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Algebraic operation on Complex Numbers - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JMI B.Tech Admission

Quick Facts

  • 21 Questions around this concept.

Solve by difficulty

The equation \small Im\left ( \frac{iz -2}{z - i} \right ) + 1 = 0, z\: \epsilon \; C, z \neq i represents a part of a circle having radius equal to :

Let the complex number z = x + iy be such tha t \frac{2z-3i}{2z+i} is purely imaginary. If x+y^{2}=0,  then y^{4}+y^{2}-y  is equal to :

For two non-zero complex numbers \mathrm{z_1 \: \: and\: \: z_2,} if \mathrm{\operatorname{Re}\left(z_1 z_2\right)=0\: \: and \: \: \operatorname{Re}\left(z_1+z_2\right)=0}, then which of the following are possible?
\mathrm{A. \operatorname{Im}\left(z_1\right)>0 \: and\: \operatorname{Im}\left(z_2\right)>0}
\mathrm{B. \operatorname{Im}\left(z_1\right)<0\: and \: \operatorname{Im}\left(z_2\right)>0}
\mathrm{C. \operatorname{Im}\left(z_1\right)>0 \: and\: \operatorname{Im}\left(z_2\right)<0}
\mathrm{D. \operatorname{Im}\left(z_1\right)<0 \: and\: \operatorname{Im}\left(z_2\right)<0}
Choose the correct answer from the options given below:


 

Concepts Covered - 2

Algebraic operation on Complex Numbers

1. Addition of Two Complex Numbers

z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as 

z1 + z2 = (a + ib) + (c + id) = (a + c) + i(b + d)

For example,  z1= (3 - 4i) and z2 = (2 + 5i), then z1 + z2 is

(3 − 4i) + (2 + 5i) = (3 + 2) + (−4 + 5)i = 5 + i


 

2. Difference of Two Complex Numbers

z1 = a + ib and z2 = c + id be any two complex numbers. Then, the difference z1 - z2 is defined as 

z1 - z2 =  (a + ib) - (c + id) =  (a - c) + i(b - d)

For example,  z1= (-5 + 7i) and z2 = (-11 + 2i), then z1 - z2 is

(−5 + 7i) − (−11 + 2i) = −5 + 7i + 11 − 2i

                                  = −5 + 11 + 7i − 2i

                                  = (−5 + 11) + (7 − 2)i

                                  =  6 + 5i


 

3. Multiplication of Two Complex Numbers

z1 = a + ib and z2 = c + id be any two complex numbers. Then, the multiplication z1・z2 is defined as 

z1・z2 = (a + ib)・ (c + id)

          = ac + iad + ibc + i2bd

          = ac + i(ad + bc) - bd

          = (ac - bd) + i(ad + bc)

For example, z1= (4 + 3i) and z2 = (2 - 5i), then z1・ z2 is

(4 + 3i)(2 − 5i) = 4(2) − 4(5i) + 3i(2) − (3i)(5i)

                        =  8 − 20i + 6i − 15(i2)

                        =  (8 + 15) + (−20 + 6)i

                        = 23 - 14i


 

4. Division of Two Complex Numbers

z1 = a + ib and z2 = c + id (and z2 is non-zero) be any two complex numbers. Then, the division \\\mathrm{\frac{z_1}{z_2}}   is defined as 

 

\\\mathrm{\frac{z_1}{z_2}=\frac{a+ib}{c+id}\cdot \frac{c-id}{c-id}}\\\mathrm{[multiplying \;numerator\;and\;denominator\;by\;c-id\;where\;one\;of\;c\;and\;d\;is\;non-zero]}\\\mathrm{\frac{z_1}{z_2}=\frac{ac-iad+ibc-i^2bd}{c^2-\left(id\right)^2}=\frac{ac+i\left(bc-ad\right)+bd}{c^2-i^2d^2}}\\\mathrm{\frac{z_1}{z_2}=\frac{ac+bd+i\left(bc-ad\right)}{c^2+d^2}}\\\mathrm{\mathbf{\frac{z_1}{z_2}=\frac{ac+bd}{c^2+d^2}+i\frac{bc-ad}{c^2+d^2}}}

 

Properties of Addition of Complex Numbers
  1. Closure law: sum of two complex number is a complex number, i.e. z1 + z2 is a complex number for all complex numbers z1 and z2.
  2. Commutative law : for any two complex numbers  z1 and z2 ,  z1 + z2 = z2 + z1
  3. Associative law : for any three complex numbers  z1, z2 and z3 ,   (z1 + z2) + z3 =   z1 + (z2 + z3)
  4. Additive identity: if the sum of a complex number z1 with another complex number z2 is z1, then z2 is called the additive identity.  We have z + 0 = z = 0 + z, so 0 is additive identity.
  5. Additive inverse : To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. i.e. z + (-z) = 0 (-z is additive inverse).

Study it with Videos

Algebraic operation on Complex Numbers
Properties of Addition of Complex Numbers

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