6. Prove that:(i) Ji = 1+1V2(ii) V-i = 1-1(iii) Ji +H=J2

1.	6. Prove that:(i) Ji = 1+1V2(ii) V-i = 1-1(iii) Ji +H=J2
Answer» i) ((1 + i) / √2)²= (1 + i)²/ 2= (1 + 2i + i2) / 2= (1 + 2i - 1) / 2= 2i / 2= i. so√i = (1 + i) / √2 ii) suppose z2= -i z is complex and therefore can be represented by x + iy (x+iy)2= -i x2- y2+2xyi = -i xy = -1/2 x2- y2= 0 x = +/-y Therefore (x,y) = (1/√2,-1/√2),(-1/√2,1/√2) And z = (1/√2)(1-i) or (1/√2)(-1+i) iii) for this add i+ii √i +√-i = √2

Answer»

i) ((1 + i) / √2)²= (1 + i)²/ 2= (1 + 2i + i2) / 2= (1 + 2i - 1) / 2= 2i / 2= i.

so√i = (1 + i) / √2

ii) suppose z2= -i

z is complex and therefore can be represented by x + iy

(x+iy)2= -i

x2- y2+2xyi = -i

xy = -1/2

x2- y2= 0

x = +/-y

Therefore (x,y) = (1/√2,-1/√2),(-1/√2,1/√2)

And z = (1/√2)*(1-i) or (1/√2)*(-1+i)

iii) for this add i+ii √i +√-i = √2

Discussion