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Q-ComplexNumber.html
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<!DOCTYPE html>
<html>
<head>
<title>Q ⟩ ComplexNumber</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1.0">
<meta name="description" content="Quantum computing in your browser.">
<meta name="copyright" content="Stewart Smith 2019–2020">
<meta name="keywords" content="
Q, Q.js, Q-js, Qjs, quantum JavaScript,
quantum, quantum physics, quantum mechanics, superposition,
quantum computer, quantum computer programming, quantum computing, QC,
quantum simulator, quantum computer simulator,
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Web, Web site, website, Web browser, browser, HTML, HTML5, JavaScript, ES6, CSS,
Chrome, Firefox, Safari, Opera, Brave, Edge, WebKit, Blink, Gecko, Mozilla,
Stewart Smith, Stewart, Stew, Stuart, Steven, Steve, Stewdio, stewartsmith, stew_rtsmith, @stew_rtsmith,
Moar, Moar Technologies Corp, MTC,
Google, IBM, Microsoft, Amazon, NASA, DWave, D-Wave,
Quil, OpenQASM,
ProjectQ, Qiskit,
Quantum Development Kit, Cirq, Strawberry Fields, t|ket>,
QCL, Quantum pseudocode, Q#, Q|SI>, Q language, qGCL, QFC, QML, LIQUi|>, Quipper,
Stanford CS 269 Q: Quantum Computer Programming">
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<meta property="og:url" content="https://quantumjavascript.app/ComplexNumber.html">
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<script src="Q/Q-ComplexNumber.js"></script>
<script src="Q/Q-Matrix.js"></script>
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</head>
<body>
<main class="api">
<p>
Source code:
<a href="https://github.com/stewdio/q.js/blob/master/source/Q-ComplexNumber.js?ts=4" target="_blank">
<code>Q-ComplexNumber.js</code>
</a>
</p>
<hr>
<h3 id="Real_numbers">Real numbers ℝ</h3>
<p>
Our regular, ordinary, everyday numbers are called
<a href="https://en.wikipedia.org/wiki/Real_number" target="_blank">real numbers</a>.
These include integers and decimals.
You can visualize real numbers as existing along an infinite number line—with zero in the middle, positive numbers counting up forever to infinity on the right, and negative numbers doing the exact opposite on the left.
</p>
<svg class="complex-plane" viewBox="0 0 600 600">
<use xlink:href="#complex-plane-real">
</svg>
<p>
When a real number is multiplied by itself the product is always positive.
For example, if we choose the number 2 we see that 2 × 2 = 4.
Similarly, had we chosen the negative number -2, the product would still be positive
because two negative numbers multiplied together also produce a positive result;
-2 × -2 = 4.
For brevity we could rewrite these equations as 2<sup>2</sup> = 4 and (-2)<sup>2</sup> = 4, respectively.
</p>
<p>
The <a href="https://en.wikipedia.org/wiki/Square_root" target="_blank">square root</a>
of a real number has two possible answers.
The square root of 4, for example, is both 2 <em>and</em> -2
because both are solutions for <span class="symbol">x</span>
in the equation <span class="symbol">x</span> = √4.
</p>
<h3 id="Imaginary_numbers">Imaginary numbers 𝕀</h3>
<p>
But suppose we wanted to find the square root of a <em>negative</em> number.
Is there any number that could solve for <span class="symbol">x</span>
in the equation <span class="symbol">x</span> = √(-4)?
Sadly, there is not. Or more precisely: there is not any <em>real</em> solution for the square root of a negative number.
</p>
<p>
In his 1991 address on “Creativity in Management”, Monty Python’s John Cleese
articulates Edward de Bono’s concept of the
<a href="https://youtu.be/Pb5oIIPO62g?t=1926" target="_blank">intermediate impossible (32:06)</a>
as a useful stepping stone towards a novel and useful solution.
<a href="https://en.wikipedia.org/wiki/Imaginary_number" target="_blank">Imaginary numbers</a>
might be considered an intermediate impossible.
The symbol <span class="symbol">i</span> is defined as the imaginary solution to the equation
<span class="symbol">x</span> = √(-1),
therefore <span class="symbol">i</span><sup>2</sup> = -1.
With this imaginary device we now have a solution to the above equation
<span class="symbol">x</span> = √(-4)
and that solution is 2<span class="symbol">i</span>.
(And also -2<span class="symbol">i</span>, of course!
We can indicate this “plus or minus” possibility as ±2<span class="symbol">i</span>.)
Let’s inspect this more closely.
</p>
<pre><code>
𝒙 = √(-4)
𝒙 = √( 4 × -1)
𝒙 = √4 × √(-1)
𝒙 = ±2 × √(-1)
𝒙 = ±2 × <span class="symbol">i</span>
𝒙 = ±2<span class="symbol">i</span>
</code></pre>
<p>
2<span class="symbol">i</span> is an
<a href="https://en.wikipedia.org/wiki/Imaginary_number" target="_blank">imaginary number</a>
that consists of a real number multiplier, 2, and
our imaginary solution to √(-1), called <span class="symbol">i</span>.
Like real numbers, imaginary numbers also exist along an infinite number line.
We plotted our real number line horizontally,
so let’s plot our imaginary number line vertically.
</p>
<svg class="complex-plane" viewBox="0 0 600 600">
<use xlink:href="#complex-plane-imaginary">
</svg>
<h3 id="Complex_numbers">Complex numbers ℂ</h3>
<p>
We just saw that multiplying a real number by <span class="symbol">i</span>
yields an imaginary number.
But what if you <em>add</em> a real number to an imaginary one?
Things get <em>complex</em>.
A <a href="https://en.wikipedia.org/wiki/Complex_number" target="_blank">complex number</a>
is a number that can be expressed in the form
<span class="symbol">a</span> + <span class="symbol">bi</span>,
where <span class="symbol">a</span> is the real component and
<span class="symbol">bi</span> is the imaginary component.
Some examples might be 1 + 2<span class="symbol">i</span> or 3 - 4<span class="symbol">i</span>.
</p>
<svg class="complex-plane" version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
viewBox="0 0 600 600" style="enable-background:new 0 0 600 600;" xml:space="preserve">
<style type="text/css">
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<line class="dashed" x1="300" y1="300" x2="300" y2="600"/>
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<g id="real-numbers">
<text transform="matrix(1 0 0 1 496 282)">4</text>
<text transform="matrix(1 0 0 1 546 282)">5</text>
<text transform="matrix(1 0 0 1 453 325)" class="symbol">π</text>
<text transform="matrix(1 0 0 1 446 282)">3</text>
<text transform="matrix(1 0 0 1 433 325)" class="symbol">e</text>
<text transform="matrix(1 0 0 1 396 282)">2</text>
<text transform="matrix(1 0 0 1 363 328)">√2</text>
<text transform="matrix(1 0 0 1 347 282)">1</text>
<text transform="matrix(1 0 0 1 296 282)">0</text>
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<text transform="matrix(1 0 0 1 240 282)">-1</text>
<text transform="matrix(1 0 0 1 190 282)">-2</text>
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<!--
“Quantum Mechanics Concepts: 1 Dirac Notation and Photon Polarisation”
https://youtu.be/pBh7Xqbh5JQ?t=352
Maybe someday add the above video’s bits about a^2 + b^2 = c^2 ?
-->
<p>
This is what the <code>ComplexNumber</code> class was created to handle.
Open up your JavaScript console and paste in the following:
</p>
<pre><code>
var
cat = new <a href="Q.html">Q</a>.ComplexNumber( 1, 2 ),
dog = new <a href="Q.html">Q</a>.ComplexNumber( 3, -4 )
cat.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '1 + 2i'</span>
dog.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '3 - 4i'</span>
</pre></code>
<p>
Now we have two variables, <code>cat</code> and <code>dog</code>, that we can operate with.
As you might guess, <code>ComplexNumber</code>
includes instance methods for common operations like
addition, subtraction, multiplication, and division.
Try the following lines individually in your JavaScript console:
</p>
<pre><code>
cat.<a href="#.prototype.add">add</a>( dog ).<a href="#.prototype.toText">toText</a>() <span class="comment">// Returns '4 - 2i'</span>
cat.<a href="#.prototype.subtract">subtract</a>( dog ).<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '-2 + 6i'</span>
cat.<a href="#.prototype.multiply">multiply</a>( dog ).<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '11 + 2i'</span>
cat.<a href="#.prototype.divide">divide</a>( dog ).<a href="#.prototype.toText">toText</a>() <span class="comment">// Returns '-0.2 + 0.4i'</span>
</pre></code>
<p>
We can now verify that <span class="symbol">i</span><sup>2</sup> = -1.
</p>
<pre><code>
var i = new <a href="Q.html">Q</a>.ComplexNumber( 0, 1 )
i.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns 'i'</span>
i.<a href="#.prototype.power">power</a>( 2 ).<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '-1'</span>
</pre></code>
<p>
Operation functions on <code>Q.ComplexNumber</code> instances generally accept as
arguments both sibling instances and pure <code>Number</code> instances, though the
value returned is always an instance of <code>Q.ComplexNumber</code>.
</p>
<hr>
<h3 id="Constructor">Constructor</h3>
<p>
<span class="constructor">ComplexNumber</span>
<code class="value-type">Function([ real: Number or <a href="Q.html">Q</a>.ComplexNumber ][, imaginary: Number ]]) => <a href="Q.html">Q</a>.ComplexNumber</code>
<br>
Expects zero to two arguments.
If the first argument is a <code>ComplexNumber</code>
then that value is cloned and any remaining arguments are ignored.
If either of the arguments are <code>undefined</code> they are assumed to be zero.
If the first argument is not a <code>ComplexNumber</code>
and either of the arguments are not <a href="Q-ComplexNumber.html#.isNumberLike">number-like</a>
then an error is thrown.
</p>
<pre><code>
var
ape = new <a href="Q.html">Q</a>.ComplexNumber(),
bee = new <a href="Q.html">Q</a>.ComplexNumber( 1 ),
elk = new <a href="Q.html">Q</a>.ComplexNumber( 1, 2 ),
fox = new <a href="Q.html">Q</a>.ComplexNumber( elk )
ape.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '0'</span>
bee.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '1'</span>
elk.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '1 + 2i'</span>
fox.<a href="#.prototype.toText">toText</a>()<span class="comment">// Returns '1 + 2i'</span>
elk.<a href="#.prototype.isEqualTo">isEqualTo</a>( fox ) <span class="comment">// Returns true</span>
elk.<a href="#this.index">index</a> === fox.<a href="#.prototype.index">index</a><span class="comment">// Returns false</span>
</pre></code>
<ul class="properties">
<li>
<dt id="this.real">real</dt>
<dd>
<code class="value-type">Number</code>
Traditionally the first argument,
a <a href="Q-ComplexNumber.html#.isNumberLike">number-like</a> value
representing the <em>real</em> component of the
<a href="Q-ComplexNumber.html#Complex_numbers">complex number</a>.
</dd>
</li>
<li>
<dt id="this.imaginary">imaginary</dt>
<dd>
<code class="value-type">Number</code>
Traditionally the second argument,
a <a href="Q-ComplexNumber.html#.isNumberLike">number-like</a> value
representing the <em>imaginary</em> component of the
<a href="Q-ComplexNumber.html#Complex_numbers">complex number</a>.
</dd>
</li>
<li>
<dt id="this.index">index</dt>
<dd>
<code class="value-type">Number</code>
An identification number assigned to the instance,
used for minding the total number of instances created.
</dd>
</li>
</ul>
<hr>
<h3>Static properties</h3>
<ul class="properties">
<li>
<dt id=".help">help</dt>
<dd>
<code class="value-type">Function ⇒ String</code>
Calls and returns the value of
<code><a href="Q.html">Q</a>.<a href="Q.html#help">help</a></code>,
passing <code><a href="Q.html">Q</a>.ComplexNumber</code> as the argument.
</dd>
</li>
<li>
<dt id=".index">index</dt>
<dd>
<code class="value-type">Number</code>
The number of instances created so far.
</dd>
</li>
</ul>
<h4>Constants and constant creation</h4>
<ul class="properties">
<li>
<dt id=".constants">constants</dt>
<dd>
<code class="value-type">Object</code>
Constants are appended <em>directly</em> to the
<code><a href="Q.html">Q</a>.ComplexNumber</code> object.
For convenience they are also appended to this
<code><a href="Q.html">Q</a>.ComplexNumber</code>.constants</code> object
to make looking up constants in the JavaScript console trivial,
and to make iterating across all constants convenient via functions like
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/entries" target="_blank">Object.entries</a></code>,
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/keys" target="_blank">Object.keys</a></code>,
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/values" target="_blank">Object.values</a></code>,
and so on.
<!-- Configured to be unwritable once appended via
<code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/defineProperty" target="_blank">Object.defineProperty</a></code>
and they are labeled in uppercase to signal to us that this is so. -->
The intention that a property act as a constant is signaled
by its labelling in all-uppercase.
</dd>
</li>
<li>
<dt id=".createConstant">createConstant</dt>
<dd>
<code class="value-type">Function( key: String, value: * )<!-- → undefined --></code>
Appends a property named by <code>key</code>
with a value of <code>value</code>
to both the
<code>Q</code> object
and its <code><a href="#.constants">constants</a></code> property.
</dd>
</li>
<li>
<dt id=".createConstants">createConstants</dt>
<dd>
<code class="value-type">Function( … )</code>
Expects an even number of arguments.
Will use each pair in the sequence of arguments to call
<code><a href="#.createConstant">createConstant</a></code>.
</dd>
</li>
<li>
<dt id=".ZERO">ZERO</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( 0, 0 )</code>.
Described as 0.
</dd>
</li>
<li>
<dt id=".ONE">ONE</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( 1, 0 )</code>.
Described as 1.
</dd>
</li>
<li>
<dt id=".E">E</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( Math.E, 0 )</code>.
Described as <span class="symbol">e</span> ≈ 2.7183.
</dd>
</li>
<li>
<dt id=".PI">PI</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( Math.PI, 0 )</code>.
Described as <a href="https://youtu.be/L1eegVTwDS0" target="_blank"><strong>exactly</strong> equal to 3</a>.
</dd>
</li>
<li>
<dt id=".I">I</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( 0, 1 )</code>.
Described as <span class="symbol">i</span>.
</dd>
</li>
<li>
<dt id=".EPSILON">EPSILON</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( <a href="Q.html">Q</a>.<a href="Q.html#.EPSILON">EPSILON</a>, <a href="Q.html">Q</a>.<a href="Q.html#.EPSILON">EPSILON</a> )</code>.
Described as ≈ 1.3323 × 10<sup>-15</sup> + 1.3323 × 10<sup>-15</sup><span class="symbol">i</span>.
</dd>
</li>
<li>
<dt id=".INFINITY">INFINITY</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( Infinity, Infinity )</code>.
Described as <span class="symbol">∞</span> + <span class="symbol">∞i</span>.
</dd>
</li>
<li>
<dt id=".NAN">NAN</dt>
<dd>
<code class="value-type"><a href="Q.html">Q</a>.ComplexNumber</code>
Initialized as
<code>new <a href="Q.html">Q</a>.ComplexNumber( NaN, NaN )</code>.
<code>Array( 16 ).join( NaN ) +' BATMAN!'</code>
</dd>
</li>
</ul>
<h4>Inspection</h4>
<ul class="properties">
<li>
<dt id=".isNumberLike">isNumberLike</dt>
<dd>
<code class="value-type">Function( n: * ) ⇒ Boolean</code>
Returns true if
<code>n</code>’s <code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/typeof" target="_blank">typeof</a></code> is equal to the <code>String</code> <code>'number'</code>
or if <code>n <a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/instanceof" target="_blank">instanceof</a> Number</code> is <code>true</code>,
otherwise returns <code>false</code>.
</dd>
</li>
<li>
<dt id=".isNaN">isNaN</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Returns <code>true</code> if either of this instance’s
<code><a href="#this.real">real</a></code>
or
<code><a href="#this.real">imaginary</a></code>
components are <code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/NaN" target="_blank">NaN</a></code>,
otherwise returns <code>false</code>.
</dd>
</li>
<li>
<dt id=".isZero">isZero</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Returns <code>true</code> if both of this instance’s
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.real">imaginary</a></code>
absolute values are equal to zero,
otherwise returns <code>false</code>.
See <a href="https://en.wikipedia.org/wiki/Signed_zero" target="_blank">“Signed zero”</a> for an explanation of why the <em>absolute</em> value of zero must be taken.
</dd>
</li>
<li>
<dt id=".isFinite">isFinite</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Returns <code>true</code> if both of this instance’s
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.real">imaginary</a></code>
values are <a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/isFinite" target="_blank">finite</a>,
otherwise returns <code>false</code>.
</dd>
</li>
<li>
<dt id=".isInfinite">isInfinite</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Returns <code>true</code> if both of this instance’s
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.real">imaginary</a></code>
values are not <a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/isFinite" target="_blank">finite</a>
and are also not <code><a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/isNaN" target="_blank">NaN</a></code>,
otherwise returns <code>false</code>.
</dd>
</li>
<li>
<dt id=".areEqual">areEqual</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Returns <code>true</code> if the arguments
<code>a</code> and <code>b</code>
are within
<code><a href="Q.html">Q</a>.<a href="Q.html#.EPSILON">EPSILON</a></code> of each other,
otherwise returns <code>false</code>.
</dd>
</li>
</ul>
<h4>Maths</h4>
<ul class="properties">
<li>
<dt id=".absolute">absolute</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Number</code>
Calls and returns the value of <code><a href="Q.html">Q</a>.<a href="Q.html#.hypotenuse">hypotenuse</a></code>
using <code>c</code>’s
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.real">imaginary</a></code>
properties as arguments.
</dd>
</li>
<li>
<dt id=".conjugate">conjugate</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns a new complex number with the values
<code>c.<a href="#this.real">real</a></code>
and
<code>c.<a href="#this.real">imaginary</a> * -1</code>.
</dd>
</li>
<li>
<dt id=".sine">sine</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the
<a href="https://en.wikipedia.org/wiki/Sine#Sine_with_a_complex_argument" target="_blank">sine</a>
of <code>c</code>.
</dd>
</li>
<li>
<dt id=".cosine">cosine</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the
<a href="https://en.wikipedia.org/wiki/Trigonometric_functions#In_the_complex_plane" target="_blank">cosine</a>
of <code>c</code>.
</dd>
</li>
<li>
<dt id=".arcCosine">arcCosine</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the
<a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane" target="_blank">arccosine</a>
of <code>c</code>.
</dd>
</li>
<li>
<dt id=".arcTangent">arcTangent</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the
<a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane" target="_blank">arctangent</a>
of <code>c</code>.
</dd>
</li>
<li>
<dt id=".power">power</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns <code>a</code> raised to the power of <code>b</code>.
</dd>
</li>
<li>
<dt id=".squareRoot">squareRoot</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the square root of <code>c</code>.
</dd>
</li>
<li>
<dt id=".log">log</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns the log of <code>c</code>.
</dd>
</li>
<li>
<dt id=".operate">operate</dt>
<dd>
<code class="value-type">Function(
name: String,
a: Number or <a href="Q.html">Q</a>.ComplexNumber,
b: Number or <a href="Q.html">Q</a>.ComplexNumber,
numberAndNumber: Function,
numberAndComplex: Function,
complexAndNumber: Function,
complexAndComplex: Function ) ⇒ *</code>
Meta function for performing operations on two values
that may each be either numbers or complex numbers.
The <code>name</code> argument indicates the name of the
operation being performed and is used in error logging
should the operation fail.
The intent is to return a <code><a href="Q.html">Q</a>.ComplexNumber</code>, though this is up to the functions
passed in as it is their return values that are returned.
</dd>
</li>
<li>
<dt id=".multiply">multiply</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Uses <code><a href="Q.html">Q</a>.ComplexNumber.<a href="#.operate">operate</a></code>
to return the result of multiplying
<code>a</code> and <code>b</code>.
</dd>
</li>
<li>
<dt id=".divide">divide</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Uses <code><a href="Q.html">Q</a>.ComplexNumber.<a href="#.operate">operate</a></code>
to return the result of dividing
<code>a</code> by <code>b</code>.
</dd>
</li>
<li>
<dt id=".add">add</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Uses <code><a href="Q.html">Q</a>.ComplexNumber.<a href="#.operate">operate</a></code>
to return the result of adding
<code>a</code> and <code>b</code>.
</dd>
</li>
<li>
<dt id=".subtract">subtract</dt>
<dd>
<code class="value-type">Function( a: Number or <a href="Q.html">Q</a>.ComplexNumber, b: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Uses <code><a href="Q.html">Q</a>.ComplexNumber.<a href="#.operate">operate</a></code>
to return the result of subtracting
<code>b</code> from <code>a</code>.
</dd>
</li>
</ul>
<hr>
<h3 id="Prototype_properties">Prototype properties</h3>
<ul class="properties">
<li>
<dt id=".prototype.clone">clone</dt>
<dd>
<code class="value-type">Function ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Returns a new instance with the values for
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.imaginary">imaginary</a></code>
copied from this instance.
</dd>
</li>
<li>
<dt id=".prototype.copy$">copy$</dt>
<dd>
<code class="value-type">Function( c: <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Copies the values for
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.imaginary">imaginary</a></code>
from the supplied <code><a href="Q.html">Q</a>.ComplexNumber</code> argument.
</dd>
</li>
</ul>
<h4>Inspections (non-destructive)</h4>
<ul class="properties">
<li>
<dt id=".prototype.isNaN">isNaN</dt>
<dd>
<code class="value-type">Function ⇒ Boolean</code>
Calls and returns the result of the
<a href="#.isNaN"><code>isNaN</code> static method</a>,
passing the calling instance as the argument.
Will return a <code>Boolean</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.isZero">isZero</dt>
<dd>
<code class="value-type">Function ⇒ Boolean</code>
Calls and returns the result of the
<a href="#.isZero"><code>isZero</code> static method</a>,
passing the calling instance as the argument.
Will return a <code>Boolean</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.isFinite">isFinite</dt>
<dd>
<code class="value-type">Function ⇒ Boolean</code>
Calls and returns the result of the
<a href="#.isFinite"><code>isFinite</code> static method</a>,
passing the calling instance as the argument.
Will return a <code>Boolean</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.isInfinite">isInfinite</dt>
<dd>
<code class="value-type">Function ⇒ Boolean</code>
Calls and returns the result of the
<a href="#.isInfinite"><code>isInfinite</code> static method</a>,
passing the calling instance as the argument.
Will return a <code>Boolean</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.isEqualTo">isEqualTo</dt>
<dd>
<code class="value-type">Function( n: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ Boolean</code>
Calls and returns the result of the
<a href="#.areEqual"><code>areEqual</code> static method</a>,
passing the calling instance as the first argument
and <code>n</code> as the second argument.
Will return a <code>Boolean</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.reduce">reduce</dt>
<dd>
<code class="value-type">Function ⇒ <a href="Q.html">Q</a>.ComplexNumber or Number</code>
If no <code><a href="#this.imaginary">imaginary</a></code> component exists,
returns a <code>Number</code> representing the <code><a href="Q.html">Q</a>.ComplexNumber <a href="#this.real">real</a></code> component,
otherwise returns the instance itself.
May return a <code>Number</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.toText">toText</dt>
<dd>
<code class="value-type">Function([ roundToDecimal: Number ]) ⇒ String</code>
Returns the value of this instance expressed as text
in a form similar to
<span class="symbol">a</span>+<span class="symbol">bi</span>.
If <code>roundToDecimal</code> is supplied, will round both the
<code><a href="#this.real">real</a></code>
and
<code><a href="#this.imaginary">imaginary</a></code>
components
by passing themselves and <code>roundToDecimal</code>
to <code><a href="Q.html">Q</a>.<a href="Q.html#.round">round</a></code>
as arguments.
Will return a <code>String</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
</ul>
<h4>Maths (non-destructive)</h4>
<ul class="properties">
<li>
<dt id=".prototype.absolute">absolute</dt>
<dd>
<code class="value-type">Function ⇒ Number</code>
Passes this instance as an argument to the
<a href="#.absolute"><code>absolute</code>
static method</a> and returns the result.
Will return a <code>Number</code> value,
thereby halting
<a href="https://en.wikipedia.org/wiki/Fluent_interface" target="_blank">“Fluent interface” method chaining</a>
for this instance.
</dd>
</li>
<li>
<dt id=".prototype.conjugate">conjugate</dt>
<dd>
<code class="value-type">Function ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as an argument to the
<a href="#.conjugate"><code>conjugate</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.power">power</dt>
<dd>
<code class="value-type">Function( n: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as the first argument
and <code>n</code> as the second argument
to the
<a href="#.power"><code>power</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.squareRoot">squareRoot</dt>
<dd>
<code class="value-type">Function ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as an argument to the
<a href="#.squareRoot"><code>squareRoot</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.log">log</dt>
<dd>
<code class="value-type">Function ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as an argument to the
<a href="#.log"><code>log</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.multiply">multiply</dt>
<dd>
<code class="value-type">Function( n: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as the first argument
and <code>n</code> as the second argument
to the
<a href="#.multiply"><code>multiply</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.divide">divide</dt>
<dd>
<code class="value-type">Function( n: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as the first argument
and <code>n</code> as the second argument
to the
<a href="#.divide"><code>divide</code>
static method</a> and returns the result.
</dd>
</li>
<li>
<dt id=".prototype.add">add</dt>
<dd>
<code class="value-type">Function( n: Number or <a href="Q.html">Q</a>.ComplexNumber ) ⇒ <a href="Q.html">Q</a>.ComplexNumber</code>
Passes this instance as the first argument
and <code>n</code> as the second argument
to the
<a href="#.add"><code>add</code>
static method</a> and returns the result.
</dd>
</li>