Difference between revisions of "Algebra of Complex Numbers"
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=== Closure law === | === Closure law === | ||
The sum of two complex numbers is a complex number. | The sum of two complex numbers is a complex number. <math>z_1+z_2</math> is a complex number for all complex numbers <math>z_1</math> and <math>z_2</math>. | ||
=== Commutative law === | === Commutative law === | ||
For any two complex numbers <math>z_1</math> and <math>z_2</math>, <math>z_1+z_2=z_2+z_1</math> | |||
=== Associative law === | === Associative law === | ||
For any three complex numbers <math>z_1 , z_2 ,z_3</math> , <math>(z_1+z_2)+z_3=z_1+(z_2+z_3)</math> | |||
=== Existence of additive identity === | === Existence of additive identity === | ||
There exists the complex number <math>0+i0</math> (denoted as <math>0</math>) called the additive identity or the zero complex number, such that for every complex number <math>z</math> ,<math>z+0=z</math> | |||
=== Existence of additive inverse === | === Existence of additive inverse === | ||
To every complex number <math>z=a+ib</math> , we have the complex number <math>-z=-a+i(-b)</math> called the additive inverse or negative of <math>z</math>. We observe that <math>z+(-z)=0</math> (the additive identity). | |||
== Difference of two complex numbers == | == Difference of two complex numbers == | ||
Revision as of 15:02, 6 November 2023
In this Section, we shall develop the algebra of complex numbers.
Addition of two complex numbers
Let and be any two complex numbers.Then the sum is defined as follows:
which is also a complex number.
Example: Let and . Hence
The addition of complex numbers satisfy the following properties:
Closure law
The sum of two complex numbers is a complex number. is a complex number for all complex numbers and .
Commutative law
For any two complex numbers and ,
Associative law
For any three complex numbers ,
Existence of additive identity
There exists the complex number (denoted as ) called the additive identity or the zero complex number, such that for every complex number ,
Existence of additive inverse
To every complex number , we have the complex number called the additive inverse or negative of . We observe that (the additive identity).