Express the following complex numbers in the standard form a + ib: (i) $1/(2 + i)^2$ (ii) (1

Home » Express the following complex numbers in the standard form a + ib: (i) (ii) (1 – i) / (1 + i)

Express the following complex numbers in the standard form a + ib:
(i) $1/(2 + i)^2$
(ii) (1 – i) / (1 + i)

Express the following complex numbers in the standard form a + ib:
(i) $1/(2 + i)^2$
(ii) (1 – i) / (1 + i)

Solution:

(i) $1 /(2+i)^{2}$
Simplify and express in the standard form of $(a+i b)$ ,
$1 /(2+i)^{2}=1 /\left(2^{2}+i^{2}+2(2)(i)\right)$ $=1 /(4-1+4 i)\left[\right.$ since, $\left.i^{2}=-1\right]$
$=1 /(3+4 i)$
Multiplying and dividing with $(3-4 i)$
$\begin{array}{l} =1 /(3+4 i) \times(3-4 i) /(3-4 i)] \\ =(3-4 i) /\left(3^{2}-(4 i)^{2}\right) \\ =(3-4 i) /\left(9-16 i^{2}\right) \\ =(3-4 i) /(9-16(-1))\left[\text { since, } i^{2}=-1\right] \\ =(3-4 i) / 25 \end{array}$
As a result, the values of $a, b$ are $3 / 25,-4 / 25$

(ii) $(1-i) /(1+i)$
Simplify and express in the standard form of $(a + ib)$ ,
$(1 - i) / (1 +i) = (1 - i) / (1+i) \times (1-i)/(1-i)$
Multiplying and dividing with $(1-i)$
$\begin{array}{l} =\left(1^{2}+i^{2}-2(1)(i)\right) /\left(1^{2}-i^{2}\right) \\ =(1+(-1)-2 i) /(1-(-1)) \\ =-2 i / 2 \\ =-i \end{array}$
As a result, the values of $a, b$ are $0,-1$