Question

Q11) find the modulus and amplitude of the complex number 1-1​

Answer 1

Answer:

Let a complex number (z) = 1/1+i

\begin{gathered}z= \frac{1(1-i)}{(1+i)(1-i)} \\ z= \frac{1-i}{ 1^{2} - i^{2} } \\ z= \frac{1-i}{1-(-1)} = \frac{1-i}{1+1 } \\ z= \frac{1-i}{2} \end{gathered}

z=

(1+i)(1−i)

1(1−i)

z=

1

2

−i

2

1−i

z=

1−(−1)

1−i

=

1+1

1−i

z=

2

1−i

now,

z = 1/2 -i/2

so real part of z = x = 1/2

Imaginary part of z = y = 1/2

∴\begin{gathered}|z|= \sqrt{ x^{2} + y^{2} } \\ |z|= \sqrt{ (1/2)^{2} + (1/2)^{2} } \\ |z|= \sqrt{ \frac{2}{ 2^{2} } } = \frac{1}{ \sqrt{2} } \end{gathered}

∣z∣=

x

2

+y

2

∣z∣=

(1/2)

2

+(1/2)

2

∣z∣=

2

2

2

=

2

1

Argument = tan⁻¹(y/x)

Argument = tan^{-1} \frac{1/2}{1/2} = tan^{-1} \frac{1}{1} = \frac{ \pi }{4}Argument=tan

−1

1/2

1/2

=tan

−1

1

1

=

4

π