Complex Numbers
1.0 Definition
1.0 Definition
A number in the form of $z=a+ib$, where $a$ and $b$ are real numbers and $i = \sqrt { - 1} $, is called a complex number $(z)$.
$a$ is denoted by ${R_e}(z)$ i.e., real part of $z$ and $b$ is denoted by $\operatorname{Im} (z)$ i.e., imaginary part of $z$.
Any complex number is
(i) purely real, if $b=0$.
(ii) purely imaginary, if $a=0$.
(iii) imaginary, if $b \ne 0$.
The complex number $0=0+0i$ is both purely real and purely imaginary but not imaginary.
Two complex numbers are said to be equal if and only if their real parts and imaginary parts are separately equal i.e., $$a + ib = c + id \Leftrightarrow a = c{\text{ and }}b = d$$
iota $(i)$ is called the imaginary unit which is equal to $$i = \sqrt { - 1} $$ Squaring both sides, we get $${i^2} = - 1$$ Similarly, $$\begin{equation} \begin{aligned} {i^3} = {i^2}.i = - i \\ {i^4} = {i^2}.{i^2} = - 1. - 1 = 1 \\\end{aligned} \end{equation} $$
From the above equations, we can say that in general, for an integer $n$, $$\begin{equation} \begin{aligned} {i^{4n}} = 1 \\ {i^{4n + 1}} = i \\ {i^{4n + 2}} = - 1 \\ {i^{4n + 3}} = - i \\\end{aligned} \end{equation} $$
Note: Sum of four consecutive powers of $i$ is zero i.e., $$\sum\limits_{k = 0}^3 {{i^{4n + k}}} = 0,{\text{ }}n{\text{ be any integer}}$$
Question 1. Find the value of
(i) ${\left[ {\frac{1}{{{i^{26}}}} + {i^{32}}} \right]^2}$
(ii) ${i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}$
(iii) $\sum\limits_{n = 1}^{13} {{i^n}} $
Solution:
(i) $${\left[ {\frac{1}{{{i^{4 \times 6}}.{i^2}}} + {i^{4 \times 8}}} \right]^2} = {\left[ {\frac{1}{{ - 1}} + 1} \right]^2} = 0$$
(ii) $$\begin{equation} \begin{aligned} {i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}} \\ {i^n}(1 + i + {i^2} + {i^3}) = {i^n}(1 + i - 1 - i) = 0 \\\end{aligned} \end{equation} $$
(iii) $$\begin{equation} \begin{aligned} \sum\limits_{n = 1}^{13} {{i^n}} = i + {i^2} + {i^3} + {i^4} + ... + {i^{13}} \\ {\text{ = }}0 + 0 + 0 + {i^{13}} = {i^{4 \times 3}}.i = i \\\end{aligned} \end{equation} $$
