# Clever use of Python's complex number type

I was thoroughly amused when I saw the Linear equations solver in 3 lines recipe in ASPN's Python Cookbook this morning. Since it's so short, I'll reproduce it here as a two line recipe ;)

def solve(eq, var='x'): """ >>> solve("x - 2*x + 5*x - 46*(235 - 24) = x + 2") 3236.0 """ c = eval(eq.replace("=", "-(") + ")", {var: 1j}) return -c.real/c.imag

The way it works is really clever. First, it uses string replacement to turn it into a single expression rather than an equality. This is effectively the same as rewriting an equation in the form of `x = 4` to `x - 4 = 0`.

x - 2*x + 5*x + 46*(235 - 24) = x + 2 x - 2*x + 5*x + 46*(253 - 24) - (x + 2)

(Please forgive me for using uncommon math expressions below, the standard notations are confusing given the way the equation above was written and I wanted to remain close to valid Python syntax.)

In order to solve such an equation, you would work through the simple math and end up with something in the form of `a*x + b`. Since this is for simple linear equations only, the author took the liberty of (ab)using the `complex` type to make this happen automagically.

Complex numbers are in the form of `a + bj`, where the *a* represents the "real" component and *b* represents the "imaginary", the *j* represents `math.sqrt(-1)`. Python's `complex` type stores the *a* in the `real` attribute and the *b* in the `imag` attribute. In this example, we are pretending that the *j* really represents something other than `math.sqrt(-1)`, namely, the value of *x* that solves our equation. In order to solve for *x*, we need to do the following transformation:

a + bj bj = -a j = (-a)/b

When the `solve` function evaluates the rewritten expression, it uses `1j` (really, `0 + 1j` -- or `complex(0, 1)`) for the value of *x*, which Python simply evaluates down to a single complex number. As stated above, when we divide the negation of the real part by the imaginary part, we get the desired solution. Neat!

UPDATE: maxim commented that a more comprehensive recipe by Rick Muller, Manipulate simple polynomials in Python, is also available in the Python Cookbook. It's not two lines, though :)