# 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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### 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 ...
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### 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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### 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 =...
287 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 ...
807 views

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### 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 ...
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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### 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 ...
498 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. ...
946 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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, ...
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### 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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### Primitive Roots of Unity

Let z be a complex number. z is an nth primitive root of unity if for a certain positive integer ...
388 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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### 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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### 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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### 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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### Counting Eisenstein primes

Introduction Eisenstein integers are complex numbers of the form a+bω Where a,b are integers, and ...
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Gaussian integers are complex numbers of the form a+bi where a and b are both integers. In ...
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### 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 ...