数学 – Python - JavaScript - 数学はここから始まる - 数 – 2次方程式と複素数 – 解と係数の関係(対称式、解の和と積)

数学読本〈1〉
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• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス(Kobo) Yahoo!)
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo) Yahoo!)
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo) Yahoo!)

数学読本〈1〉数・式の計算/方程式/不等式 (松坂 和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.2(2次方程式と複素数)、解と係数の関係の問9.を取り組んでみる。

9. 
   
   1. 
      
      $\begin{array}{}\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}=-\frac{4}{-2}=2\\ \mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}=\frac{5}{2}\end{array}$

      対称式を工夫。

      $\begin{array}{}{\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}-\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2\\ ={\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2-2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2\\ ={\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2-4\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\\ =4-4·\; <!--\; middle\; dot\; -->\frac{5}{2}\\ =4-10\\ =-6\end{array}$
   2. 
      
      $\begin{array}{}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^3+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^3\\ =\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)\left({\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2-\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2\right)\\ =\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)\left({\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2-3\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)\\ =2·\; <!--\; middle\; dot\; -->\left(4-\frac{15}{2}\right)\\ =8-15\\ =-7\end{array}$
   3. 
      
      $\begin{array}{}\frac{1}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}+\frac{1}{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}\\ =\frac{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}\\ =2·\; <!--\; middle\; dot\; -->\frac{2}{5}\\ =\frac{4}{5}\end{array}$
   4. 
      
      $\begin{array}{}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^4-{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^4\\ ={\left({\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2\right)}^2-3{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2\\ ={\left({\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2-2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2-3{\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2\\ ={\left(4-5\right)}^2-3·\; <!--\; middle\; dot\; -->\frac{25}{4}\\ =1-\frac{75}{4}\\ =-\frac{71}{4}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, solve

print('9.')

x = symbols('x')
eq = 2 * x ** 2 - 4 * x + 5
a, b = solve(eq, x)
ts = [(a - b) ** 2,
      a ** 3 + b ** 3,
      1 / a + 1 / b,
      a ** 4 - a ** 2 * b ** 2 + b ** 4]

for i, t in enumerate(ts, 1):
    print(f'({i})')
    pprint(t.simplify())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample9.py
9.
(1)
-6

(2)
-7

(3)
4/5

(4)
-71/4

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample9.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[x => 2 * x ** 2 - 4 * x + 5, 'red']];
    
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =