Review of Complex numbers

1

Exponential Form:

𝑧=𝑥+ 𝑖𝑦 𝑧=𝑟 𝑒𝑖𝜙=|𝑧|𝑒𝑖 𝜙Rectangular Form:

Real

Imag

x

y

f

r=|z|

𝑧

𝑅𝑒 (𝑧 )=𝑥

𝐼𝑚 (𝑧 )=𝑦

The real and imaginary parts of a complex number in rectangular form are real numbers:

Real

Imag

x=Re(z)

y=Im(z)

𝑧

𝑥+ 𝑖𝑦=𝑅𝑒 (𝑧 )+𝑖𝐼𝑚(𝑧)

Therefore, rectangular form can be equivalently written as:

Real & Imaginary Parts of Rectangular Form

Real

Imag

x

f

r=|z|

𝑧

The real and imaginary components of exponential form can be found using trigonometry:

cos𝜙=𝑎𝑑𝑗h𝑦𝑝

=𝑥𝑟

Geometry Relating the Forms

y

Real

Imag

f

r=|z|

𝑧

sin 𝜙=𝑜𝑝𝑝h𝑦𝑝

=𝑦𝑟

𝑥=𝑟 cos𝜙

𝑦=𝑟 sin 𝜙

→

→

𝐼𝑚 (𝑧 )=𝑦=𝑟 sin 𝜙=|𝑧|sin 𝜙

Geometry Relating the Forms: Real & Imaginary Parts

Real

Imag

𝜙

r=|z|

𝑧The real and imaginary parts of a complex number can be expressed as follows:

Geometry Relating the Forms: Quadrants

In exponential form, the positive angle, , is always defined from the positive real axis. If the complex number is not in the first quadrant, then the “triangle” has lengths which are negative numbers.

cos𝜃=𝑎𝑑𝑗h𝑦𝑝

=¿ 𝑥∨ ¿𝑟

¿

Real

Imag

x

y

f

r=|z|

𝑧

𝜃

𝑥=−𝑟 cos𝜃=𝑟 cos𝜙Real

Imag

𝑥>0𝑦>0

𝑥>0𝑦<0

𝑥<0𝑦<0

𝑥<0𝑦>0

Real

Imag

x

y

r=|z|

𝑧

𝑟2=|𝑧|2=𝑥2+𝑦 2

Use Pythagorean Theorem

𝑟=|𝑧|=√𝑥2+𝑦2

to find in terms of and :

Geometry Relating the Forms: in terms of and

Geometry Relating the Forms: in terms of and

tan𝜙= 𝑟 sin 𝜙𝑟 cos𝜙

=𝐼𝑚(𝑧)𝑅𝑒(𝑧)

= 𝑦𝑥

Real

Imag

x

y

f

r=|z|

𝑧

tan𝜃=𝑜𝑝𝑝𝑎𝑑𝑗

adj

opp

𝜃

hypUse trigonometry

to find in terms of and

𝜙=tan−1( 𝑦𝑥 )

Summary of Algebraic Relationships between Forms

Real

Imag

x

y

f

r=|z|

𝑧 𝑥=𝑟 cos𝜙

𝑦=𝑟 sin 𝜙

𝑟=|𝑧|=√𝑥2+𝑦2

𝜙=tan−1( 𝑦𝑥 )

𝑒𝑖𝜙=cos𝜙+ 𝑖 sin𝜙

Euler’s Formula

𝑧=𝑥+ 𝑖𝑦

¿𝑟 cos𝜙+𝑖𝑟 sin 𝜙=|𝑧|cos𝜙+𝑖|𝑧|sin 𝜙

)

¿𝑟 𝑒𝑖𝜙=¿ 𝑧∨𝑒𝑖𝜙

Rectangular Form:

Exponential Form:

Consistency argument

If these represent the same thing, then the assumed Euler relationship says:

𝑧=𝑥+ 𝑖𝑦 𝑧=𝑟 𝑒𝑖𝜙=¿ 𝑧∨𝑒𝑖𝜙

11

𝑒𝑖𝜙=exp (𝑖𝜙 )=cos𝜙+𝑖 sin𝜙

Euler’s Formula

𝑒𝑖𝜔 0𝑡=exp (𝑖𝜔0𝑡)=cos𝜔0 𝑡+𝑖 sin𝜔0 𝑡

Can be used with functions:

Addition and subtraction of complexnumbers is easy in rectangular form

12

Addition & Subtraction of Complex Numbers

𝑧1=𝑎+𝑖𝑏 𝑧 2=𝑐+𝑖𝑑

𝑧=𝑧1+𝑧 2=𝑎+𝑖𝑏  +𝑐+𝑖𝑑¿ (𝑎+𝑐)+𝑖 (𝑏+𝑑)  

Addition and subtraction are analogous to vector addition and subtraction

Real

Imag

ab

d

c

𝑧 2

𝑧1

�⃗�1=𝑎 �̂�+𝑏 �̂� �⃗� 2=𝑐 �̂�+𝑑 �̂�

�⃗�= �⃗�1+𝑧 2=(𝑎+𝑐 ) �̂�+(𝑏+𝑑) �̂�

x

y

ab

d

c

�⃗�1

�⃗�1

�⃗� 2

�⃗�

𝑧

Multiplication of Complex Numbers

13

Multiplication of complex numbers is easy in exponential form

𝑧1=𝑟1𝑒𝑖 𝜙 𝑧 2=𝑟2𝑒

𝑖𝜃

𝑧=𝑧1 𝑧2=𝑟1𝑒𝑖 𝜙𝑟 2𝑒

𝑖 𝜃

¿𝑟1𝑟2𝑒𝑖(𝜙+𝜃 )

¿¿ 𝑧 1∨¿ 𝑧 2∨𝑒𝑖(𝜙+𝜃 )

Multiplication by a complex number, , can be thought of as scaling by and rotation by

Real

Imag

𝑧

𝑧𝑟 𝑒𝑖 𝜃

𝜃Magnitude scaled by

Angle rotated counterclockwise by

14

Division of Complex NumbersDivision of complex numbers is

easy in exponential form

𝑧1=𝑟1𝑒𝑖 𝜙 𝑧 2=𝑟2𝑒

𝑖𝜃

𝑧=𝑧1

𝑧2

=𝑟1𝑒

𝑖 𝜙

𝑟2𝑒𝑖 𝜃  

¿𝑟1

𝑟2

𝑒𝑖(𝜙−𝜃 )

¿¿ 𝑧 1∨¿

¿ 𝑧2∨¿𝑒𝑖(𝜙−𝜃 )¿¿

Division of complex numbers is sometimes easy in rectangular form

𝑧=𝑎+𝑖𝑏𝑐+𝑖𝑑

¿𝑎+𝑖𝑏𝑐+ 𝑖𝑑

𝑐− 𝑖𝑑𝑐− 𝑖𝑑

¿𝑎𝑐+𝑏𝑑+𝑖 (𝑏𝑐−𝑎𝑑)

𝑐2+𝑑2

¿ 𝑎𝑐+𝑏𝑑𝑐2+𝑑2 +𝑖

(𝑏𝑐−𝑎𝑑)𝑐2−𝑑2

¿𝑅𝑒(𝑧 )+𝑖𝐼𝑚(𝑧 )¿∨𝑧∨𝑒𝑖(𝜙−𝜃 )

Multiply by 1 using the complex conjugate of the denominator

Complex Conjugate

Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.: change i -i

𝑧=𝑥+ 𝑖𝑦𝑧∗=𝑥−𝑖 𝑦

𝑧=𝑟 𝑒𝑖𝜙

𝑧∗=𝑟 𝑒− 𝑖𝜙

Real

Imag

x

y

f

r=|z|

𝑧

𝑧∗

The complex conjugate is a reflection about the real axis

The product of a complex number and its complex conjugate is REAL.

Common Operations with the Complex Conjugate

Addition of the complex number and its complex conjugate results in a real number

𝑧+𝑧∗=𝑥+ 𝑖𝑦+𝑥−𝑖𝑦¿2 𝑥

𝑧 𝑧∗=𝑟 𝑒𝑖 𝜙𝑟 𝑒−𝑖 𝜙

¿𝑟 2𝑒𝑖(𝜙−𝜙 )

¿𝑟 2

¿¿ 𝑧∨¿2 ¿

Real

Imag

x

y

f

r=|z|

𝑧

𝑧∗

x