How to add, subtract, multiply, and divide complex numbers
Complex-number arithmetic uses the algebra we already know, together with the rule that i squared equals negative 1. We combine real parts with real parts and imaginary parts with imaginary parts. For division, a conjugate lets us turn the denominator into a real number.
Add and subtract matching parts
For subtraction, distribute the negative sign across both parts of the second number.
The sum of and is . Their difference is .
Multiply the binomials and simplify i squared
The product of two imaginary terms is real: . Multiplication by a real number works by distributing, and multiplication by zero gives zero.
Use a conjugate to make the denominator real
The conjugate of is . Their product is , because the imaginary middle terms cancel. To divide by a nonzero complex number, multiply both numerator and denominator by its conjugate.
Divide seven plus three i by nine minus four i
The result is back in standard form, with a real coefficient for each part.