SOLUTION: Prove that √i = (1+i)/√2
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Question 1040725: Prove that √i = (1+i)/√2
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
an imaginary number can be represented by a+bi where a and b are real numbers, then
:
(a+bi)^2 = i
:
(a^2 - b^2) + (2ab)i = 0 + 1i
:
note that i^2 = -1
:
now equate the real and imaginary parts and we have
:
1) a^2 - b^2 = 0
2) 2ab = 1
:
a^2 - b^2 = 0 means that a = + or - b
:
if a = -b then equation 2 becomes -2b^2 = 1, this can not be solve for b a real number
:
so we use a = b and equation 2 becomes 2a^2 = 1 and a = b = 1/square root(2) or a = b = -1/square root(2)
:
*************************************************************************
therefore
:
a+bi = (1/square root(2)) + (1/square root(2))i = (1/square root(2))(1+i)
:
a+bi = (-1/square root(2))(1+i)
*************************************************************************
note that there are two answers to the square root(i)
:
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