Numbers which involve .i are called imaginary numbers, and .i is the imaginary unit. We now need to define addtion, subtraction, multiplication, division, and multiplicative and additive identities to complete the definition of the complex number field. The imaginary part of a sum is the sum of the imaginary parts. Subtraction is similarly trivial. The real part of a difference is the difference of the real parts and the imaginary part of a difference is the difference of the imaginary parts.
Multiplication is more complex.
It can be effected by using the famous FOIL method from algebra. Multiply the first terms, add to that the product of the outer, add to that sum the product of the inner, and to that subtotal, add the product of the last. The first term products are real, but the outer and inner both have a linear factor of .i in them. To understand division we first define the complex conjugate of a complex number. The defintion of the squared modulus can be used to factor the sum of two squares into the product of a complex number and its complex conjugate.