A New Coordinate
System for Complex Numbers
Python supports complex numbers.
A complete complex number can be written as complex(3,4) or 3 + 4j.
A number with “j” appended ⁖ 4j is the same as complex(0,4).
# -*- coding: utf-8 -*- # python # a complex number cc = complex(3, 4) # alternative notation cc2 = 3 + 4j # same as complex(3, 4) # complex numbers are printed with a parenthesis print(cc) # (3+4j) print(cc2) # (3+4j) print(cc==cc2) # True print(cc.real) # 3.0 print(cc.imag) # 4.0
http://docs.python.org/lib/typesnumeric.html
Basic arithmetic operations.
# -*- coding: utf-8 -*- # python # length of a complex number. That is, Sqrt[ i^2 + j^2] print( abs(complex(3, 4)) ) # 5.0 # complex number addition. (same as vector addition) print ( complex(2, 3) + complex(4, 5) ) # (6+8j) # multiplication of complex numbers print ( complex(1, 0) * complex(0, 1) ) # (1j) # scalar multiplication. That is, scale it. print ( complex(3, 4) * 2) # (6+8j) # adding a scalar adds to the real part print ( complex(3, 4) + 1) # (4+4j)
More more advanced operations, you'll need to use module “cmath”. Examples:
# -*- coding: utf-8 -*- # python import cmath z1 = complex(0, 1) # gets length print( abs(z1) ) # 1.0 # gets the angle. return in radians, between [-π, π] print( cmath.phase(z1) ) # 1.57079632679 # get polar coordinates. Returns this form (length, angle). print( cmath.polar(z1) ) # (1.0, 1.57079632679) # polar to rectangular. Input is (length, ‹angle in radians›). Returns a complex number z2 = cmath.rect(1, cmath.pi) print(z2) # (-1+1.22464679915e-16j) really is just -1 + 0j # constant π print(cmath.pi) # 3.14159265359 # constant e print(cmath.e) # 2.71828182846
http://docs.python.org/2/library/cmath.html
In Ruby, Complex number is represented by the object “Complex”. Example:
# -*- coding: utf-8 -*- # ruby # a complex number cc = Complex(3, 4) # when printed, it's shown as (‹a›+‹b›i) p cc # (3+4i) # input comlex number in polar form. The input is (length, ‹angle in radians›) p Complex.polar(1, Math::PI) # ⇒ (-1.0+1.2246467991473532e-16i) p cc.real # ⇒ 3.0 p cc.imag # ⇒ 4.0
Basic arithmetic operations.
# -*- coding: utf-8 -*- # ruby # length of a Complex number. That is, Sqrt[ i^2 + j^2] p Complex(3, 4).abs # ⇒ 5.0 # Complex number addition. (same as vector addition) p Complex(2, 3) + Complex(4, 5) # ⇒ (6+8i) # multiplication of complex numbers p Complex(1, 0) * Complex(0, 1) # ⇒ (0+1i) # scalar multiplication. That is, scale it. p Complex(3, 4) * 2 # ⇒ (6+8j) # adding a scalar adds to the real part p Complex(3, 4) + 1 # ⇒ (4+4i)
http://www.ruby-doc.org/core-1.9.3/Complex.html
More advanced operations. Examples:
# -*- coding: utf-8 -*- # ruby z1 = Complex(0, 1) # get length p z1.abs # ⇒ 1.0 # get the angle. return in radians p z1.angle # ⇒ 1.5707963267948966 # get polar coordinates. Returns a array [length, angle]. p z1.polar # ⇒ [1, 1.5707963267948966] # polar to rectangular. Input is (length, ‹angle in radians›). Returns a complex number p Complex.polar(1, Math::PI) # ⇒ (-1.0+1.2246467991473532e-16i) really is just (-1+0i) # constant π p Math::PI # ⇒ 3.141592653589793 # constant e p Math::E # ⇒ 2.718281828459045
http://www.ruby-doc.org/core-1.9.3/Math.html
Perl doesn't support complex numbers. But there are packages for it. One of them is “Math::Complex”.
Here's a excerpt from its documentation:
SYNOPSIS
use Math::Complex;
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
DESCRIPTION
This package lets you create and manipulate complex numbers. By default,
*Perl* limits itself to real numbers, but an extra "use" statement
brings full complex support, along with a full set of mathematical
functions typically associated with and/or extended to complex numbers.
If you wonder what complex numbers are, they were invented to be able to
solve the following equation:
x*x = -1
…
From this, we observe the general Perl programer's understanding of mathematics, and also this package's author's understanding of it. And, as well as the fanaticality and maturity from the writing style.