Question

Find the square root of following complex numbers
a)1+i​

Answer 1

Answer:

54 is squares root of 1+i

Answer 2

1+I= √1+√(-1)=

Let assume that x+ iy is the root of 1 -i.

Then,

1-i = (x+iy)²

1 -i= x²-y² +i(2xy)

Now comparing real and imaginary part we get

x²-y² = 1 ______(1)

2xy= -1 _______(2)

Now we know

(x² + y²)²=( x²-y²)+ 4x²y²

(x²+y²)²=1+ (2xy)² (using 1)

(x²+y²)²=1+ 1 (using 2)

x²+y² = √2____(3)

Adding 1 & 3.

x² = (1+√2)/2

x = ±√{(1+√2)/2}

Now , y= ±√{ (√2 - 1)/2}

two roots are

√{(1+√2)/2} - i√{ (√2 - 1)/2} or

-√{(1+√2)/2} + i√{ (√2 - 1)/2}

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