Complex numbers in Swift

Swift 5 MIT LiCENSE build status

CAVEAT: Swift Numerics vs. this module

With apple/swift-numerics complex number support on swift is official at last. You should consider using ComplexModule of Numerics instead of this. I am switching to swift-numerics myself whereever I can. But there are still a few things that make you want to use this module in spite of that.

  • swift-numerics relies 100% on swift package manager. You cannot use it on Swift Playgrounds.
  • ComplexModule may be too swifty on some respects.
    • ComplexModule adopts point at infinity. While this is mathmatically more correct, technically it may cause unexpected results because real operation on complex numbers is no longer isomorphic to real operations on real numbers. For instance, Complex(-1.0, 0.0) / Complex(0.0, 0.0) is Complex(+infinity, 0.0), not Complex(-infinity, nan) like many other platforms.


Complex numbers in Swift and Swift Package Manager.


import Complex
let z0 = 1.0 + 1.0.i    // (1.0+1.0.i)
let z1 = 1.0 - 1.0.i    // (1.0-1.0.i)
z0.conj // (1.0-1.0.i)
z0.i    // (-1.0+1.0.i)
z0.norm // 2
z0 + z1 // (2.0+0.0.i)
z0 - z1 // (0.0+2.0.i)
z0 * z1 // (2.0+0.0.i)
z0 / z1 // (0.0+1.0.i)


complex.swift implements all the functionality of std::complex in c++11, arguably more intuitively.

like C++11

  • Protocol-Oriented * Complex numbers are Complex<R> where R is a type of .real and .imag that conforms to the ComplexElement protocol or GaussianIntElement protocol.
    • In addition to basic arithmetic operations like +, -, *, / and abs(), Complex<T:RealType> gets libm functions like exp(), log(), sin(), cos().

unlike C++11

  • Instead of defining the constant i, Double and Complex have a property .i which returns self * Complex(0,1) so it does not pollute the identifier i, too popularly used for iteration to make it a constant.
  • Following functions are provided as compouted properties:
    • z.abs for abs(z)
    • z.arg for arg(z)
    • z.norm for norm(z)
    • z.conj for conj(z)
    • z.proj for proj(z)
  • Construct a complex number via polar notation as:
    • Complex(abs:magnitude, arg:argument)



$ git clone
$ cd swift-complex # the following assumes your $PWD is here
$ swift build



$ swift run --repl


$ scripts/


$ swift build && swift -lComplex

and in your repl,

Welcome to Apple Swift version 4.2 (swiftlang-1000.11.37.1 clang-1000.11.45.1). Type :help for assistance.
  1> import Complex
  2> Complex.sqrt(1.i)
$R0: Complex.Complex<Double> = {
  real = 0.70710678118654757
  imag = 0.70710678118654757


Xcode project is deliberately excluded from the repository because it should be generated via swift package generate-xcodeproj . For convenience, you can

$ scripts/prep-xcode

And the Workspace opens up for you with Playground on top. The playground is written as a manual.

iOS and Swift Playground

Unfortunately Swift Package Manager does not support iOS. To make matters worse Swift Playgrounds does not support modules.

Fortunately Playgrounds allow you to include swift source codes under Sources directory. Just run:

$ scripts/

and you are all set. iOS/Complex.playground now runs on Xcode and Playgrounds on macOS, and Playgrounds on iOS (Well, it is supposed to iPadOS but it is still labeled iOS).

From Your SwiftPM-Managed Projects

Add the following to the dependencies section:

  url: "", from: "5.0.0"

and the following to the .target argument:

  name: "YourSwiftyPackage",
  dependencies: ["Complex"])

Now all you have to do is:

import Complex

in your code. Enjoy!


Swift 5 or better, OS X or Linux to build.


  • Swift Tools 4.0.0
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Last updated: Tue Apr 02 2024 13:37:42 GMT-0900 (Hawaii-Aleutian Daylight Time)