Complex Numbers
Real Numbers may be plotted on the Real Number Line.
1
3
−2
−1
𝑒
√2
0
1
2
𝜋
3
Figure 1: Real Number Line
Each Real Number may be interpreted as a Scalar multiplying the Unit Real Number 1.
Imaginary Numbers are the square roots of negative numbers, √−5, √−16, √−1, etc.
So
√−5 = √−1√5
The Unit Imaginary Number is defined as:
𝑗 = √−1
This is the same number that mathematical texts denote as 𝑖. Electrical Engineers use the letter 𝑗 since the
letter 𝑖 is customarily used to represent Electrical Current.
Since 𝑗 = √−1 we conclude:
2
𝑗 2 = (√−1 ) = −1
𝑗 3 = 𝑗𝑗 2 = −𝑗
𝑗 4 = 𝑗 2 𝑗 2 = (−1)(−1) = 1
𝑗 5 = 𝑗𝑗 4 = 𝑗
The reciprocal of the imaginary unit number is:
1
1 𝑗
𝑗
𝑗
=
= 2=
= −𝑗
𝑗
𝑗 𝑗
𝑗
−1
1
1 𝑗
𝑗
𝑗
𝑗
=
= 2=
= =𝑗
−𝑗
−𝑗 𝑗
−𝑗
−(−1) 1
1
Any imaginary number can be expressed as a multiple of the imaginary unit number:
√−5 = √(−1)5 = √−1 √5 = 𝑗√5
√−16 = √(−1)16 = √−1 √16 = 𝑗4
All imaginary numbers may be plotted on the Imaginary Number Line:
𝑗2
𝑗
𝑗0.375
0
−𝑗
−𝑗1.5
−𝑗2
Figure 2: Imaginary Number Line
Each Imaginary Number may be interpreted as a Scalar multiplying the Unit Imaginary Number 𝑗.
2
Complex Numbers have both a Real Part and an Imaginary Part. Any Complex Number can be
displayed on the Complex Plane using Rectangular Notation as shown in Figure 1.
Im
𝑧 = 𝑎 + 𝑗𝑏
𝑗𝑏
𝑎
Re
Figure 3: Rectangular Form of Complex Numbers.
The Rectangular Form of a complex number is
𝑧 = 𝑎 + 𝑗𝑏
where 𝑎 is the Real Part of the Complex Number, 𝑏 is the Imaginary Part of the Complex Number and 𝑗
is the Imaginary Unit (the square root of −1).
Re{𝑧} = Re{𝑎 + 𝑗𝑏} = 𝑎
Im{𝑧} = Im{𝑎 + 𝑗𝑏} = 𝑏
The Real and Imaginary Parts may be positive or negative.
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The same Complex Number may be expressed in Polar Notation as shown in Figure 2.
Im
𝑧 = 𝑎 + 𝑗𝑏
𝑟
𝜃
𝑏
𝑎
Re
Figure 4: Polar Form of Complex Numbers.
The length 𝑟 is called the Magnitude of the Complex Number and the angle 𝜃 is called the Angle (or
Phase Angle) of the Complex Number.
The two forms of a Complex Number are related by the following equations.
Pythagorean Theorem:
𝑟 = |𝑧| = √𝑎2 + 𝑏 2
Definition of tangent:
𝜃 = tan−1 (
𝑏
Im{𝑧}
) = tan−1 (
)
𝑎
Re{𝑧}
Definition of cosine:
𝑎 = 𝑟 cos 𝜃
Definition of sine:
𝑏 = 𝑟 sin 𝜃
The Phase Angle is traditionally measured in Degrees and is always measured with respect to the Positive
Real Axis as shown in Figure 3.
Im
𝑧 = −𝑎 + 𝑗𝑏
𝑟
𝑏
𝜃
𝑎
Re
Figure 5: Measurement of the Phase Angle.
Complex Number Arithmetic: Given two complex numbers
𝑧1 = 𝑎1 + 𝑗𝑏1
𝑧2 = 𝑎2 + 𝑗𝑏2
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Complex Number Addition is defined
𝑧3 = 𝑧1 + 𝑧2
Substituting the numbers 𝑧1 and 𝑧2 reveals:
𝑧3 = 𝑎1 + 𝑗𝑏1 + 𝑎2 + 𝑗𝑏2
The real parts may be grouped together, and the imaginary parts grouped together as follows:
𝑧3 = (𝑎1 + 𝑎2 ) + 𝑗(𝑏1 + 𝑏2 )
The Real Part of the Sum is
Re{𝑧3 } = 𝑎1 + 𝑎2
and the Imaginary Part of the sum is
Im{𝑧3 } = 𝑏1 + 𝑏2
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Complex Number Multiplication is defined
𝑧3 = 𝑧1 𝑧2
Substituting the expressions for 𝑧1 and 𝑧2 :
𝑧3 = (𝑎1 + 𝑗𝑏1 )(𝑎2 + 𝑗𝑏2 )
Multiply the two factors using the laws of algebra:
𝑧3 = 𝑎1 𝑎2 + 𝑎1 𝑗𝑏2 + 𝑗𝑏1 𝑎2 + 𝑗𝑏1 𝑗𝑏2
Recall that 𝑗 2 = −1 and group the real and imaginary parts:
𝑧3 = (𝑎1 𝑎2 − 𝑏1 𝑏2 ) + 𝑗(𝑎1 𝑏2 + 𝑏1 𝑎2 )
The Real Part of the Product is
Re{𝑧3 } = 𝑎1 𝑎2 − 𝑏1 𝑏2
and The Imaginary Part of the product is
Im{𝑧3 } = 𝑎1 𝑏2 + 𝑏1 𝑎2
6
Complex Number Division is defined
𝑧3 =
𝑧3 =
𝑧1
𝑧2
𝑎1 + 𝑗𝑏1
𝑎2 + 𝑗𝑏2
To simplify this expression, multiply the numerator and the denominator by the Complex
Conjugate of the denominator.
𝑧3 =
(𝑎1 + 𝑗𝑏1 )(𝑎2 − 𝑗𝑏2 )
(𝑎2 + 𝑗𝑏2 )(𝑎2 − 𝑗𝑏2 )
Multiply the factors using the laws of algebra:
𝑧3 =
𝑎1 𝑎2 − 𝑎1 𝑗𝑏2 + 𝑗𝑏1 𝑎2 − 𝑗𝑏1 𝑗𝑏2
𝑎2 𝑎2 − 𝑎2 𝑗𝑏2 + 𝑗𝑏2 𝑎2 − 𝑗𝑏2 𝑗𝑏2
Recall that 𝑗 2 = −1 and group the real and imaginary parts:
𝑧3 =
(𝑎1 𝑎2 + 𝑏1 𝑏2 ) + 𝑗(𝑏1 𝑎2 − 𝑎1 𝑏2 )
𝑎22 + 𝑏22
𝑧3 =
𝑎1 𝑎2 + 𝑏1 𝑏2
𝑏1 𝑎2 − 𝑎1 𝑏2
+
𝑗
𝑎22 + 𝑏22
𝑎22 + 𝑏22
The Real Part of the Quotient is
Re{𝑧3 } =
𝑎1 𝑎2 + 𝑏1 𝑏2
𝑎22 + 𝑏22
and the Imaginary Part of the Quotient is
Im{𝑧3 } =
𝑏1 𝑎2 − 𝑎1 𝑏2
𝑎22 + 𝑏22
The Complex Conjugate of a number is found by changing the sign of the Imaginary
Part. The complex conjugate is indicated by an asterisk.
𝑧1 = 𝑎1 + 𝑗𝑏1
𝑧1∗ = 𝑎1 − 𝑗𝑏1
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The Complex Exponential
Consider the complex exponential using 𝑒, the base of the natural logarithm, 𝑒 𝑗𝜃 . This
complex exponential may be expanded using a Taylor Series.
∞
𝑒 𝑗𝜃 = ∑
𝑛=0
( 𝑗𝜃)𝑛
𝑛!
This summation may be expanded;
𝑒 𝑗𝜃 =
( 𝑗𝜃)0 ( 𝑗𝜃)1 ( 𝑗𝜃)2 ( 𝑗𝜃)3 ( 𝑗𝜃)4 ( 𝑗𝜃)5
+
+
+
+
+
+⋯
0!
1!
2!
3!
4!
5!
𝑒 𝑗𝜃 =
𝑒
𝑗𝜃
1 𝑗𝜃 𝜃 2 𝑗𝜃 3 𝜃 4 𝑗𝜃 5
+ −
−
+
+
+⋯
1 1! 2!
3!
4!
5!
𝜃2 𝜃4 𝜃6
𝜃3 𝜃5 𝜃7
= [1−
+
−
+⋯]+𝑗[𝜃 −
+
−
+⋯]
2! 4! 6!
3! 5! 7!
Recall the series expansions
cos 𝜃 = 1 −
𝜃2 𝜃4 𝜃6
+
−
+⋯
2! 4! 6!
𝜃3 𝜃5 𝜃7
sin 𝜃 = 𝜃 −
+
−
+⋯
3! 5! 7!
Thus
𝑒 𝑗𝜃 = cos 𝜃 + 𝑗 sin 𝜃
Recall the rectangular form of the Complex Number
𝑧 = 𝑎 + 𝑗𝑏
Where
𝑎 = 𝑟 cos 𝜃
𝑏 = 𝑟 sin 𝜃
So
𝑧 = 𝑟 cos 𝜃 + 𝑗 𝑟 sin 𝜃
𝑧 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃)
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And the Complex Number may be expressed in Polar Form using the complex
exponential.
𝑧 = 𝑟𝑒 𝑗𝜃
Im
𝑧 = 𝑟𝑒 𝑗𝜃
𝑟
𝜃
𝑏
𝑎
Re
Figure 6: Polar Form of Complex Numbers.
The multiplication and division of complex numbers is more easily carried out in Polar
Form. Given
𝑧1 = 𝑟1 𝑒 𝑗𝜃1
𝑧2 = 𝑟2 𝑒 𝑗𝜃2
Then
𝑧1 𝑧2 = 𝑟1 𝑒 𝑗𝜃1 𝑟2 𝑒 𝑗𝜃2 = 𝑟1 𝑟2 𝑒 𝑗(𝜃1 +𝜃2 )
𝑧1 𝑟1 𝑒 𝑗𝜃1
𝑟1
=
= 𝑒 𝑗(𝜃1 −𝜃2 )
𝑗𝜃
𝑧2
𝑟2 𝑒 2 𝑟2
𝑧1∗ = 𝑟1 𝑒 −𝑗𝜃1
𝑗 = 𝑒 𝑗90
°
1
°
= −𝑗 = 𝑒 −𝑗90
𝑗
Steinmetz Form
A short-hand notation for complex numbers:
𝑧 = 𝑟∠𝜃
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There are four ways to express Complex Numbers:
Rectangular Form:
𝑧 = 𝑎 + 𝑗𝑏
Trigonometric Form:
𝑧 = 𝑟 cos 𝜃 + 𝑗 𝑟 sin 𝜃
Exponential Form:
𝑧 = 𝑟𝑒 𝑗𝜃
Polar Form:
𝑧 = 𝑟∠𝜃
Complex Numbers may be converted between the forms using
Pythagorean Theorem:
𝑟 = √𝑎2 + 𝑏 2
Definition of tangent:
𝜃 = tan−1 (
𝑏
)
𝑎
Definition of cosine:
𝑎 = 𝑟 cos 𝜃
Definition of sine:

𝑏 = 𝑟 sin 𝜃
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