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Algebraic operation on Complex Numbers - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • 38 Questions around this concept.

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12i2+i+4i3+2i=

The reciprocal of 3+7i is

What should be added to complex no  2+3i to get 1i

If a,a,b  and b are real numbers, then, (a+iba+ib) will also be real number if:

 

If z×(3+4i)=2+3i, then the value of z is:

For what values of a and b, a2i=b+(a4)i

The real values of X and Y for which the equation (x4+2xi)(3x2+yi)=(35i)+(1+2yi) is satisfied , are 

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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=(34i) and z2=(2+5i), then z1+z2 is

(34i)+(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 z1z2 is defined as

z1z2=(a+ib)(c+id)=(ac)+i(bd)
For example, z1=(5+7i) and z2=(11+2i), then z1z2 is

(5+7i)(11+2i)=5+7i+112i=5+11+7i2i=(5+11)+(72)i=6+5i

3. Multiplication of Two Complex Numbers
z1=a+ib and z2=c+id be any two complex numbers. Then, the multiplication z1z2 is defined as

z1z2=(a+ib)(c+id)=ac+iad+ibc+i2bd=ac+i(ad+bc)bd=(acbd)+i(ad+bc)
For example, z1=(4+3i) and z2=(25i), then z1z2 is

(4+3i)(25i)=4(2)4(5i)+3i(2)(3i)(5i)=820i+6i15(i2)=(8+15)+(20+6)i=2314i

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 z1z2   is defined as 

z1z2=a+ibc+idcidcid [multiplying numerator and denominator by cid where one of c and d is non zero]z1z2=aciad+ibci2bdc2(id)2=ac+i(bcad)+bdc2i2 d2z1z2=ac+bd+i(bcad)c2+d2z1z2=ac+bdc2+d2+ibcadc2+d2

Properties of Addition of Complex Numbers

1. Closure law: The sum of two complex numbers 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 the 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).

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Algebraic operation on Complex Numbers
Properties of Addition of Complex Numbers

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