Question

Express the given complex number in the form of $a + ib\ :\ \left( 1 - i \right)^{4}$

Answer 1

In the given question ,we need to express the given complex number in the form of $a + ib$
Mathematically, a complex number is a number that can be expressed in the form of $a + ib$ where $a$ and $b$are real numbers and $i$ symbol represents the imaginary unit . The set of complex numbers is basically denoted by $C$.
Formula used :
$\left( a – b \right)^{2} = a^{2} + b^{2} – 2ab$

Complete answer: Given $\left( 1 – i \right)^{4}$
$\left( 1 – I \right)^{4} = \left( \left( 1 – i \right)^{2} \right)^{2}$
By expanding,
We get,
$= \left( 1 – i \right)^{2}\left( 1 – i \right)^{2}$
Using the formula, we can expand it
$= {(1}^{2} + i^{2} – 2 \times 1 \times i)(1^{2} + i^{2} – 2 \times 1 \times i)$
$= \left( 1 + i^{2} – 2i \right)\left( 1 + i^{2} – 2i \right)$
$= \left( 1 – 1 – 2i \right)\left( 1 – 1 – 2i \right)$
$= \left( 0 – 2i \right)\left( 0 – 2i \right)$
On further simplifying,
We get,
$= \left( - 2i \right)\left( - 2i \right)$
By multiplying,
We get,
$= 4i^{2}$
By putting $i^{2} = - 1$
$= 4\left( - 1 \right)$
By multiplying,
We get,
$= - 4$
We need to express the value in the form of $a + ib\$
$= - 4 + 0$
$= - 4 + 0i$
Thus $\left( i– 4 \right)^{2} = - 4 + 0i$
**Final answer :
$\left( I – 4 \right)^{2} = - 4 + 0i$ **

Note:
We already know that $i^{2} = - 1\$. Example for Complex number is $2 + 3i$ . Complex number consists of two parts namely the real part and the imaginary part. It is the sum of real numbers and Imaginary numbers. In the general form $a + ib\$ Here $a$ is the Real part and ${ib}$ is the imaginary part. It also helps to find the square root of negative numbers. Imaginary part is denoted by $Im(z)\$ and the real part is denoted by $Re(z)$ .