Operations on Complex numbers(worksheet)

[Pages:8]Operations on Complex numbers(worksheet)

A complex number is represented as a+ib where a is real part and b is imaginary part. The complex number a+ib can also be represented as ordered pair(a,b) and plotted as a point in Argand plane. In this plane horizontal axis is called real axis and vertical axis is called imaginary axis.

Addition and Subtraction of complex numbers:

To add or subtract complex numbers , we combine their real parts and imaginary parts separately. Real part is added with real part and imaginary part is combined with imaginary part.

Here are the examples: Example1. Let 1= 3+4i , 2=-5-2i then find iii) 31-42 Solution: 1 + 2= (3+4i)+(-5-2i)

= 3-5+4i-2i => -2+2i ii) 1 - 2 = (3+4i)-(-5-2i)

= 3+4i+5+2i

i) 1 + 2

ii) 1 - 2

= 8+ 6i iii) 31-42 = 3(3+4i)-4(-5-2i)

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

= 9+12i+20+8i => 29+20i

Some facts about i(iota) in complex numbers:

1) i = -1 2) 2 =(-1)2 = -1

3) 3 = 2 = -1*i = -i 4) 4 = (2)2 = (-1)2 = 1 5) 5 = (2)2 i = (-1)2i = i

... ... and so on

Using these facts we can simplify even very large powers of i as

shown below. It is better to break large powers as a multiple of 4 because 4 = 1

Example2. Simplify the following: a) 243

b) 540

240+3 = 240 3

(4)135

(4)60*3= 160 (-)

(1)135

1(-i) = -i

1

)7

(2)3*i

(-1)3 = (-1) i = -i

This way any power of i can be reduced to either 1,-1 or i ,-i

Multiplication of complex numbers: Multiplication of two complex numbers is commutative and associative.

i) Commutative 12=21 ii) Associative (12)3= 1(23)

Multiplication of two complex numbers is done in usual way using FOIL and then simplified to get final result.

Example3. Multiply the given complex numbers

1=-2+i 2=2+3i : 12= (-2+i)(2+3i)

= -2(2)+i(2)-2(3i)+i(3i)

(using FOIL)

= -4+2i-6i+32

= -4-4i+3(-1)

= -4-3-4i => -7-4i

Conjugate of a complex number: Conjugate of a complex number is

obtained by replacing i with ?i. If z=a+ib is a complex number then its conjugate is represented by and is given as = a-ib.

Properties of Conjugate complex number:

1. = z (double conjugate of a complex number is number itself)

2. 1+2 = 1+2

3. 1-2 = 1- 2

4. 12 = 12

5.

(12)

=

1 2

Multiplicative inverse (reciprocal) of a complex number: To find multiplicative inverse of a complex number we multiply the numerator and denominator by conjugate of denominator.

Example4. Find multiplicative inverse of 2+3i .

Solution: Let z= 2+3i

Multiplicative

inverse

would

be

-1

=

1 2+3

To simplify it further we rationalize the denominator. That means

multiply the numerator and denominator with its conjugate 2-3i.

-1

=

1 2+3

2-3 2-3

=

2-3 (2+3)(2-3)

=

2-3 22-(3)2

=

2-3 4-92

=

=

2-3 4+9

=

2 13

-

3 13

2-3 4-9(-1)

Division of two complex numbers: While dividing two complex numbers we rationalize the denominator using conjugate using same process as we used for multiplicative inverse above.

Example5. Perform the operation for given complex numbers and

express the result in standard form a+ib. 1= (1 + )2 2= 3-i

1

2

Solution: First we need to simplify 1.

1= (1 + )2 = (1+i)(1+i) = 1+2i+2 = 1+2i-1 = 2i

1 2

=

2 3-

1 2

=

2 3-

3+ 3+

=

2(3+) (3-)(3+)

=

6+22 32-2

=

6+2(-1) 9-(-1)

=

6-2 10

=

-2 10

+160

=

-1 5

+

3 5

Square root of a complex number: To find square root of a complex

number we use following process.

- Set the square root of given complex number equal to x+iy

-Take square of both sides

- Simplify and set their real and imaginary parts equal to each other.

- Solve for x and y.

Example6: Find the square root of following complex number.

5+12i Solution: Let 5 + 12 = x+iy

(5 + 12)2 = ( + )2 5+12i = 2 + 2 + 22 5+12i = 2 + 2 + (-1)2 5+12i = 2 - 2 +2xyi

Equating real and imaginary parts, we get 5 = 2 - 2 , 12= 2xy

Using identity,(2 + 2)2 = (2 - 2)2 +422 (2 + 2)2 = (5)2 + (12)2 (2 + 2)2 = 169 2+ 2 = 13

We already have 2 - 2 = 5 Solving(adding) these two equations, we get 22 =18

X = ?9 = ?3 2 = 4 => y = ?2 Since we have equation, 12= 2xy which is positive so both x and y

should be of same sign.

Either x=3, y=2 OR x=-3 , y=-2 5 + 12 = ?(3 + 2)

Example7. Find real values of x and y for which the complex numbers

-3+i2 and 2 + + 4 are conjugate of each other. Solution: Since -3+i2 and 2 + + 4 are complex conjugates , so

-3+i2 = 2++4 -3+i2 = 2 + - 4

Equating real and imaginary parts we get,

-3 = 2 + , 2 = -4

From second equation we get y

=

-4 2

and plug in it into first equation

-3

=

2

-

4 2

-32 = 4 -4

4 +3 2-4 =0

(2 + 4)(2 - 1) = 0

Since (2 + 4) doesn't yield any real values of x and y so (2 - 1)=0

2= 1 => x = ?1

Using

y

=

-4 2

,

we

get

y

=

-4

Solutions are either x=1,y=-4 OR x=-1,y=-4

Example8. Express (1 - 2)-3 is the standard form a+ib.

Solution:

(1 - 2)-3

=

1 (1-2)3

=

1 1-83-6+122

=

1 1-8(-)-6+12(-1)

=

1 1+8-6-12

=

1 -11+2

-11-2 -11-2

=

-11-2 (-11)2-(2)2

=

-11-2 121+4

=

-11 125

-

2 125

Practice problems:

For given complex numbers, perform the indicated operation and write answer in standard form a+ib for 1)- 4)

1= 5+4i 2= 4+5i

1) 1 + 2

2) 1 - 2

3) 12

4)

1 2

5) Find multiplicative inverse of z= 5 +3i

6) Find square root of complex number z = -15-8i

7) Find real values of x and y if (x+iy)(2-3i)=4+i

Answers:

1)9+9i

2) 1-i

3) 41i

4)

40 41

-

9 41

5)

5 14

-

3 14

6) ?(1 - 4)

7)

x=

5 13

,

y=

14 13

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