maths > complex-number

Properties of Complex Multiplication

    what you'll learn...


 »  Complex Multiplication is closed.
    →  z1×z2

 »  Complex Multiplication is commutative.
    →  z1×z2=z2×z1

 »  Complex Multiplication is associative.
    →  z1×(z2×z3)=(z1×z2)×z3

 »  Complex Multiplicative identity is 1.
    →  z1×1=z1

 »  Complex Multiplicative inverse of z is 1z.
    →  z×1z=1

 »  Complex Multiplication is distributive over complex addition.
    →  z1×(z2+z3)=z1×z2+z1×z3

 »  Modulus of product is product of modulus.
    →  |z1×z2|=|z1|×|z2|

 »  argument of product is sum of arguments of multiplicand and multiplier.
    →  arg(z1×z2)=arg(z1)+arg(z2)

closure law

closure property illustration

The word 'closure' means, closed and not open

Given complex numbers z1=a1+b1i and z2=a2+b2i, a1,a2,b1,b2R,


It is noted that a1a2-b1b2 and a1b2+a2b1 are real numbers (by closure law of real-number addition, subtraction, and multiplication) and so, z1×z2is a complex number.

Consider z1÷z2

The complex number division is equivalently multiplication by the conjugate. So, z1÷z2=z1×z2¯/|z2|2 is a complex number


Closure property of Multiplication and Division: For any given complex numbers z1,z2, z1×÷z2.

Product or quotient of two complex number is another complex number - closed

Commutative Law

commutative property illustration

The word 'commute' means 'to go to and fro between two places on a regular basis'.

Consider the multiplication and division of complex numbers. z3=z1×z2 given as a3+ib3=(a1a2-b1b2)+i(a2b1+a1b2), where a1,a2,b1,b2.
z4=z2×z1 =(a2a1-b2b1)+i(a1b2+a2b1)

It is noted that z3=z4 as the real and imaginary parts of the two complex numbers are equal by commutative law of real number multiplication.

z1×z2=z2×z1 will have the same

Commutative property is not defined for inverse operations. Division is the inverse of multiplication.

Division is the inverse of multiplication.

The commutative property has to be used in the following way.
    =(1z2)×z1 (commutative property of multiplication)

Given for two complex numbers z1,z2; z1×z2=3.1+i2.9. Find the value of z2×z1.

The answer is '3.1+i2.9'


Commutative Property of Complex Multiplication: for any complex number z1,z2

Complex numbers can be swapped in complex multiplication - commutative.

associative law

Associative property illustration

The word 'Associate' means 'to connect with; to join'.

When three complex numbers are multiplies,
 •  Person A associates the middle complex number with first complex number and, to the result of that, he multiplies the third complex number (z1×z2)×z3
 •  Person B associates the middle complex number with third complex number and to that she multiplies the first complex number z1×(z2×z3).

We can work out this in terms of real and imaginary parts of each number. It will prove that (z1×z2)×z3 =z1×(z2×z3)

Associative property is not applicable to division. Instead, Division is handled as inverse of multiplication and assiciative property of multiplication is applicable. i.e. For (z1÷z2)÷z3, it will be handled (z1×1z2)×1z3, then associative property of multiplication applies.

Given z1,z2, what will be the result of z1×(z2×z1)?

The answer is 'z12×z2'.


Associative property of Complex Multiplication: For Any complex number z1,z2,z3,

Order of multiplication can be changed - associative

multiplicative identity

The word "identical" means "exactly alike; same in every detail"

Consider the complex number 1+i0. The result of multiplying that with a complex number z1=a1+ib1 is 'z1'.

The product is identical to the number being multiplied.

Multiplicative identity is 'an element, when multiplied, will result in product identical to the multiplicand'.

Consider z1=2+3i What will be the product z1×(1+i0)?

The answer is 'z1'


Multiplicative Identity in Complex Numbers: For any complex number z1, there exists 1=1+i0, such that

1=1+i0 is the multiplicative identity.

multiplicative inverse

The word 'inverse' means 'opposite; converse'.

Consider two complex numbers z1=a+ib and z2=1a+ib

z1×z2 is 1 multiplicative identity.


1z or (a-ib)/(a^2+b^2) is the multiplicative inverse.

What is the multiplicative inverse of 2-i?

The answer is '2+i5'


Multiplicative Inverse: For any complex number z=a+ib, there exists 1z=a-iba2+b2 such that

For any complex number z there exists 1z : multiplicative inverse

distributive property

distributive property illustration

The word 'distribute' means 'to share; to spread'

Consider three complex numbers z1,z2,z3, and the following

add first z2+z3 and multiply that by z1. That is, z1×(z2+z3)
multiply z1×z2 and z1×z3 and added the results. That is z1×z2+z1×z3
These two results will be the same".

The complex numbers addition and multiplication is considered to be operations on numerical expressions of real numbers.



Given z1z2=3+4i and z2z3=1-3i what is (z1+z3)z2?

The answer is '4+i'


Distributive Property of Complex Numbers: For any given complex numbers z1,z2,z3

Product with sum of complex numbers is sum of, products with the complex numbers - distributive.

modulus in multiplication

Given z1=a1+ib1 and z2=a2+ib2. The modulus of product |z1×z2| is '|z1|×|z2|'.

Given z1=a1+ib1 and z2=a2+ib2, the modulus of product

Given z1=a1+ib1 and z2=a2+ib2 what is the modulus of quotient |z1÷z2|?

The answer is '|z1|÷|z2|'.


Modulus of Product: For complex numbers z1,z2


Modulus of product is product of modulus.

argument in multiplication

Given z1=a1+ib1 and z2=a2+ib2, the argument of product arg(z1×z2) is 'text(arg)z_1 + text(arg)z_2'.


Given z1=a1+ib1 and z2=a2+ib2, what is the argument of quotient arg(z1÷z2)?

The answer is 'argz1-argz2'.


Argument of Product: For complex numbers z1,z2

argument of product is sum of arguments.


The outline of material to learn "complex numbers" is as follows.

Note : Click here for detailed overview of Complex-Numbers

  →   Complex Numbers in Number System

  →   Representation of Complex Number (incomplete)

  →   Euler's Formula

  →   Generic Form of Complex Numbers

  →   Argand Plane & Polar form

  →   Complex Number Arithmetic Applications

  →   Understanding Complex Artithmetics

  →   Addition & Subtraction

  →   Multiplication, Conjugate, & Division

  →   Exponents & Roots

  →   Properties of Addition

  →   Properties of Multiplication

  →   Properties of Conjugate

  →   Algebraic Identities