algoadvance

Given two strings representing two complex numbers, you need to return a string representing their product. The complex numbers are in the form “a+bi” where a and b are integers, and i is the imaginary unit.

Clarifying Questions

1. Input Format: Are the inputs guaranteed to be valid complex numbers in the specified format without spaces?
   • Yes, inputs are guaranteed to be in the format “a+bi”.
2. Output Format: Should the output also be formatted as “a+bi”?
   • Yes, the output should be in the same “a+bi” format.
3. Constraints: Are there any specific constraints on the values of a and b?
   • No specific constraints as long as a and b are integers.

Strategy

To solve this problem, you can follow these steps:

1. Parsing the Input: Extract the real and imaginary parts from the input strings.
2. Complex Number Multiplication: Recall that if ( z1 = a + bi ) and ( z2 = c + di ), their product is computed as: [ z1 \times z2 = (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i ]
3. Format the Output: Convert the result back to the string format “a+bi”.

Code

Here’s an implementation in Python:

def complexNumberMultiply(num1: str, num2: str) -> str:
    # Helper function to parse the complex number
    def parse_complex(num):
        real, imaginary = num.split('+')
        return int(real), int(imaginary[:-1])  # remove 'i' at the end

    # Parse the complex numbers
    a, b = parse_complex(num1)
    c, d = parse_complex(num2)
    
    # Perform complex number multiplication
    real_part = a * c - b * d
    imag_part = a * d + b * c
    
    # Format the result back to the "a+bi" form
    return f"{real_part}+{imag_part}i"

# Example usage:
num1 = "1+1i"
num2 = "1+1i"
print(complexNumberMultiply(num1, num2))  # Output: "0+2i"

Explanation

• Parsing:
  • parse_complex splits the input string by ‘+’, converts the real part directly to an integer, and the imaginary part after removing the trailing ‘i’.
• Multiplication:
  • Uses the formula ( (ac - bd) + (ad + bc)i ) to calculate the real and imaginary parts.
• Formatting:
  • Formats the result as required by joining the real and imaginary parts with a ‘+’ and an ‘i’.

Time Complexity

The time complexity of this solution is ( O(1) ) because it involves a constant amount of work irrespective of the input size. Parsing, multiplication, and formatting steps are all constant-time operations given the fixed format of the input.

Try our interview co-pilot at AlgoAdvance.com

• Got blindsided by a question you didn’t expect?

• Spend too much time studying?

• Or simply don’t have the time to go over all 3000 questions?