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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

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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