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# Imaginary Numbers: Power of i

**Complex Numbers :Power of i**

**We did not define nth root of a certain number a if n is even and a is negative. For instance square root of -9 was not defined . In particular we did not defineSQRT(a) if a is a negative since there is no real number whose square root is negative.Thus we cannot solve an equation such asX^2= -81 using only real numbers.We may extend the real number systemto a larger system called the complex numbers system.To do this we first define the imaginary numberii :**

**i^2 =- 1or i = SQRT(-1)**

**Therefore-i =i^3**

**Sincei^3=( i^2)( i )=( -1) (i ) =-i**

**i^4 =1**

**Sincei^4=( i^2) (i^2)=(-1)(-1)=1**

**i^5 = i**

**Sincei^5=( i^4)(i ) = (1) (i)=i**

**Sample Exercises**

**Number One :Show thati^27=-i**

**i^27 =(i^24 )(i^2)(i)**

**Since i^24=( i^4)^6=1^6=1 and i^2=-1**

**Thereforei^27=(1) (-1)(i)=-i**

**Number Two:Findi^105**

**i^105 =(i^104 )(i)**

**Sincei^104=(i^4)^26 = 1^26=1**

**Threforei^105= ( 1)(i)= i**

**Number Three:Find i^307**

**i^307 =( i^304)( i^2) (i)**

**Sincei^304=(i^4)^76=1^76= 1 andi^2=-1**

**Thereforei^307=(1) (-1) (i)=-i**

**Number Four : Find i^1002**

**i^1002 = ( i^1000) (i^2)**

**Since i^1000= ( i^4)^250=1^250=1andi^2= -1**

**Therefore i^1002=(1) (-1)=-1**

**SOURCE :**

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