If we have two complex numbers and then we can multiply them together and then we can multiply them together then we can multiply them together

Example

and and

It is simpler to represent complex multiplication in polar form. If and then and then then

- The magnitude is . Multiplication of two complex number produces magnitude of the product length of the two factors. . Multiplication of two complex number produces magnitude of the product length of the two factors.
- The direction is . Multiplication of two complex number produces sum of angles of the two factors. . Multiplication of two complex number produces sum of angles of the two factors.

Graphically, complex multiplication is represented by polar form representation illustrated below. Geometrically multiplying a complex number
with a complex number
will
*
scale
*
the length of
by the amount of
and then
*
rotate
*
it with amount of
about the origin counterclockwise.
with a complex number
will
*
scale
*
the length of
by the amount of
and then
*
rotate
*
it with amount of
about the origin counterclockwise.
will
*
scale
*
the length of
by the amount of
and then
*
rotate
*
it with amount of
about the origin counterclockwise.
by the amount of
and then
*
rotate
*
it with amount of
about the origin counterclockwise.
and then
*
rotate
*
it with amount of
about the origin counterclockwise.
about the origin counterclockwise.

### Complex Number Calculator

### Properties of Complex Multiplication

- Commutative
- Associative
- Distributive
- Complex number is the identity element of complex multiplication because . is the identity element of complex multiplication because . .
- Factor theorem: the product of two complex number cannot be zero unless at least one of the two is itself a zero. If is zero complex number, then implies either or , or both. is zero complex number, then implies either or , or both. implies either or , or both. or , or both. , or both.
- Multiplication of a real number with a complex number will affect both real and imaginary part. Let be a real number and is a complex number. Then, be a real number and is a complex number. Then, is a complex number. Then,
- Multiplication of imaginary number with a complex number produces which geometrically means a counterclockwise rotation by a right angle. with a complex number produces which geometrically means a counterclockwise rotation by a right angle. produces which geometrically means a counterclockwise rotation by a right angle. which geometrically means a counterclockwise rotation by a right angle.

**
See Also
**
:
Complex Division
,
Complex Addition
,
Complex Subtraction

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