Question

Modulus and argument of the
complex number
number z=1+i/1-i is​

Answer 1

Given ,

The complex number is

• z = (1 + i)/(1 - i)

On simplifying , we get

$\tt \implies \frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}$

$\tt \implies\frac{ {(1 + i)}^{2} }{ {(1)}^{2} - {(i)}^{2} }$

$\tt \implies\frac{ {(1)}^{2} + {(i)}^{2} + 2(1)(i) }{1 - ( - 1)}$

$\tt \implies \frac{1 + ( - 1) + 2i}{2}$

$\tt \implies\frac{2i}{2}$

$\tt \implies i$

The given complex number in standard form is (0 + i)

On comparing with rCosΦ + irSinΦ) , we get

rCosΦ = 0 --- (i)

rSinΦ = 1 --- (ii)

Squaring eq (i) and eq (ii) , we get

(r)²Cos²(Φ) = 0 --- (iii)

(r)²Sin²(Φ) = (1)² --- (iv)

Adding eq (iii) and eq (iv) , we get

1 = (r)²{Cos²(Φ) + Sin²(Φ)}

1 = (r)²{1}

(r)² = 1

r = 1 , because r > 0

Hence , the modulus of complex number is 1

Put the value of r = 1 in eq (i) , we get

(1)CosΦ = 0

CosΦ = 0

CosΦ = Cos(90)

Φ = 90

Hence , the argument or amplitude of complex number is 1 and 90°