Most general-purpose programming languages have either no support or limited support for complex numbers. Your typical options are learning some specialized tool like MATLAB or finding a third-party library. Python is a rare exception because it comes with complex numbers built in.
Python lets you use complex numbers in arithmetic expressions and call functions on them just like you would with other numbers in Python. It leads to elegant syntax that reads almost like a math textbook.
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Notice how Python displays complex numbers by default. Their textual representation contains an enclosing pair of parentheses, a lowercase letter j, and no whitespace. Additionally, the imaginary part comes second.
In mathematics, complex numbers are a superset of real numbers, which means that every real number is also a complex number whose imaginary part is equal to zero. Python models this relationship through a concept called the numeric tower, described in PEP 3141:
While it may seem of little value, the complex conjugate has a few useful arithmetic properties that can help calculate the division of two complex numbers with pen and paper, among many other things.
For now, remembering a single rule will let you apply your primary school knowledge of arithmetic to calculate basic operations involving complex numbers. The rule to remember is the definition of the imaginary unit, which satisfies the following equation:
You can raise complex numbers to a power using the binary exponentiation operator (**) or the built-in pow() but not the one defined in the math module, which only supports floating-point values:
This fact leads to one of the coolest features of the complex data type in Python, which embodies a rudimentary implementation of a two-dimensional vector for free. While not all operations work the same way in both of them, vectors and complex numbers share many similarities.
Note that rotation becomes more natural when you express your complex numbers in polar coordinates, which already describe the angle. You can then take advantage of the exponential form to make the calculations more straightforward:
Mathematically, two complex numbers are equal when they have identical values irrespective of the adopted coordinate system. However, converting between polar and rectangular coordinates typically introduces rounding errors in Python, so you need to watch out for minute differences when comparing them.
The Python data model defines a set of special methods that you can implement to make your classes compatible with certain built-in types. Say you were working with points and vectors and wanted to get the angle between two bound vectors. You might calculate their dot product and do some trigonometry. Alternatively, you can take advantage of complex numbers.
A Point has the x and y coordinates, while a Vector connects two points. You might remember cmath.phase(), which calculates the angular distance of a complex number. Now, if you treated your vectors as complex numbers and knew their phases, then you could subtract them to obtain the desired angle.
The ease of using complex numbers in Python makes them a surprisingly fun and practical tool. You saw two-dimensional vectors implemented practically for free, and you were able to analyze sound frequencies thanks to them. Complex numbers let you elegantly express mathematical formulas in code without much boilerplate syntax standing in the way.
This module provides access to mathematical functions for complex numbers. Thefunctions in this module accept integers, floating-point numbers or complexnumbers as arguments. They will also accept any Python object that has either a__complex__() or a __float__() method: these methods are used toconvert the object to a complex or floating-point number, respectively, andthe function is then applied to the result of the conversion.
While Argand (1806) is generally credited with the discovery, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line.
Indeed, we already know that all non-zero complex numbers can be expressed in polar coordinates in a unique way. In particular, any number of the form $e^ix$ (with real $x$), which is non-zero, can be expressed as: \[ e^ix = r(\cos \theta + i \sin \theta) \] where $\theta$ is its principal angle from the positive real axis (with, say, $0 \le \theta radius (with $r>0$). We make no assumption about the values of $r$ and $\theta$, except the fact that they are functions of $x$ (which may or may not contain $x$ as variable). They will be determined in the course of the proof.
If anything, the exponential form sure makes it easier to see that multiplying two complex numbers is really the same as multiplying magnitudes and adding angles, and that dividing two complex numbers is really the same as dividing magnitudes and subtracting angles. 2ff7e9595c
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