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537. Complex Number Multiplication

Description

A complex number can be represented as a string on the form "real+imaginaryi" where:

  • real is the real part and is an integer in the range [-100, 100].
  • imaginary is the imaginary part and is an integer in the range [-100, 100].
  • i2 == -1.

Given two complex numbers num1 and num2 as strings, return a string of the complex number that represents their multiplications.

 

Example 1:

Input: num1 = "1+1i", num2 = "1+1i"
Output: "0+2i"
Explanation: (1 + i) * (1 + i) = 1 + i2 + 2 * i = 2i, and you need convert it to the form of 0+2i.

Example 2:

Input: num1 = "1+-1i", num2 = "1+-1i"
Output: "0+-2i"
Explanation: (1 - i) * (1 - i) = 1 + i2 - 2 * i = -2i, and you need convert it to the form of 0+-2i.

 

Constraints:

  • num1 and num2 are valid complex numbers.

Solutions

Solution 1: Simulation

Thinking

Complex multiplication is \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\). The input is already a+bi, so four integers suffice.

Strip the trailing i, split on +, apply the formula, and format. No polynomial expansion is required.

We can convert the complex number string into its real part \(a\) and imaginary part \(b\), and then use the formula for complex number multiplication \((a_1 + b_1i) \times (a_2 + b_2i) = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i\) to calculate the result.

The time complexity is \(O(1)\), and the space complexity is \(O(1)\).

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class Solution:
    def complexNumberMultiply(self, num1: str, num2: str) -> str:
        a1, b1 = map(int, num1[:-1].split("+"))
        a2, b2 = map(int, num2[:-1].split("+"))
        return f"{a1 * a2 - b1 * b2}+{a1 * b2 + a2 * b1}i"

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class Solution {
    public String complexNumberMultiply(String num1, String num2) {
        int[] x = parse(num1);
        int[] y = parse(num2);
        int a1 = x[0], b1 = x[1], a2 = y[0], b2 = y[1];
        return (a1 * a2 - b1 * b2) + "+" + (a1 * b2 + a2 * b1) + "i";
    }

    private int[] parse(String s) {
        var cs = s.substring(0, s.length() - 1).split("\\+");
        return new int[] {Integer.parseInt(cs[0]), Integer.parseInt(cs[1])};
    }
}

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class Solution {
public:
    string complexNumberMultiply(string num1, string num2) {
        int a1, b1, a2, b2;
        sscanf(num1.c_str(), "%d+%di", &a1, &b1);
        sscanf(num2.c_str(), "%d+%di", &a2, &b2);
        return to_string(a1 * a2 - b1 * b2) + "+" + to_string(a1 * b2 + a2 * b1) + "i";
    }
};

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func complexNumberMultiply(num1 string, num2 string) string {
    x, _ := strconv.ParseComplex(num1, 64)
    y, _ := strconv.ParseComplex(num2, 64)
    return fmt.Sprintf("%d+%di", int(real(x*y)), int(imag(x*y)))
}

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function complexNumberMultiply(num1: string, num2: string): string {
    const [a1, b1] = num1.slice(0, -1).split('+').map(Number);
    const [a2, b2] = num2.slice(0, -1).split('+').map(Number);
    return `${a1 * a2 - b1 * b2}+${a1 * b2 + a2 * b1}i`;
}

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