REVIEW OF COMPLEX NUMBERS:
Introduction :
- Mathematician L.Euler named a number 'i' as Iota whose square is -1 ie i=√-1 .This Iota or i is defined as imaginery unit.
- It is because of i ,we can interpret the square root of a negative number as a product of a real number with i. example. √-9 =3i
Defination :
Any number that can be written in the form of a+ bi where a,b are real numbers and i=√-1 is called a complex number.
Operations and Examples :
- Addition:(a + bi) + (c + di) = (a + c) + (b + d)i. for example
( 3 + 4i ) + (5 + 6i ) = ( 3+5) + (4 +6) i = 8 + 10i
- Subtraction:(a + bi) - (c + di) = (a - c) + (b - d).for example
( 3 + 4i ) - (5 + 6i ) = ( 3-5) + (4 -6) i = -2 - 2i
- Multiplication:(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad).for example
(3+ 4i) (5 + 6i) = 15 + 20i +18i + 24 i^2 = -9 + 38i
Exponential and Polar Forms of Complex Numbers
The relations e^jθ = cosθ + j sinθ and e^jθ = cosθ – j sinθ are known as the Euler’s identities.
Q)Convert the following complex number to exponential and polar form: a) 3 + j4
Ans ) 3+j4 = √(3^2 + 4^2) e^(j*(inverse tan 4/3)= 5∠53.1°
