Square Roots of Complex Numbers without trigonometry or exponential functions

 

M, N, D and E do not equal 0 

D and E are real numbers

i * i = - 1

- i * - i = - 1

D + E i = ( M + N i ) ^ 2 = M ^ 2 - N ^ 2 + 2 M N i

D = M ^ 2 - N ^ 2

E = 2 M N

N = E / 2 M

D = M ^ 2 - ( E / 2 M ) ^ 2

You should not multiply or divide both sides of an equation by a variable unless you exclude cases of division by zero but that has already been done in the preconditions already set

D M ^ 2 = M ^ 4 - ( E ^ 2 ) / 4

0 = 4 M ^ 4 - 4 D M ^ 2 - E ^ 2

if a x ^ 2 + b x + c = 0 then x = ( - b +- [ b ^ 2 - 4 a c ] ^ 0.5 ) / ( 2 a )

x = M ^ 2

a  = 4

b = - 4 D

c = - E ^ 2

M ^ 2 = ( 4 D +- [ ( - 4 D ) ^ 2 - 4 * 4 * (- E ^ 2) ] ^ 0.5 ) / ( 2 * 4 )

M ^ 2 = ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8

M = + - [ ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8 ] ^ 0.5

N = +- 0.5 E / [ ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8 ] ^ 0.5

Although there are eight possible solutions for N and four possible solutions for M based on what sign is chosen based on the plus or minus signs, only certain combinations of M and N values work and there should theoritically be exactly two solutions for the square root of D + Ei and not all combinations of solutions for M and N will work so you should check your solutions

if M is a pure imaginary number then N will also be a pure imaginary number and this would not prevent a solution

if M is a real number then N will also be a real number

Example

D + E i = 1 +  i

M =  + - [ ( 4 * 1 +- [ 16 + 16 ] ^ 0.5 ) / 8 ] ^ 0.5

Real solutions for M and N

In this case when M is positive then N is positive and when M is negative then N is negative because both D and E are positive

https://duckduckgo.com/?q=(+(+4+*+1+%2B+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5&ia=calculator

https://duckduckgo.com/?q=0.5+%2F+(+(+(+4+*+1+%2B+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5+)&ia=calculator

M = approximately +- 1.09868411347

N = +- 0.5 / 1.09868411347 approximately

N = approximately +- 0.455089860562

M ^ 2 - N ^ 2 = 1.09868411347 ^ 2 - 0.455089860562 ^ 2 = 1.00000000001 = D = = 1 approximately

2MN = 2*1.09868411347*0.455089860562 = 1 = E approximately

https://duckduckgo.com/?q=2*1.09868411347*0.455089860562&ia=calculator

https://duckduckgo.com/?q=(+1.09868411347+%5E+2+)+-+(+0.455089860562+%5E+2+)&ia=calculator

Pure imaginary solutions for M and N

Flip Flopped Solutions for the absolute values of M and N compared to real solutions

These have the same absolute values as before except M and N are switched and M and N are pure imaginary numbers instead of real numbers.  Only two out of these four solutions work based on choosing the sign correctly.  I recommend simply solving for the real solutions for M and N because figuring out the proper sign may be easier for most people that way.

M = +- 0.455089860562 i

N = +- 1.09868411347 i

https://duckduckgo.com/?q=(+(+4+*+1+-+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5&ia=calculator

https://duckduckgo.com/?q=0.5+%2F+(+(+(+4+*+1+-+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5+)&ia=calculator

Copyright Carl Janssen 2022 March 18