1.2.2. Subtraction and division

Subtraction and division are derived from addition and multiplication. Subtraction uses the binary operator '-' and takes the form:

difference = minuend - subtrahend

           dividend
quotient = --------
           divisor
           
quotient = dividend ÷ divisor
           
quotient = dividend / divisor

Given two complex numbers:

x = a + ci
y = b + di 

Subtraction

Subtract real parts and subtract imaginary parts

x - y = (a - c) + (b - d)i

When the imaginary parts are both zero, this becomes:

x - y = a - c

Division

It is not possible to directly divide a complex number. To calculate a quotient (result of division), multiply the dividend (top) and the divisor (bottom) by the complex conjugate of the divisor. The complex conjugate of c + di is c - di. On an Argand diagram, the complex conjugate is a reflection in the real axis.

      a + bi   (a + bi)(c - di)
x/y = ------ = ----------------
      c + di   (c + di)(c - di)

      a·c + b·d + (b·c - a·d)i
    = ------------------------
             c2 + d2

When the imaginary parts are zero for both x and y, the above reduces to:

x/y = (a·c)/c² = a/c

Division by zero is not defined.

The axioms for subtraction and division are the following:

Laws for quotients

-(x/y) = -x/y = x/(-y) = -(-x)/(-y)
(-x)/(-y) = x/y
x/y = z/w if and only if x·w = y·z

Principle of fractions

x/y = (k·x)/(k·y) for any non-zero number k.