Polytechnic semester 1 maths chapter 1 complex numbers. Study note

 Polytechnic semester 1 maths chapter 1 complex numbers. Study note.

Complex numbers:-

1)Find the real and imaginary part from given complex numbers?

i)z=3i+1

 

Real part of (z), Re(z)=1

Imaginary part of (z), Im(z)=3

ii)z=1+i

Ans:Re(z)=1

        Im(z)=1

iii)z=-3-2i

Ans:Re(z)=-3

        Im(z)=-2

Conjugate of a complex number:-

Q)Find the conjugate of given complex numbers?

1)z=2+3i

        symbol of conjugate is    _

                    _                               z

Answer :    z=2-3i

2)z=-5-3i

         _

Ans: z=-5+3i

(Tip:change the symbol of imaginary part eg:-3i when change it on +3i)

Modulus of a complex number:-

Symbol of nodulus is |z|

                      ______

Equation =√x²+y²

1)find the modulus of z=2+3i

                  ______     _____

Ans:|z|=√2²+3²  =√4+9 =√13

Amplitude of a complex number:-

   θ=tan-¹(y/x)

1)Find the amplitude of z=√3+I

Ans:θ=tan-¹(I/√3)

Polar form (modulus amplitude form)of a complex number:-

1)Find the polar form of z=√3+i

                ________     __

Ans:|z|=√(√3)²+1²=√4=2

θ=tan-¹(1/√3)

Polar form =r(cosθ+isinθ)

2[cos(tan-¹(1/√3))+isin(tan-¹(1/√3))]

Addition of complex number:-

1)add the following

(2+3i), (5i+4)

Ans:(2+3i)+(5i+4)

        (2+4)+i(3+5)

         6+8i

Subtraction of complex number:-

1)subtract (1+i) from (2+3i)

Ans:(2+3i)-(1+i)

        (2-1)+i(3-1)

            1+2i

Multiplication of complex number:-

1)Find the product of 3+3i and (5+2i)

Ans:(3+3i)×(5+2i)=3(5+2i)+3i(5+2i)

                               =3×5+3×2i+3i×5+3i×2i    

                               =15+6i+15i+6i²            [i²=-1]

                               =15+21i+6×-1

                               =15+6+21i

                               =9+21i