Complex number multiplication
by associative, commutative, distributive laws of real numbers, considering as a variable, and applying
Complex Number Conjugate
» Conjuage of
Conjugate means "coupled or related". Conjugate of a complex number makes the number real by addition or multiplication.
Complex Number Division
by multiplicative identity, multiplicative inverse, distributive properties.
Consider two complex numbers and . Then is ''. This is from the associative and distributive laws of real numbers extended to numbers with .
and . What is ?
Given and what is ?
The answer is ''
Multiplication of two complex numbers : For any complex number and
Multiplication of two complex numbers follows numerical expression laws and properties with handled as per the property
In numerical expressions or algebraic expressions, we can manipulate the expressions without modifying the value of the expression
• add and subtract : eg
• multiply and divide : eg :
. These manipulations help to arrive at a different form of expressions or help to solve.
In case of complex numbers, , one modification stands out "convert the complex number to real number". This can be achieved either by addition or multiplication with the number .
For a given complex number , the connected number that give a real number on multiplication is . It is named as conjugate of z and represented as or
The word 'conjugate' means 'coupled; joined ; related in reciprocal or complementary'
Conjugate of a Complex NumberFor a complex number the conjugate of is given as .
Conjugate of a complex number is the number with the same real part and negative of imaginary part.
Consider two complex numbers and . Then, in form is ''
and . What is ?
Given and then is ''.
Division of two complex numbers : For any complex number and
Division of two complex numbers follows numerical expression laws and properties with handled as to arrive at form.
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