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z^4+i=0 equation

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Numerical solution:

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The solution

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 4        
z  + I = 0
z4+i=0z^{4} + i = 0
Simplify
Detail solution
Given the equation
z4+i=0z^{4} + i = 0
Because equation degree is equal to = 4 and the free term = -i complex,
so the real solutions of the equation d'not exist

All other 4 root(s) is the complex numbers.
do replacement:
w=zw = z
then the equation will be the:
w4=−iw^{4} = - i
Any complex number can presented so:
w=reipw = r e^{i p}
substitute to the equation
r4e4ip=−ir^{4} e^{4 i p} = - i
where
r=1r = 1
- the magnitude of the complex number
Substitute r:
e4ip=−ie^{4 i p} = - i
Using Euler’s formula, we find roots for p
isin⁡(4p)+cos⁡(4p)=−ii \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = - i
so
cos⁡(4p)=0\cos{\left(4 p \right)} = 0
and
sin⁡(4p)=−1\sin{\left(4 p \right)} = -1
then
p=πN2−π8p = \frac{\pi N}{2} - \frac{\pi}{8}
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for w
Consequently, the solution will be for w:
w1=−12−24−i24+12w_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
w2=12−24+i24+12w_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
w3=−24+12+i12−24w_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
w4=24+12−i12−24w_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
do backward replacement
w=zw = z
z=wz = w

The final answer:
z1=−12−24−i24+12z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
z2=12−24+i24+12z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
z3=−24+12+i12−24z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
z4=24+12−i12−24z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
The graph
Rapid solution [src] 
            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
z1 = -   /   - - -----  - I*  /   - + ----- 
       \/    2     4        \/    2     4
z1=−12−24−i24+12z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} 
          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
z2 =   /   - - -----  + I*  /   - + ----- 
     \/    2     4        \/    2     4
z2=12−24+i24+12z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} 
            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
z3 = -   /   - + -----  + I*  /   - - ----- 
       \/    2     4        \/    2     4
z3=−24+12+i12−24z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} 
          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
z4 =   /   - + -----  - I*  /   - - ----- 
     \/    2     4        \/    2     4
z4=24+12−i12−24z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} 
z4 = sqrt(sqrt(2)/4 + 1/2) - i*sqrt(1/2 - sqrt(2)/4)
Sum and product of roots [src] 
sum
       ___________          ___________        ___________          ___________          ___________          ___________        ___________          ___________
      /       ___          /       ___        /       ___          /       ___          /       ___          /       ___        /       ___          /       ___ 
     /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  1   \/ 2          /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  1   \/ 2  
-   /   - - -----  - I*  /   - + -----  +   /   - - -----  + I*  /   - + -----  + -   /   - + -----  + I*  /   - - -----  +   /   - + -----  - I*  /   - - ----- 
  \/    2     4        \/    2     4      \/    2     4        \/    2     4        \/    2     4        \/    2     4      \/    2     4        \/    2     4
(24+12−i12−24)+(((−12−24−i24+12)+(12−24+i24+12))+(−24+12+i12−24))\left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) + \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) 
=
0
00 
product
/       ___________          ___________\ /     ___________          ___________\ /       ___________          ___________\ /     ___________          ___________\
|      /       ___          /       ___ | |    /       ___          /       ___ | |      /       ___          /       ___ | |    /       ___          /       ___ |
|     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  1   \/ 2  | |     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  1   \/ 2  |
|-   /   - - -----  - I*  /   - + ----- |*|  /   - - -----  + I*  /   - + ----- |*|-   /   - + -----  + I*  /   - - ----- |*|  /   - + -----  - I*  /   - - ----- |
\  \/    2     4        \/    2     4   / \\/    2     4        \/    2     4   / \  \/    2     4        \/    2     4   / \\/    2     4        \/    2     4   /
(−12−24−i24+12)(12−24+i24+12)(−24+12+i12−24)(24+12−i12−24)\left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) 
=
I
ii 
i
Numerical answer [src] 
z1 = 0.38268343236509 + 0.923879532511287*i
z2 = -0.923879532511287 + 0.38268343236509*i
z3 = -0.38268343236509 - 0.923879532511287*i
z4 = 0.923879532511287 - 0.38268343236509*i
z4 = 0.923879532511287 - 0.38268343236509*i
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