+0j)
Similarly, functions for other ratios cos(), tan(), asin(), acos() and atan() are defined in cmath module.
cmath.exp() − Similar to math.exp(), this function returns ex where x is a complex number and e is 2.71828
>>> cmath.exp(a) (-1.1312043837568135+2.4717266720048188j)
cmath.log10() − This function calculates log value of complex number taking base as 10
>>> a = 1+2j >>> cmath.log10(a) (0.3494850021680094+0.480828578784234j)
cmath.sqrt() − This function returns square root of complex number.
>>> cmath.sqrt(a) (1.272019649514069+0.7861513777574233j)
In this article we learned important features of Python’s complex number data type and how arithmetic operations can be done on it. We also explored various functions defined in cmath module.