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Question: Find real and imaginary parts of the complex number \[z = \dfrac{{3{i^{20}} - {i^{19}}}}{{2i - 1}}\]...

Find real and imaginary parts of the complex number z=3i20−i192i−1z = \dfrac{{3{i^{20}} - {i^{19}}}}{{2i - 1}}.

Explanation

Solution

We convert the power of iota (i) in the lowest form. Thereafter, we rationalise the denominator of iota(i).

Formula used: (i)4=1,  i2=−1,  i3=−i{(i)^4} = 1,\,\,{i^2} = - 1,\,\,{i^3} = - i
z=a+ibz = a + ib (standard form of complex Number), with the real part and the imaginary part.

Complete step by step answer:
(1) We have, z=3(i)20−i192i−1z = \dfrac{{3{{(i)}^{20}} - {i^{19}}}}{{2i - 1}}
⇒z=3(i4)5−(i4)4(i)(i)22i−1\Rightarrow z = \dfrac{{3{{({i^4})}^5} - {{({i^4})}^4}\left( i \right){{(i)}^2}}}{{2i - 1}}
⇒z=3(1)5−(1)4×(i)(−1)22i−1\Rightarrow z = \dfrac{{3{{(1)}^5} - {{(1)}^4} \times (i){{( - 1)}^2}}}{{2i - 1}}
⇒z=3+i2i−1       (∵i4=1)\Rightarrow z = \dfrac{{3 + i}}{{2i - 1}}\,\,\,\,\,\,\,(\because {i^4} = 1)
(2) In step 1 simplification, we see that iota(i) comes in denominator.
Therefore on rationalize,
z=3+i2i−1×2i+12i+1z = \dfrac{{3 + i}}{{2i - 1}} \times \dfrac{{2i + 1}}{{2i + 1}} ,z=(3+i)(2i+1)(2i−1)(2i+1)z = \dfrac{{(3 + i)(2i + 1)}}{{(2i - 1)(2i + 1)}}
Using algebraic identity (a+b)(a−b)=a2−b2(a + b)(a - b) = {a^2} - {b^2} in denominator
z=3(2i+1)+i(2i+1)(2i)2−(1)2z = \dfrac{{3(2i + 1) + i(2i + 1)}}{{{{(2i)}^2} - {{(1)}^2}}}
=6i+3+2(i)2+i4(i)2−1= \dfrac{{6i + 3 + 2{{(i)}^2} + i}}{{4{{(i)}^2} - 1}}
6i+3+2(−1)+i4(−1)−1\dfrac{{6i + 3 + 2( - 1) + i}}{{4( - 1) - 1}}
=6i+3−2+1i4(−1)−1= \dfrac{{6i + 3 - 2 + 1i}}{{4( - 1) - 1}}
=7i+1−4−1= \dfrac{{7i + 1}}{{ - 4 - 1}}
=7i+1−5= \dfrac{{7i + 1}}{{ - 5}}
z=−15−i75z = \dfrac{{ - 1}}{5} - i\dfrac{7}{5}
(4) z=−15−i75z = \dfrac{{ - 1}}{5} - i\dfrac{7}{5} is the same form as z=a+ibz = a + ib.
Hence, z=−15−i75z = \dfrac{{ - 1}}{5} - i\dfrac{7}{5} is the required standard form.
(5) On comparing z=−15−i75z = \dfrac{{ - 1}}{5} - i\dfrac{7}{5} with z=a+ibz = a + ib we have
a=−15  and  b=−75a = \dfrac{{ - 1}}{5}\,\,and\,\,b = - \dfrac{7}{5}.
Where is called real part of the complex number and is called imaginary part of complex number

Hence, real part =15 = \dfrac{1}{5}, Imaginary part =−75 = - \dfrac{7}{5}

Additional Information: A complex number is a number that can be expressed in the form a + bi, a{\text{ }} + {\text{ }}bi,\,where a and ba{\text{ }}and{\text{ }}bare real numbers, and iirepresents the imaginary unit, satisfying the equation i2=−1{i^2} = - 1because no real number satisfies this equation, ii is called an imaginary number.

Note: When we are comparing real and imaginary parts then in standard complex numbers, imaginary part is the coefficient of and the part without iota(i) is called real.