C++ 模板實作復數四則運算：運算符重載與泛型設計實戰

實驗需求

實現復數類別 Complex，支援：

1. 實部/虛部構造復數
2. 加、減、乘、除運算
3. 提取實部、虛部
4. 共軛復數
5. 標準格式輸出 (a+bi) 或 (a-bi)

泛型類別設計

template <class Elem>
class Complex {
private:
    Elem reality;   // 實部
    Elem falsehood; // 虛部
public:
    Complex(Elem r, Elem f) : reality(r), falsehood(f) {}

    // 運算符重載
    Complex operator+(const Complex& c) const {
        return Complex(reality + c.reality, falsehood + c.falsehood);
    }
    Complex operator-(const Complex& c) const {
        return Complex(reality - c.reality, falsehood - c.falsehood);
    }
    Complex operator*(const Complex& c) const {
        return Complex(
            reality * c.reality - falsehood * c.falsehood,
            reality * c.falsehood + falsehood * c.reality
        );
    }
    Complex operator/(const Complex& c) const {
        Elem denom = c.reality * c.reality + c.falsehood * c.falsehood;
        if (denom == 0) {
            cout << "分母不能為 0" << endl;
            return Complex(0, 0);
        }
        return Complex(
            (reality * c.reality + falsehood * c.falsehood) / denom,
            (falsehood * c.reality - reality * c.falsehood) / denom
        );
    }

    // 存取器
    Elem Reality() const { return reality; }
    Elem Falsehood() const { return falsehood; }

    // 共軛
    void Conjugate() { falsehood = -falsehood; }

    // 輸出
    void Output() const {
        if (falsehood >= 0)
            cout << "(" << reality << "+" << falsehood << "i)";
        else
            cout << "(" << reality << falsehood << "i)";
    }
};

測試主程式

int main() {
    Complex<float> c1(1, 2), c2(2, 3);
    cout << "測試結果：" << endl;
    cout << "c1 = (1+2i), c2 = (2+3i)" << endl;

    cout << "c1 + c2 = "; (c1 + c2).Output(); cout << endl;
    cout << "c1 - c2 = "; (c1 - c2).Output(); cout << endl;
    cout << "c1 * c2 = "; (c1 * c2).Output(); cout << endl;
    cout << "c1 / c2 = "; (c1 / c2).Output(); cout << endl;

    cout << "c1 實部: " << c1.Reality() << ", 虛部: " << c1.Falsehood() << endl;
    cout << "c2 實部: " << c2.Reality() << ", 虛部: " << c2.Falsehood() << endl;

    cout << "c1 共軛: "; c1.Conjugate(); c1.Output(); cout << endl;
    cout << "c2 共軛: "; c2.Conjugate(); c2.Output(); cout << endl;
    return 0;
}

關鍵知識點

• 運算符重載：回傳值而非引用，避免臨時對象生命週期問題
• const 正確性：運算符標記 const，參數用 const Complex&
• 復數乘除公式：
  • (a+bi)(c+di) = (ac-bd) + (ad+bc)i
  • (a+bi)/(c+di) = (ac+bd)/(c²+d²) + (bc-ad)/(c²+d²)i
• 模板優勢：支援 float/double/long double 任意精度

改進建議

1. 除法分母檢查應拋出異常而非輸出錯誤
2. 增加 operator==、operator<< 輸出流重載
3. 提供極坐標構造/轉換函數
4. 使用 std::complex 替代生產環境代碼