{"id":2206,"date":"2020-01-04T08:06:12","date_gmt":"2020-01-04T08:06:12","guid":{"rendered":"https:\/\/www.askpython.com\/?p=2206"},"modified":"2022-08-06T13:09:35","modified_gmt":"2022-08-06T13:09:35","slug":"python-complex-numbers","status":"publish","type":"post","link":"https:\/\/www.askpython.com\/python\/python-complex-numbers","title":{"rendered":"Python Complex Numbers"},"content":{"rendered":"\n

A Complex Number is any number of the form a + bj<\/code>, where a<\/code> and b<\/code> are real numbers, and j*j<\/code> = -1.<\/p>\n\n\n\n

In Python, there are multiple ways to create such a Complex Number.<\/p>\n\n\n\n

\n\n\n\n

Create a Complex Number in Python<\/h2>\n\n\n\n
  • We can directly use the syntax a + bj<\/code> to create a Complex Number.<\/li><\/ul>\n\n\n
    \n>>> a = 4 + 3j\n>>> print(a)\n(4+3j)\n>>> print(type(a))\n<class 'complex'>\n<\/pre><\/div>\n\n\n
    • We can also use the complex<\/code> Class to create a complex number<\/li><\/ul>\n\n\n
      \n>>> a = complex(4, 3)\n>>> print(type(a))\n<class 'complex'>\n>>> print(a)\n(4+3j)\n<\/pre><\/div>\n\n\n
      Real and Imaginary Parts in Complex Number<\/h3>\n\n\n\n
      Every complex number (a + bj<\/code>) has a real part (a<\/code>), and an imaginary part (b<\/code>).<\/p>\n\n\n\n
      To get the real part, use number.real<\/code>, and to get the imaginary part, use number.imag<\/code>.<\/p>\n\n\n
      \n>>> a\n(4+3j)\n>>> a.real\n4.0\n>>> a.imag\n3.0\n<\/pre><\/div>\n\n\n
      Conjugate of a Complex Number<\/h3>\n\n\n\n
      The conjugate of a complex number a + bj<\/code> is defined as a - bj<\/code>. We can also use number.conjugate()<\/code>method to get the conjugate.<\/p>\n\n\n
      \n>>> a\n(4 + 3j)\n>>> a.conjugate()\n(4-3j)\n<\/pre><\/div>\n\n\n
      Arithmetic Operations on Complex Numbers<\/h2>\n\n\n\n
      Similar to real numbers, Complex Numbers also can be added, subtracted, multiplied and divided. Let us look at how we could do this in Python.<\/p>\n\n\n
      \na = 1 + 2j\nb = 2 + 4j\nprint('Addition =', a + b)\nprint('Subtraction =', a - b)\nprint('Multiplication =', a * b)\nprint('Division =', a \/ b)\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \nAddition = (3+6j)\nSubtraction = (-1-2j)\nMultiplication = (-6+8j)\nDivision = (2+0j)\n<\/pre><\/div>\n\n\n
      NOTE<\/strong>: Unlike real numbers, we cannot compare two complex numbers. We can only compare their real and imaginary parts individually, since they are real numbers. The below snippet proves this.<\/p>\n\n\n
      \n>>> a\n(4+3j)\n>>> b\n(4+6j)\n>>> a < b\nTraceback (most recent call last):\n  File "<stdin>", line 1, in <module>\nTypeError: '<' not supported between instances of 'complex' and 'complex'\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Phase (Argument) of a Complex Number<\/h2>\n\n\n\n
      We can represent a complex number as a vector consisting of two components in a plane consisting of the real<\/code> and imaginary<\/code> axes. Therefore, the two components of the vector are it’s real part and it’s imaginary part.<\/p>\n\n\n\n
      \"Complex
      Complex Number Vector<\/figcaption><\/figure><\/div>\n\n\n\n
      The angle between the vector and the real axis is defined as the argument<\/code> or phase<\/code> of a Complex Number.<\/p>\n\n\n\n
      It is formally defined as :<\/p>\n\n\n\n
      phase(number) = arctan(imaginary_part \/ real_part)<\/strong><\/em><\/p>\n\n\n\n
      where the arctan function is the tan inverse mathematical function.<\/p>\n\n\n\n
      In Python, we can get the phase of a Complex Number using the cmath<\/code> module for complex numbers. We can also use the math.arctan<\/code> function and get the phase from it’s mathematical definition.<\/p>\n\n\n
      \nimport cmath\nimport math\n\nnum = 4 + 3j\n\n# Using cmath module\np = cmath.phase(num)\nprint('cmath Module:', p)\n\n# Using math module\np = math.atan(num.imag\/num.real)\nprint('Math Module:', p)\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \ncmath Module: 0.6435011087932844\nMath Module: 0.6435011087932844\n<\/pre><\/div>\n\n\n
      Note that this function returns the phase angle in radians<\/code>, so if we need to convert to degrees<\/code>, we can use another library like numpy<\/code>.<\/p>\n\n\n
      \nimport cmath\nimport numpy as np\n\nnum = 4 + 3j\n\n# Using cmath module\np = cmath.phase(num)\nprint('cmath Module in Radians:', p)\nprint('Phase in Degrees:', np.degrees(p))\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \ncmath Module in Radians: 0.6435011087932844\nPhase in Degrees: 36.86989764584402\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Rectangular and Polar Coordinates<\/h2>\n\n\n\n
      A Complex Number can be written in Rectangular Coordinate or Polar Coordinate formats using the cmath.rect()<\/code> and cmath.polar()<\/code> functions.<\/p>\n\n\n
      \n>>> import cmath\n>>> a = 3 + 4j\n>>> polar_coordinates = cmath.polar(a)\n>>> print(polar_coordinates)\n(5.0, 0.9272952180016122)\n\n>>> modulus = abs(a)\n>>> phase = cmath.phase(a)\n>>> rect_coordinates = cmath.rect(modulus, phase)\n>>> print(rect_coordinates)\n(3.0000000000000004+3.9999999999999996j)\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Constants in the cmath Module<\/h2>\n\n\n\n
      There are special constants in the cmath module. Some of them are listed below.<\/p>\n\n\n
      \nprint('\u03c0 =', cmath.pi)\nprint('e =', cmath.e)\nprint('tau =', cmath.tau)\nprint('Positive infinity =', cmath.inf)\nprint('Positive Complex infinity =', cmath.infj)\nprint('NaN =', cmath.nan)\nprint('NaN Complex =', cmath.nanj)\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \n\u03c0 = 3.141592653589793\ne = 2.718281828459045\ntau = 6.283185307179586\nPositive infinity = inf\nPositive Complex infinity = infj\nNaN = nan\nNaN Complex = nanj\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Trigonometric Functions<\/h2>\n\n\n\n
      Trigonometric functions for a complex number are also available in the cmath<\/code> module.<\/p>\n\n\n
      \nimport cmath\n\na = 3 + 4j\n\nprint('Sine:', cmath.sin(a))\nprint('Cosine:', cmath.cos(a))\nprint('Tangent:', cmath.tan(a))\n\nprint('ArcSin:', cmath.asin(a))\nprint('ArcCosine:', cmath.acos(a))\nprint('ArcTan:', cmath.atan(a))\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \nSine: (3.853738037919377-27.016813258003936j)\nCosine: (-27.034945603074224-3.8511533348117775j)\nTangent: (-0.0001873462046294784+0.999355987381473j)\nArcSin: (0.6339838656391766+2.305509031243477j)\nArcCosine: (0.9368124611557198-2.305509031243477j)\nArcTan: (1.4483069952314644+0.15899719167999918j)\n<\/pre><\/div>\n\n\n
      Hyperbolic Functions<\/h2>\n\n\n\n
      Similar to Trigonometric functions, Hyperbolic Functions for a complex number are also available in the cmath<\/code> module.<\/p>\n\n\n
      \nimport cmath\n\na = 3 + 4j\n\nprint('Hyperbolic Sine:', cmath.sinh(a))\nprint('Hyperbolic Cosine:', cmath.cosh(a))\nprint('Hyperbolic Tangent:', cmath.tanh(a))\n\nprint('Inverse Hyperbolic Sine:', cmath.asinh(a))\nprint('Inverse Hyperbolic Cosine:', cmath.acosh(a))\nprint('Inverse Hyperbolic Tangent:', cmath.atanh(a))\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \nHyperbolic Sine: (-6.5481200409110025-7.61923172032141j)\nHyperbolic Cosine: (-6.580663040551157-7.581552742746545j)\nHyperbolic Tangent: (1.000709536067233+0.00490825806749606j)\nInverse Hyperbolic Sine: (2.2999140408792695+0.9176168533514787j)\nInverse Hyperbolic Cosine: (2.305509031243477+0.9368124611557198j)\nInverse Hyperbolic Tangent: (0.11750090731143388+1.4099210495965755j)\n<\/pre><\/div>\n\n\n
      Exponential and Logarithmic Functions<\/h2>\n\n\n
      \nimport cmath\na = 3 + 4j\nprint('e^c =', cmath.exp(a))\nprint('log2(c) =', cmath.log(a, 2))\nprint('log10(c) =', cmath.log10(a))\nprint('sqrt(c) =', cmath.sqrt(a))\n<\/pre><\/div>\n\n\n
      Output<\/strong>:<\/p>\n\n\n
      \ne^c = (-13.128783081462158-15.200784463067954j)\nlog2(c) = (2.321928094887362+1.3378042124509761j)\nlog10(c) = (0.6989700043360187+0.4027191962733731j)\nsqrt(c) = (2+1j)\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Miscellaneous Functions<\/h2>\n\n\n\n
      There are some miscellaneous functions to check if a complex number is finite, infinite or nan<\/code>. There is also a function to check if two complex numbers are close.<\/p>\n\n\n
      \n>>> print(cmath.isfinite(2 + 2j))\nTrue\n\n>>> print(cmath.isfinite(cmath.inf + 2j))\nFalse\n\n>>> print(cmath.isinf(2 + 2j))\nFalse\n\n>>> print(cmath.isinf(cmath.inf + 2j))\nTrue\n\n>>> print(cmath.isinf(cmath.nan + 2j))\nFalse\n\n>>> print(cmath.isnan(2 + 2j))\nFalse\n\n>>> print(cmath.isnan(cmath.inf + 2j))\nFalse\n\n>>> print(cmath.isnan(cmath.nan + 2j))\nTrue\n\n>>> print(cmath.isclose(2+2j, 2.01+1.9j, rel_tol=0.05))\nTrue\n\n>>> print(cmath.isclose(2+2j, 2.01+1.9j, abs_tol=0.005))\nFalse\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      Conclusion<\/h2>\n\n\n\n
      We learned about the Complex Numbers module, and various functions associated with the cmath<\/code> module.<\/p>\n\n\n\n
      References<\/h2>\n\n\n\n