Complex numbers

Package: SYMBOLIC_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

A complex number has two parts: a real part, an ordinary number, and an imaginary part, a multiple of i, the number whose square is -1. They are what the square root of a negative number turns out to be, and they appear in electronics, signal processing and the roots of equations. Writing a number flush against i makes it imaginary.

3i // 3i
2+3i // 2+3i
1i*1i // -1

Arithmetic is exact, in the same sense the rest of the engine is: both parts are exact fractions, never floating-point approximations.

(2+3i)+(1-1i) // 3+2i
(1+1i)*(1-1i) // 2

That exactness is what lets a value be recognised as really real. (1+i)(1-i) is 2, not 2 with a residue of imaginary noise, so the imaginary part disappears when it should.

i is still a variable name

i is not a reserved word. It is one of the most common variable names there is, so claiming it would break more than it fixed.

:i = 5
:i + 1 // 6
3 * i // 15

The imaginary literal is recognised only when the letter is written flush against a number, with no space: 3i is imaginary, 3 i and 3 * i are a number times a variable. Write 1i for the imaginary unit on its own.

Square roots of negative numbers

These have answers now.

sqrt(-4) // 2i
sqrt(-2) // sqrt(2)*i

The first is exact. The second keeps its surd, because the square root of two is irrational and rounding it would be the one thing this engine will not do.

Taking a complex number apart

Each part of a complex number can be read on its own. re gives the real part and im the imaginary part (as a plain number, without the i). conj gives the conjugate, the same number with its imaginary part negated, which is what multiplies with the original to give a real answer.

re(2+3i) // 2
im(2+3i) // 3
conj(2+3i) // 2-3i
abs(3+4i) // 5

abs gives the modulus, which is exact whenever it is rational.

Roots that are complex

Every quadratic has two roots, and now they are both returned.

solve(x^2+1=0, x) // [-i, i]
solve(x^2+2x+5=0, x) // [-1-2i, -1+2i]
solve(x^2+2=0, x) // [-sqrt(2)*i, sqrt(2)*i]

Factoring stays over the rationals

Factoring is only defined once you say over what, and the default here is the rational numbers, which is the usual convention.

factor(x^2+1) // x^2+1

x^2+1 does factor over the complex numbers, as (x-i)(x+i), but returning that by default would mean factor changed what it meant depending on the input. Use solve when you want the roots.