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| We know | | We know |
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| <math>\frac{1}{z}=\frac{a}{a^2+b^2}+i \frac{-b}{a^2+b^2}</math> | | <math>\frac{1}{z}=\frac{a}{a^2+b^2}+i \frac{-b}{a^2+b^2}</math> for <math>z=a+ib</math> |
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| Hence | | Hence for <math>z=2-i</math> <math>a=2 ; b=-1</math> |
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| <math>\frac{1}{z}=\frac{a}{a^2+b^2}+i \frac{-b}{a^2+b^2}</math> | | <math>\frac{1}{2-i}=\frac{2}{2^2+(-1)^2}+i \frac{-(-1)}{2^2+(-1)^2}</math> |
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| | <math>\frac{1}{2-i}=\frac{2}{4+1}+i \frac{1}{4+1}=\frac{2}{5}+i \frac{1}{5}=\frac{2+i}{5}</math> |
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| | <math>\frac{z_1}{z_2}=(6+3i) \times \frac{2+i}{5}=\frac{1}{5}(12+6i+6i+3i^2)=\frac{1}{5}(12+12i+3(-1))=\frac{1}{5}(9+12i)</math> |
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| == Power of i == | | == Power of i == |
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| == The square roots of a negative real number == | | == The square roots of a negative real number == |
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).
Difference of two complex numbers
Given any two complex numbers the difference is defined as follows
Example:
Multiplication of two complex numbers
Let and be any two complex numbers.Then the product is defined as follows:
The multiplication of complex numbers satisfy the following properties:
Closure law
The product 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 multiplicative identity
There exists the complex number (denoted as ) called the multiplicative identity, such that for every complex number ,
Existence of multiplicative inverse
For every non-zero complex number , we have the complex number (denoted by or ) called the multiplicative inverse of such that (the multiplicative identity).
Distributive law
For any three complex numbers ,
Division of two complex numbers
Given any two complex numbers where , the quotient is defined by
Example: Let and
We know
for
Hence for
Power of i
The square roots of a negative real number