Question

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

Answer 1

Hint: 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 \[Re(z)\] .