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CBSE NOTES ⇒ Class 11th ⇒ Mathematics ⇒ 5. Complex Number and Quadratic Equations

## 5. Complex Number and Quadratic Equations

### 7. Algebra of Complex Numbers

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

z1 + z2 = (a+ib)+(c+id) = (a + c) + i (b + d), which is again a complex number.

Ex: z1 =2 + i3 and z2 = 7 + i5

z1+ z2 = (2+7) +i(3+5)

= 9+i8

Alternative Method:

z1+ z2 = 2+i3 +7+i5

= 2+7+i3+i5

= 9+i(3+5)

= 9+i8

(B) Subtraction of Complex Numbers

Given any two complex numbers z1 and z2, the difference z1z2 is defined as follows:

z1z2 = (a+ib) – (c+id) = (a–c)+i(b–d)

For example: z1= 4+i3,  z2 =3 + i7

z1z2 =(4–3)+i(3–7)= 1–i4

Note: {4 and 3are like term and i3 and i7 are another like term}

Alternative Method:

z1z2 =(4+i3) – (3+i7)

=4+i3 – 3 – i7

=4– 3+i3 – i7

=1+i(3 – 7)

=1-i4

(C) Multiplication of Complex numbers:

z1 × z2 = (a +ib) (c+id)

=a(c+id) +ib(c+id)

= ac + iad + ibc +(bd)

For example: z­1= 2+3i, z2=5 +2i

1× z2 = (2+3i)( 5 +2i)

= (2×5 – 3×2)+ i(2×2+3×5)

= 10 – 6 + i(4+15)

= 4 + 19i

Alternative Method:

1× z2 = (2+3i)( 5 +2i)

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

= 10 + 4i + 15i + 6i2

= 10 + 4i + 15i + 6(–1)

= 10 – 6 + 19i

= 4 + 19i

(D)Some Important identities:

1. (z1 + z2)2 = z12 + 2 z1 z2 + z22
2. (z1 – z2)2 = z12 – 2 z1 z2 + z22
3. (z1 + z2)3 = z13 + 3 z12 z2 + 3 z1z22 + z23
4. (z1 – z2)3 = z13 – 3 z12 z2 + 3 z1z22 – z23
5. z12 – z22 = (z1 + z2)( z1 – z2)

Important Study materials for classes 06, 07, 08,09,10, 11 and 12. Like CBSE Notes, Notes for Science, Notes for maths, Notes for Social Science, Notes for Accountancy, Notes for Economics, Notes for political Science, Noes for History, Notes For Bussiness Study, Physical Educations, Sample Papers, Test Papers, Mock Test Papers, Support Materials and Books. Mathematics Class - 11th

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## Study Materials List:

##### CBSE NOTES ⇒ Class 11th ⇒ Mathematics
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle of Mathematical Induction
5. Complex Number and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability

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