Given two strings representing two complex numbers. You need to return a string representing their multiplication. Note i2 = -1 according to the definition.

Example 1:
Input: “1+1i”, “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: “1+-1i”, “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.

Note:
The input strings will not have extra blank.
The input strings will be given in the form of a+bi, where the integer a and b will both belong to the range of [-100, 100]. And the output should be also in this form.

The Javascript Function to Compute the Complex Number Multiplication

In Javascript, we can use the split function, which allows us to separate the complex number string into two parts: the real and the imaginary. We can use the slice(0, -1) to remove the last character ‘i’ from the imaginary. Then, we just need to compute the real and imaginary of the result complex number respectively using simple math multiplication law.

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/**
 * @param {string} a
 * @param {string} b
 * @return {string}
 */
var complexNumberMultiply = function(a, b) {
    const process = (x) => {
        x = x.split('+');
        const a = x[0];
        const b = x[1].slice(0, -1);
        return [a, b];
    }
    const aa = process(a);
    const bb = process(b);
    const xx = aa[0] * bb[0] - aa[1] * bb[1];
    const yy = aa[0] * bb[1] + aa[1] * bb[0];
    return `${xx}+${yy}i`;
};

/**
 * @param {string} a
 * @param {string} b
 * @return {string}
 */
var complexNumberMultiply = function(a, b) {
    const process = (x) => {
        x = x.split('+');
        const a = x[0];
        const b = x[1].slice(0, -1);
        return [a, b];
    }
    const aa = process(a);
    const bb = process(b);
    const xx = aa[0] * bb[0] - aa[1] * bb[1];
    const yy = aa[0] * bb[1] + aa[1] * bb[0];
    return `${xx}+${yy}i`;
};

The time and space complexity is both O(1) constant – assuming spliting a complex number string is trivial.

The Complex Number Multiplication in C++

In C++, we can use the sscanf function to extract values from the string by analyzing the string using format/templates. Then the rest are similar.

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class Solution {
public:
    string complexNumberMultiply(string a, string b) {
        int a1, a2, b1, b2;
        sscanf(a.c_str(), "%d+%di", &a1, &a2);
        sscanf(b.c_str(), "%d+%di", &b1, &b2);
        int xx = a1 * b1 - a2 * b2;
        int yy = a1 * b2 + a2 * b1;
        return std::to_string(xx) + "+" + std::to_string(yy) + "i";
    }
};

class Solution {
public:
    string complexNumberMultiply(string a, string b) {
        int a1, a2, b1, b2;
        sscanf(a.c_str(), "%d+%di", &a1, &a2);
        sscanf(b.c_str(), "%d+%di", &b1, &b2);
        int xx = a1 * b1 - a2 * b2;
        int yy = a1 * b2 + a2 * b1;
        return std::to_string(xx) + "+" + std::to_string(yy) + "i";
    }
};

–EOF (The Ultimate Computing & Technology Blog) —