Multiplication and division rules for mod and argument of two complex numbers

Multiplication and division rules for mod and argument of two complex numbers

Multiplication Rules for Mod and Argument of Complex Numbers

Modulus Product Rule

• When you multiply two complex numbers, the modulus (think of it as the absolute value) of the resulting complex number is equal to the product of the moduli of the original complex numbers.

• For example, if you have complex numbers z1 and z2, with moduli

  z1

  and

  z2

  , then

  z1*z2

  =

  z1

  *

  z2

  .

Argument Sum Rule

• The argument (also known as the phase or the angle) of the product of two complex numbers is found by adding the arguments of the original complex numbers.
• If the arguments of z1 and z2 are arg(z1) and arg(z2) respectively, then arg(z1*z2) = arg(z1) + arg(z2).
• Be cautious! Arguments are periodic with period 2π, so sometimes we adjust our result to stay in the range from -π to +π.

Division Rules for Mod and Argument of Complex Numbers

Modulus Division Rule

• When you divide one complex number by another, the modulus of the resulting complex number is equal to the modulus of the numerator divided by the modulus of the denominator.

• If z1 and z2 are your complex numbers, then

  z1/z2

  =

  z1

  /

  z2

  .

Argument Subtraction Rule

• The argument of the quotient of two complex numbers is found by subtracting the argument of the denominator from the argument of the numerator.
• If you’re dividing z1 by z2, then arg(z1/z2) = arg(z1) - arg(z2).
• Be cautious! Arguments are periodic with period 2π, so sometimes we adjust our result to stay in the range from -π to +π.