数学 – JavaScript - 新しい数とその表示 - 複素数と複素平面 – 複素平面(複素数の絶対値)

数学読本〈3〉

学習環境/開発環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• OS X El Capitan - Apple (OS)
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• kjs-math-number (JavaScript Library)
• Safari(Web browser)

数学読本〈3〉平面上のベクトル/複素数と複素平面/空間図形/2次曲線/数列 (松坂 和夫(著)、岩波書店)の第10章(新しい数とその表示 - 複素数と複素平面)、10.1(複素平面)、複素数の絶対値、問2、3、4.を取り組んでみる。

問2.

1. 
   $2·\sqrt{10}\sqrt{20}\sqrt{2}=40$
2. 
   $\frac{\sqrt{5}\sqrt{13}}{\sqrt{2}·5}=\frac{\sqrt{130}}{10}$

問3.

$\begin{array}{l}{\left|\alpha +\beta \right|}^2+{\left|\alpha -\beta \right|}^2\\ =\left(\alpha +\beta \right)\overline{\left(\alpha +\beta \right)}+\left(\alpha -\beta \right)\overline{\left(\alpha -\beta \right)}\\ =\left(\alpha +\beta \right)\left(\overline{\alpha }+\overline{\beta }\right)+\left(\alpha -\beta \right)\left(\overline{\alpha }-\overline{\beta }\right)\\ =\alpha \overline{\alpha }+\beta \overline{\beta }\\ ={\left|\alpha \right|}^2+{\left|\beta \right|}^2\end{array}$

問4.

$\begin{array}{l}{\left|\alpha -\beta \right|}^2=\left(\alpha -\beta \right)\overline{\left(\alpha -\beta \right)}\\ =\left(\alpha -\beta \right)\left(\overline{\alpha }-\overline{\beta }\right)\\ ={\left|\alpha \right|}^2+{\left|\beta \right|}^2-2\left(\alpha \overline{\beta }+\overline{\alpha }\beta \right)\\ \\ {\left|1-\overline{\alpha }\beta \right|}^2=\left(1-\overline{\alpha }\beta \right)\left(1-\alpha \overline{\beta }\right)\\ =1+{\left|\alpha \right|}^2{\left|\beta \right|}^2-2\left(\alpha \overline{\beta }+\overline{\alpha }\beta \right)\\ \\ \left|\alpha \right|=1\\ {\left|\alpha \right|}^2+{\left|\beta \right|}^2=1+{\left|\beta \right|}^2\\ 1+{\left|\alpha \right|}^2{\left|\beta \right|}^2=1+{\left|\beta \right|}^2\\ \\ \left|\beta \right|=1\\ {\left|\alpha \right|}^2+{\left|\beta \right|}^2={\left|\alpha \right|}^2+1\\ 1+{\left|\alpha \right|}^2{\left|\beta \right|}^2={\left|\alpha \right|}^2+1\end{array}$

number.js で確認。

JavaScript

コード(Emacs)

(function () {
    'use strict';
    var div_output = document.querySelector('#output0'),
        button_calc = document.querySelector('#calc0'),
        nl = '<br>',
        calc;

    calc = function () {
        var a,
            b,
            c,
            d,
            left,
            right,
            n,
            one = 1,
            two = 2,
            output0 = '',
            output1 = '';

        output0 += '問2' + nl;
        a = new Complex(0, -2);
        b = new Complex(3, 1);
        c = new Complex(2, -4);
        d = new Complex(1, 1);
        n = a.mul(b).mul(c).mul(d).abs();

        output0 += '(1) <math><mn>' + n + '</mn></math>' + nl;

        a = new Complex(-1, 2);
        b = new Complex(3, -2);
        c = new Complex(1, 1);
        d = new Complex(3, -4);
        n = a.mul(b).div(c).div(d).abs();

        output1 += ' (2) <math><mn>' + n + '</mn></math>' + nl + nl;

        a = new Complex(Math.floor(Math.random() * 100),
                        Math.floor(Math.random() * 100));
        b = new Complex(Math.floor(Math.random() * 100),
                        Math.floor(Math.random() * 100));

        left = Math.pow(a.add(b).abs(), 2);
        left = left.add(Math.pow(a.sub(b).abs(), 2));

        right = two.mul(Math.pow(a.abs(), 2).add(Math.pow(b.abs(), 2)));

        output1 +=
            '問3' + nl +
            'a = ' + a + ', b = ' + b + nl +
            '左辺 = 右辺: ' + left.isEqual(right) + nl +
            '差: ' + left.sub(right) + nl + nl;

        a = new Complex(0.6, 0.8);
        b = new Complex(Math.floor(Math.random() * 100),
                        Math.floor(Math.random() * 100));

        left = a.sub(b).abs();
        right = one.sub(a.conjugate().mul(b)).abs();
        output1 +=
            '問4' + nl +
            'a = ' + a + ', b = ' + b + nl +
            '左辺 = 右辺: ' + left.isEqual(right) + nl +
            '差: ' + left.sub(right) + nl + nl;

        a = new Complex(Math.floor(Math.random() * 100),
                        Math.floor(Math.random() * 100));
        b = new Complex(0.6, 0.8);

        left = a.sub(b).abs();
        right = one.sub(a.conjugate().mul(b)).abs();
        output1 +=
            'a = ' + a + ', b = ' + b + nl +
            '左辺 = 右辺: ' + left.isEqual(right) + nl +
            '差: ' + left.sub(right) + nl;
        
        div_output.innerHTML = output0 + output1;
    };

    calc();

    button_calc.onclick = calc;
}());

問2
(1) $40$
(2) $1.1401754250991378$

問3
a = 28+66i, b = 31+25i
左辺 = 右辺: false
差: -1.8189894035458565e-12

問4
a = 0.6+0.8i, b = 62+56i
左辺 = 右辺: true
差: 0

a = 58+33i, b = 0.6+0.8i
左辺 = 右辺: true
差: 0