Complex Number

Complex Number

Four form of Complex Number – Rectangular

• Z = X + jY or Z = X + Yj– Trigonometry

• Z = |Z|( Cosθ + j Sinθ)– Euler (exponential)

• Z=|Z| – Polar / Phase

• Z = |Z| < (general math)– time domain, we write : V(t) = Vm Sin (ωt+ )– frequency domain we write V(ωj)=Vm <

Complex Number (Rectangular Form)

• Complex numbers, real and imaginary parts, complex conjugates A complex number is determined by two real numbers, its real and imaginary parts. We write:

• Z=A+Bi OR Z=A+iB (General Math)• Z=A+Bj OR Z=A+jB (Electrical or Mech

Engginering)

• where X and Y are real and i2 or j2 = −1 Same action

for i

• A the real part and B is the imaginary part, and we write A =Re{z}, B =Im{z}.

Complex number presented in Argand diagram

• 1. Commutative Lawz1 + z2 = z2 + z1

z1z2 = z2z1

• 2. Associative Lawz1 + (z2 + z3) = (z1 + z2) + z3

z1(z2z3) = (z1z2)z3

• 3. Distributif Law z1(z2 + z3) = z1z2 + z1z3

Characteristic Complex Number in Algebra

ADDER AND SUBTRACTION

Adder and Subtraction

• Two complex numbers Z1 = A1 + jB1

Z2 = A2 + jB2

• are summed by adding their real parts and imaginary parts to obtain

Z1 + Z2 = (A1+A2 ) + j(B1+B2 )

ANDZ1 - Z2 = (A1-A2 ) + j(B1-B2 )

• The complex number 0+ i 0 or 0 is known as the additive identity element and −Z is known as the additive inverse of Z or inverse of Z with respect to addition.

Example Adder and Subtraction

• Z1= 3 + 2j• Z2= 3 - j

– Adder Z1 + Z2

(3 + 2j) + (3 - j) = (3 + 3) + j(2 + (-1)) = 6 + j

– Subtraction Z1 - Z2

(3 + 2j ) - (3 − j) = (3 − 3) + j(2 – (-1)) = 3j

MULTIPLICATION

Multiplication

• Two complex numbers Z1 = A1 + jB1 and Z2 = A2 + jB2 are multiplied by using the ordinary rules of algebra and z1 · z2 =(A1 + jB1) · (A2 + jB2 )

= A1A2 + jA1B2 + jB1A2 + j2B1B2

=(A1A2 − B1B2)+ j (A1B2 + B1A2)where we have used the fact that j2 =

−1.

Example Multiplication

• Z1 = 2+3j• Z2 = 3+2j• Multiplication Z1.Z2

(2 + 3j).(3 + 2j) = (2.3 – 3.2) + j(2.2+3.3) = (6 – 6) + (4j + 9j) = 13j

DIVISION

Division

• Two complex numbers Z1 = A1 + jB1 and Z2 = A2 + jB2 are divided by using the ordinary rules of algebra and, then we use following technique to obtain the quotient. We multiply by giving complex conjungate:

• Z1/Z2= A1+jB1 x A2 - jB2

A2+jB2 A2- jB2

• Z1/Z2= (A1.A2 + B1.B2 ) + j.(A2 .B1 - A1.B2 )

A22+B2

2

Example

• Divider Z1/Z2

COMPLEX CONJUGATE

Complex Conjugate

• Complex numbers possess a new property compared to real numbers and this is the complex conjugate. If z = a + jb then the complex conjugate is defined as

z’ = a − jb or z = a - jbIt is very useful since the following are real:

z + z’= (a + a) + (jb − jb) = 2az – z’= (a - a) + (jb-(-jb))= 2jb

z.z’= (a + jb)(a − jb) = a2+ jab − jab −(jb)2

= a2- (-1.b2)= a2+b2

The complex number z and its complex conjugate z*.

Example

• If z = (3 + 2i), find z + z*(3 + 2i) + (3 + 2i)* = (3 + 2i) + (3 − 2i)

= 3 + 2i + 3 − 2i= 3 + 3 + 2i − 2i= 6 (2a) prove

ROOT SQUARE

Principal Square Root

• A square root, also called a radical or surd, of  is x a number r such that r2=x.

• that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are  -3 and 3 , since -32 = 32 = 9. Any nonnegative real number  has a unique nonnegative square root ; this is called the principal square root and is written 

or

Square roots of a Complex Number

• Any nonzero complex number Z also has two square roots. For example, using the imaginary unit j , the two square roots of -9 are ± =  ±3j. The principal square root of a number Z is denoted  (as in the positive real case)

• The two square roots of A+jB are R1 = (X +jY) R2 = -(X +jY)

With Y = and X = B/(2.Y)

R2= A2+B2

Example

Calculate the square root of 12 + 16j.Step 1: The given problem is in the form of (A+jB)                R2 = (A² + B²)                   = (12² + 16²)                  = (144 + 256)                  = 400                R = 20 Step 2: For finding Y we have to use the formula.                Y2 = ((R - A) / 2)                   = ((20 - 12)/2)                  = (8 / 2)                   = 4                Y = 2Step 3: Substitute the value of B and Y in X.                X = B / 2Y                   = 16 / 2.2                  = 16 / 4                 X = 4Step 4: To find the square root of 12 + 16i substitute X and Y value in R1 and R2.                R1 = X + jY = 4 + 2j                R2 = -(X + jY) = -4 – 2j

Exercise 1

1. Add or subtract the following complex numbers.(Z1 + Z2 or Z1 - Z2)

a) (1+j) +(3+j) c) (-1+3j) + ½ (2+2j)b) (2+5j) – (2 – 4j) d) ¼ (2 – 6j) – ⅛(8 - 3j)

2. Multiply the following complex numbers. (Z1 . Z2)a) (−1 + 3j).(2 + 2j) c) (2 − i)(3 + 4i)b) (2 − 5j).(8 − 3j) d) (2+4j) + ½ (1+4j)

3. Perform the following divisions (Z1 / Z2)a) c)

b) d)

4. Combine the following complex numbers and their conjugates.

a) If z = (3 + 2j), find z + z∗ c) If z = (3 − 2j), find z.z∗ b) If z = (−1 + 3j), find z - z∗ d) If z = (4 − 3j), find z.z*

5. Find the square root the following complex numbersa) 3 + 4j c) 15 + 20jb) 12 + 9j

TRIGONOMETRY AND POLAR FORM OF COMPLEX NUMBER

Polar Form

• The standard form of a complex number is

but this can be shown to be equivalent to the form

which is called the polar form of a complex number

Basic Trigonometric Functions

• The basic trigonometric functions can be defined in terms of a right triangle. For the angle at one apex of the right triangle the functions can be defined by: