maths > complex-number

Properties of Complex Addition

    what you'll learn...


Understanding Properties of Complex Arithmetic
 »  The properties of real number arithmetic is extended to include i=-1 as a number that cannot be added or multiplied to other real-numbers.Properties of Complex numbers extend on properties of Real numbers.

 »  Complex Addition is closed.
    →  z1+z2

 »  Complex Addition is commutative.
    →  z1+z2=z2+z1

 »  Complex Addition is associative.
    →  z1+(z2+z3)=(z1+z2)+z3

 »  Complex Additive identity is 0.
    →  z1+0=z1

 »  Complex Additive inverse of z is -z
    →  z1+(-z1)=0

 »  Complex modulus of sum is less than or equal to sum of moduluses
    →  |z1+z2||z1|+|z2|

Understanding Properties of Complex Arithmetic

complex number system is :

 •  Real number system extended to include solutions to polynomials.

 •  Real number system added with an additional number -1

 •  Model of amplitude and phase of sine waves and elements interacting with sine waves

To understand properties of the complex numbers, the following are to be learned
 •  Properties of addition and subtraction
 •  Properties of multiplication and division
 •  Properties of exponents and roots
 •  properties like commutative, distributive, associative, etc.

One Need not learn each of these afresh. Instead, just extend the definitions and properties of real numbers as though complex number is a numerical expression.

Complex number is a numerical expression with -1 as an element. When learning rules and properties of complex number, visualize the complex number as a numerical expression of real numbers.


Complex Arithmetic Fundamentals: The properties of real number arithmetic is extended to include -1 as a number that cannot be added or multiplied to other numbers.

Complex arithmetic is an extension of Real numbers arithmetic.

closure law

closure property illustration

The word 'closure' means, closed and not open

The word "span" means "Full extent of something"

The span of complex numbers is
a+ib with a,b

That is, real and imaginary parts taking any real number.

The span of complex numbers or the set of complex numbers is represented as .

When any two complex numbers are added, the result is another complex number.


Closure Property of Addition and Subtraction: For any given complex numbers z1,z2, z1±z2.

Sum or difference of two complex numbers is another complex number - closure property

commutative law

commutative property illustration

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

Consider the addition and subtraction of complex numbers. z3=z1+z2 given as a3+ib3=(a1+a2)+i(b1+b2), where a1,a2,b1,b2.
z4=z2+z1, and a4+ib4=(a2+a1)+i(b2+b1)

Since the real numbers are commutative, a1+a2=a2+a1, and similarly for imaginary part. This proves that z1+z2=z2+z1

The commutative property is not defined for real subtraction, and so not for complex subtraction.

Subtraction is the inverse of addition.

The commutative property has to be used in the following way.
    =(-z2)+z1(commutative property of addition.)

Given 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 addition: for any complex number z1,z2

Complex numbers can be swapped in complex addition - commutative.

Associative Law

Associative property illustration

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

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

If the real and imaginary parts of the numbers are substituted, and associative property of real numbers is applied, it is proven that (z1+z2)+z3 =z1+(z2+z3)

Associative property is not applicable to subtraction, but the property can be used as when subtraction is handled as inverse of addition. For (z1-z2)-z3, it will be considered as (z1+(-z2))+(-z3).

Given z1,z2, what will be the result of z1+z2+z1?

The answer is '2z1+z2'. This algebraic modification uses associative property of addition.


Associative Property of Complex Addition: For Any complex number z1,z2,z3,

Order of addition can be changed - associative.

additive identity

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

Consider the complex number 0+i0. The result of adding that with a complex number z1 gives result as 'z1'. The sum is identical to the complex number being added.

An additive identity is 'an element, when added, will result in sum identical to the addend.'.

Consider z1=e+πi. What will be the sum z1+0?

The answer is 'z1+0=z1=e+πi'


Additive Identity in complex numbers: For any complex number z1, there exists 0=0+i0,0 such that

0=0+i0 is additive identity.

additive inverse

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

Consider two complex numbers z1=a+ib and z2=-a-ib.

= additive identity

-z is called the additive inverse for z as z+(-z)=0.

What is the additive inverse of 3.5+2.1i?

The answer is '-3.5-2.1i'


Additive Inverse: For any complex number z=a+ib, there exists -z=-a-ib such that

For any complex number z there exits -z : additive inverse.

Modulus in addition

Given z1=a1+ib1 and z2=a2+ib2, the modulus of sum |z1+z2| is '|z1|+|z2|'.

|z1+z2||z1|+|z2| is proven with the representation on complex plane. z1,z2,z1+z2 form a triangle. Length of the sides of the triangle are |z1|,|z2|,|z1+z2|. Sum of any two sides of the triangle is greater than the third side.

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

The answer is '|z1|+|z2|'. Subtraction is the inverse of addition. So, |z1-z2| = |z1+(-z2)|. |z1|+|-z2| |z1|+|z2|. Note : |-z2|=|z2|


Modulus of Sum: For complex numbers z1,z2

Modulus of sum is less than or equal to the sum of modulus.


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