Questions tagged [complex-numbers]

This challenge involves the manipulation of complex numbers, including parsing and printing them as well as performing complex arithmetic. This tag also encompasses generalised complex numbers like quaternions.

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25 votes
16 answers
3k views

Convert to base i - 1

Given \$ i = \sqrt{-1} \$, a base-\$ (i - 1) \$ binary number \$ N \$ with \$ n \$ binary digits from \$ d_{0} \$ to \$ d_{n - 1} \$ satisfies the following equation. $$ N = d_{n - 1} (i - 1) ^ {n - 1}...
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10 votes
2 answers
349 views

Greatest Common Gaussian Divisor

Gaussian integers are complex numbers \$x+yi\$ such that \$x\$ and \$y\$ are both integers, and \$i^2 = -1\$. The norm of a Gaussian integer \$N(x+yi)\$ is defined as \$x^2 + y^2 = |x+yi|^2\$. It is ...
45 votes
27 answers
2k views

Simplify ijk-string

Related: Multiply Quaternions Challenge Given a string made of ijk, interpret it as the product of imaginary units of quaternion and simplify it into one of the ...
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25 votes
10 answers
2k views

Floor of complex number

Background Complex floor is a domain extension of the mathematical floor function for complex numbers. This is used in some APL languages to implement floor , ...
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20 votes
15 answers
3k views

Complex Fibonacci numbers

The Binet formula is a closed form expression for the \$n\$'th Fibonacci number: $$F_n = \frac {\phi^n - (1-\phi)^n} {\sqrt 5}$$ where \$\phi = \frac {1 + \sqrt 5} 2\$ is the golden ratio. This ...
10 votes
10 answers
320 views

Counting shortest paths on a triangular grid

Background An Eisenstein integer is a complex number of the form \$ z = a + b\omega \$ where \$a, b\$ are integers and \$\omega\$ is the third root of unity \$\frac{1-\sqrt3i}{2}\$. The Eisenstein ...
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3 votes
1 answer
313 views

Gaussian integer division reminder [closed]

Gaussian integer is a complex number in the form \$x+yi\$, where \$x,y\$ are integer and \$i^2=-1\$. The task is to perform such operation for Gaussian integers \$a,b\$, that \$a=q \cdot b+r\$ and \$|...
33 votes
10 answers
3k views

The qvolume of an integer

It is ancient knowledge that every non-negative integer can be rewritten as the sum of four squared integers. For example the number 1 can be expressed as \$0^2+0^2+0^2+1^2\$. Or, in general, for any ...
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11 votes
11 answers
3k views

Quaternion square root

Background Quaternion is a number system that extends complex numbers. A quaternion has the following form $$ a + bi + cj + dk $$ where \$ a,b,c,d \$ are real numbers and \$ i,j,k \$ are three ...
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18 votes
20 answers
3k views

Hermitian matrix?

Note that this challenge requires no handling or understanding of complex numbers. Given a non-empty square matrix where every element is a two-element (Re,Im) integer list, determine (giving any ...
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11 votes
8 answers
2k views

Eigenvalues of a Matrix

Given a square matrix, output the matrix's eigenvalues. Each eigenvalue should be repeated a number of times equal to its algebraic multiplicity. The eigenvalues of a matrix ...
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10 votes
22 answers
2k views

Exponent of complex numbers

Given two integers, which may be negative, zero, or positive, a and b (taken in any reasonable format, including inputting a ...
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23 votes
6 answers
572 views

Factorize a Gaussian integer

A Gaussian integer is a complex number whose real and imaginary parts are integers. Gaussian integers, like ordinary integers, can be represented as a product of Gaussian primes, in a unique manner. ...
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9 votes
15 answers
1k views

Compute a complex power

The Rundown Given any input x and y, perform a complex operation, and print a corresponding result. How your program should work Given an input x and y in the form z = x+yi, find zi-z If the ...
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36 votes
21 answers
6k views

Draw a graph of \$y=(-n)^x\$

Challenge Given an input of an integer, \$n\$ (where \$0<n<50\$), output the graph of \$y=\mathrm{Re}((-n)^x)\$ from \$x = -3\$ to \$x = 3\$ inclusive. Where \$\mathrm{Re}(p)\$ is the real ...
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9 votes
3 answers
309 views

Imaginary Parts of Non-Trivial Riemann Zeroes

Introduction According to the Riemann Hypothesis, all zeroes of the Riemann zeta function are either negative even integers (called trivial zeroes) or complex numbers of the form ...
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35 votes
25 answers
3k views

Complex Binary Numbers

Let's create a simple, surjective mapping from positive integers to Gaussian integers, which are complex numbers where the real and imaginary parts are integers. Given a positive integer, for example ...
20 votes
13 answers
3k views

Codegolf the permanent

The challenge is to write codegolf for the permanent of a matrix. The permanent of an \$n\times n\$ matrix \$A = a_{i,j}\$) is defined as $$\text{perm}(A) = \sum_{\sigma \in S_n} \prod^n_{i=1} a_{i,\...
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17 votes
7 answers
844 views

Gauss to Eisenstein

Given a Gaussian integer \$a+bi\$ where \$a\$,\$b\$ are integers and \$i = \exp\left(\pi i/2\right)\$ is the imaginary unit, return the closest (w.r.t to the Euclidean distance) Eisenstein integer \$k+...
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8 votes
3 answers
543 views

Counting Eisenstein primes

Introduction Eisenstein integers are complex numbers of the form a+bω Where a,b are integers, and ...
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26 votes
10 answers
2k views

Parse a Quaternion

If you don't know already, a quaternion is basically a 4-part number. For the purposes of this challenge, it has a real component and three imaginary components. The imaginary components are ...
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67 votes
9 answers
5k views

Addition in base -1+i

Gaussian integers are complex numbers of the form a+bi where a and b are both integers. In ...
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11 votes
3 answers
3k views

Evaluate the Riemann Zeta Function at a Complex Number

Introduction I found this question that was closed because it was unclear, yet it was a nice idea. I'll do my best to make this into a clear challenge. The Riemann Zeta function is a special function ...
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11 votes
8 answers
566 views

Primitive Roots of Unity

Let z be a complex number. z is an nth primitive root of unity if for a certain positive integer ...
17 votes
2 answers
614 views

Output quater-imaginary base numbers in binary

Write a function or program that outputs Quater-imaginary base displayed as binary digits. The number base is 2i, where i is the square root of -1. See Complex Number for more details on i. Each digit ...
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16 votes
5 answers
434 views

Add and multiply perplexing numbers

The split-complex numbers, also known as "perplex numbers" are similar to the complex numbers. Instead of i^2 = -1, however, we have ...
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17 votes
5 answers
961 views

What's the voltage over each component?

The picture below shows a RLC circuit. A RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. (1) In order to ...
12 votes
4 answers
631 views

Multiply Pauli Matrices

The Pauli matrices are a set of 2x2 matrices which appear very commonly in quantum physics (no, you don't need to know any quantum physics for this challenge). If we include the identity in the set, ...
15 votes
4 answers
1k views

Solve a 2x2 Eigensystem

For those with a little linear algebra background, the challenge is as simple as this: determine the eigenvalues and eigenvectors of a given complex 2x2 matrix. You may skip ahead to The Challenge for ...
7 votes
11 answers
2k views

Wait, but what's e^N?

Objective: The objective is to calculate e^N for some real or imaginary N (e.g. 2, -3i, 0.7, 2.5i, but not 3+2i). This is code golf, so, shortest code (in bytes) wins. So, for example: N = 3, e^N =...
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29 votes
54 answers
5k views

Find i^n, given n

The Challenge In as few characters as possible, find the value of \$i^n\$, given \$n\$, a positive integer greater than 0. This should be outputted as a String. For those that don't know, \$i\$ is ...
  • 425
16 votes
10 answers
2k views

Multiply Quaternions

Write a named function or program that computes the quaternion product of two quaternions. Use as few bytes as possible. Quaternions Quaternions are an extension of the real numbers that further ...
  • 139k
23 votes
13 answers
2k views

is_gaussian_prime(z)?

Task Write a function that accepts two integers \$a,b\$ that represent the Gaussian integer \$z = a+bi\$ (complex number). The program must return true or false depending on whether \$a+bi\$ is a ...
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48 votes
14 answers
4k views

Too Fast, Too Fourier: FFT Code Golf

Implement the Fast Fourier Transform in the fewest possible characters. Rules: Shortest solution wins It can be assumed that the input is a 1D array whose length is a power of two. You may use the ...
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2 votes
5 answers
499 views

Perform basic operations on complex numbers in a language without native support for complex numbers

Your code has to provide the following functions: read(x) Reads a complex number from the standard input. It has to accept and evaluate something in the form ...
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3 votes
7 answers
840 views

Resolve quadratic equation

Challenge: Write the smallest program (in characters) that resolves quadratic equations i.e. ax² + bx + c = 0 Rules: Given 3 numbers in R comma-delimited on ...
  • 957
24 votes
3 answers
6k views

Generate Newton fractals

You all know the Newton method to approximate the roots of a function, don't you? My goal in this task is to introduce you into an interesting aspect of this algorithm. Newton's algorithm converges ...
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