What is

**COMPLEX NUMBER**?
A number in the form a +

*i*b, where a and b are real numbers and**denotes the imaginary unit.For example √-1,√-2 etc. are imaginary numbers We denote √-1 by the Greek letter ‘***i**i*’, called**iota**and in complex number imaginary number play an important role.**= √-1 ,**

*i*

*i*^{2 }= -1,

*i*^{3 }=

*i*

^{2 }.

*i*

^{1 }= -

*i*,

*i*^{4 }=

*i*

^{2 }

*i*

^{2 }= (-1) (-1) = 1

Imaginary number

**value of***i**i*^{5},*i*^{6},**i**^{7},*i*^{8}till*i*^{n}term all will be expressed in same value like*i*or -1 or –*i*or 1.Example i^{7}will be written i^{4 }. i^{3 }=1.-i = -i |

Now if

*i*

^{17}we can’t do same every time so there is a

**TRICK**. We divide the power of imaginary number by ‘

**’.**

*4*
SO FOLLOW THE STEPS HOW TO SOLVE.

#

**First Step****i**

^{17}you have seen imaginary power is 17 divide it by 4 it will give remainder 1.

#

**Second Step**
Now the 1 you got will be the power of you imaginary number

*i*^{1}and that will be your answer.**CONCLUSION**– If the imaginary power is more than 4 you have to just divide it by 4 and the remainder your new imaginary power and it will be your answer.

Example i^{39} on dividing by 4 we get remainder 3 i^{3} = -i. |

#

**HOW TO FROM COMPLEX NUMBER**
We represent Complex
Number ‘Z’ and if we write Z=a+ib. so this is called a complex number.

Were a and b both are
Real.

A is called real part of
complex number and we represent it by Re(Z).

B is called imaginary
part of complex number and we represent it by Img(Z).So any number is a
complex number.

Any number we can write in form of complex number.

Example Z = 5 So we can write 5 by = 5 + i0 Here Re(5) and Img(0) |

**# PRPORTY OF A COMPLEX NUMBER**

##
** How to add two complex number*

Suppose z

_{1}=a_{1}+ib_{1 }and z_{2}=a_{2}+ib_{2 }
When we add

z

_{1}+ z_{2 }= (a_{1}+ib_{1}) + (a_{2}+ib_{2})**Note**- Real part will add to real part and imaginary part with imaginary part.

(a

_{1}+a_{2}) + i(b_{1}+b_{2})
Now if we have to add three or more complex
number Z

_{3}=a_{3}+ib_{3}
z

_{1}+ z_{2}+z_{3 }= (a_{1}+a_{2}+a_{3}) + i(b_{1}+b_{2}+b_{3}) and same till n^{th }term.Example: (3 + 2i) + (1 + 7i) = (4 + 9i) |

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