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3 solutions
Same way man!!!
If u take square of both term and then solve it, it gives "i" answer...
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Huh? This is not a 'solve it' question it is a quasi 'simplify' question - rewrite the expression in simplest form with a real denominator. btw - if you square the expression the numerator is -2i and the denominator is 2i so the square of the expression is -1. That is what one gets as (-i)^2 = -1. Nice explanation Nihar.
If we let "?" = X.... Multiply both "sides" by 1+i and get:
1 - i = X + iX
Square both "sides" and get:
(1 - i)(1 - i) = (X + iX)(X + iX)
1 - 2i + i^2 = X^2 + 2iX^2 + (i^2)(X^2)
1 - 2i - 1 = X^2 + 2iX^2 - X^2
-2i = 2iX^2
Divide both "sides" by 2i:
-1 = X^2
X = sqrt(-1)
X = ?, so
X = i
Can someone please explain how this is incorrect? Thanks in advance!
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I did this. Why my result isnt the same?
(1-i)/(1+i)=a
((1-i)^2)/((1+i)^2)=a^2
(1-2i+i^2)/(1+2i+i^2)=a^2
(1-2i-1)/(1+2i-1)=a^2
(-2i)/(2i)=a^2
-1=a^2
a=sqrt(-1)=i PD: Please excuse the test format. :D
When you take the square root of a number, here − 1 , you have to remember that there's a positive AND negative solution: X 2 = − 1 ↔ X = ± − 1
Shit man by mistakely selected a wrong option! :'(
I got 1/i, which is I because I multiplied top & bottom by (1+i)
Multiplying both sides by the conjugate of the denominator works. But here is another cute solution:
Note that 1 = − i 2 . Then z = 1 + i 1 − i = 1 + i − i − i 2 = 1 + i ( − i ) ( 1 + i ) = − i .
Or, draw 1 − i and 1 + i as points in the complex plane and connect them to the origin. It is easy to see that from 1 + i to 1 − i requires a rotation of 9 0 ∘ anti-clockwise, without changing the length. This is precisely what multiplication by − i does. Therefore, z = − i .
Nice solution!
That was simple.
Everyone knows that i 2 = − 1 .
Now,
1 + i 1 − i = ( 1 + i ) ( 1 − i ) ( 1 − i ) 2 = 1 − i 2 1 − 2 i + i 2 = 1 − ( − 1 ) 1 − 2 i − 1 = 1 + 1 − 2 i = 2 − 2 i = − i
1 + i 1 − i = ( 1 + i ) ( 1 − i ) ( 1 − i ) 2 = 1 − i 2 1 − 2 i + i 2 = 1 − ( − 1 ) 1 − 2 i − 1 = 1 + 1 − 2 i = 2 − 2 i = − i