数学 - Python - 代数学 - 連立方程式と高次方程式 - 複素数、等式、実部と虚部

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第5章(連立方程式と高次方程式)、問6の解答を求めてみる。

1. $\begin{array}{l} {\begin{cases} a + b = 4 \\ a b = 2 \end{cases} \\ b = 4 - a \\ a (4 - a) = 2 \\ a^{2} - 4 a + 2 = 0 \\ a = 2 \pm \sqrt{2} \\ b = 4 - (2 \pm \sqrt{2}) \\ = 2 \mp \sqrt{2} \end{array}$
  
  （複号同順）
2. $\begin{array}{l} {(a + b i)}^{2} \\ = (a^{2} - b^{2}) + 2 a b i \\ {\begin{cases} a^{2} - b^{2} = 15 \\ 2 a b = - 8 \end{cases} \\ b = - \frac{4}{a} \\ a^{2} - \frac{16}{a^{2}} = 15 \\ a^{4} - 15 a^{2} - 16 = 0 \\ a^{2} = \frac{15 + \sqrt{225 + 64}}{2} \\ = \frac{15 + \sqrt{289}}{2} \\ = \frac{15 + 17}{2} \\ = 16 \\ a = \pm 4 \\ b = - \frac{4}{\pm 4} = \mp 1 \end{array}$
  
  （複号同順）
3. $\begin{array}{l} (a^{2} + 2 a b + 3 b^{2}) + (3 a^{2} - a b - 2 b^{2}) i = 19 \\ {\begin{cases} a^{2} + 2 a b + 3 b^{2} = 19 \\ 3 a^{2} - a b - 2 b^{2} = 0 \end{cases} \\ (3 a + 2 b) (a - b) = 0 \\ 3 a + 2 b = 0 \\ b = - \frac{3}{2} a \\ a^{2} - 3 a^{2} + \frac{27}{4} a^{2} = 19 \\ a^{2} = 4 \\ a = \pm 2 \\ b = \mp 3 \\ a - b = 0 \\ b = a \\ a^{2} + 2 a^{2} + 3 a^{2} = 19 \\ a^{2} = \frac{19}{6} \\ a = b = \pm \sqrt{\frac{19}{6}} \end{array}$
  
  （複号同順）

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, solve, Rational, I, sqrt print('6.') a, b = symbols('a:b', real=True) class TestImaginaryNumbers(TestCase): def test1(self): self.assertEqual( solve(((a + b) + a * b * I) - (4 + 2 * I)), [{a: 2-sqrt(2), b: 2+sqrt(2)}, {a: 2+sqrt(2), b: 2-sqrt(2)}] ) def test2(self): self.assertEqual( solve((a + b * I) ** 2 - (15 - 8 * I)), [{a: -4, b: 1}, {a: 4, b: -1}] ) def test3(self): self.assertEqual( solve(((1 + 3 * I) * a ** 2 + (2 - I) * a * b + (3 - 2 * I) * b ** 2) - 19), [{a: -2, b: 3}, {a: 2, b: -3}, {a: -sqrt(Rational(19, 6)), b: -sqrt(Rational(19, 6))}, {a: sqrt(Rational(19, 6)), b: sqrt(Rational(19, 6))}] ) if __name__ == "__main__": main()

% ./sample6.py -v 6. test1 (__main__.TestImaginaryNumbers) ... ok test2 (__main__.TestImaginaryNumbers) ... ok test3 (__main__.TestImaginaryNumbers) ... ok ---------------------------------------------------------------------- Ran 3 tests in 0.350s OK %

Kamimura's blog

2020年5月10日日曜日

数学 - Python - 代数学 - 連立方程式と高次方程式 - 複素数、等式、実部と虚部

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