Complex Numbers

LessonAbout 1.5 h1 of 3 in Advanced Engineering Mathematics

Key Ideas

A complex number combines real and imaginary parts.
Complex numbers have rectangular, polar, trigonometric, and exponential forms.
Euler formulas convert complex values between exponential and trigonometric forms.

Definition of Complex Numbers

A complex number is a number expressible in the standard form \(a + bi\) or \(a + jb\), where \(a\) and \(b\) are real numbers, \(a\) is the real part, \(b\) is the imaginary part, and \(i\) (or \(j\)) represents the imaginary unit \(\sqrt{-1}\).

Real part is denoted as \(a\).
Imaginary part is denoted as \(b\).
Imaginary unit satisfies \(i^2 = -1\).

Imaginary Unit Powers

\[ i = \sqrt{-1}, \qquad\qquad i^2 = -1, \qquad\qquad i^3 = -i, \qquad\qquad i^4 = 1 \]

Example. Simplify the expression \(i^{1997} + i^{1999}\), where \(i\) is the imaginary unit.

Solution. Divide the exponents by \(4\) to find the remainders:

\[ 1997 = 4(499) + 1 \implies i^{1997} = i^1 = i \]

\[ 1999 = 4(499) + 3 \implies i^{1999} = i^3 = -i \]

Substitute the simplified powers back into the expression:

\[ i^{1997} + i^{1999} = i + (-i) = 0 \]

Answer: \(0\)

Forms of Complex Numbers: Rectangular, Trigonometric, Polar, Exponential

A complex number can be expressed in four standard forms depending on whether Cartesian or polar coordinates are utilized in the complex plane.

Rectangular form uses real and imaginary components.
Trigonometric form uses modulus and angle functions.
Polar form uses magnitude and angle shorthand.
Exponential form uses Euler transcendental representation.

Forms and Conversions of Complex Numbers

\[ z = a + jb = r(\cos \theta + j \sin \theta) = r \angle \theta = r e^{j\theta} \]

\[ r = \sqrt{a^2 + b^2}, \qquad\qquad \theta = \arctan\left(\dfrac{b}{a}\right) \]

Example. Convert the complex number \(3 + j4\) into polar form and exponential form.

Solution. Calculate the modulus \(r\) and argument \(\theta\):

\[ r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \]

\[ \theta = \arctan\left(\dfrac{4}{3}\right) \approx 53.13^\circ = 0.9273\text{ rad} \]

Write in polar form \(r \angle \theta\) and exponential form \(r e^{j\theta}\):

Answer: Polar form is \(5 \angle 53.13^\circ\); Exponential form is \(5 e^{j0.9273}\).

Addition and Subtraction of Complex Numbers

Addition and subtraction of complex numbers are performed in rectangular form by combining the real parts together and the imaginary parts together.

Addition and Subtraction Formulas

\[ (a_1 + jb_1) + (a_2 + jb_2) = (a_1 + a_2) + j(b_1 + b_2) \]

\[ (a_1 + jb_1) - (a_2 + jb_2) = (a_1 - a_2) + j(b_1 - b_2) \]

Example. Simplify the expression \((3 - i)^2 - 7(3 - i) + 10\).

Solution. Expand the squared term and distribute the scalar:

\[ (3 - i)^2 = 9 - 6i + i^2 = 9 - 6i - 1 = 8 - 6i \]

\[ -7(3 - i) = -21 + 7i \]

Combine all real and imaginary components:

\[ (8 - 6i) + (-21 + 7i) + 10 = (8 - 21 + 10) + i(-6 + 7) = -3 + i = -(3 - i) \]

Answer: \(-(3 - i)\)

Multiplication of Complex Numbers

Multiplication can be carried out in rectangular form using binomial expansion (\(j^2 = -1\)) or in polar form by multiplying the moduli and adding the arguments.

Multiplication Formulas

\[ (r_1 \angle \theta_1)(r_2 \angle \theta_2) = r_1 r_2 \angle (\theta_1 + \theta_2) \]

\[ (a_1 + jb_1)(a_2 + jb_2) = (a_1 a_2 - b_1 b_2) + j(a_1 b_2 + a_2 b_1) \]

Example. Multiply \((3 - 2i)(4 + 3i)\).

Solution. Expand as a product of binomials:

\[ (3 - 2i)(4 + 3i) = 12 + 9i - 8i - 6i^2 \]

Substitute \(i^2 = -1\) and combine terms:

\[ 12 + i - 6(-1) = 12 + i + 6 = 18 + i \]

Answer: \(18 + i\)

Division of Complex Numbers

Division in rectangular form is accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator. In polar form, divide the moduli and subtract the arguments.

Division Formulas

\[ \dfrac{r_1 \angle \theta_1}{r_2 \angle \theta_2} = \dfrac{r_1}{r_2} \angle (\theta_1 - \theta_2) \]

\[ \dfrac{a_1 + jb_1}{a_2 + jb_2} = \dfrac{(a_1 + jb_1)(a_2 - jb_2)}{a_2^2 + b_2^2} \]

Example. Rationalize and divide \(\dfrac{4 + 3i}{2 - i}\).

Solution. Multiply numerator and denominator by the conjugate \(2 + i\):

\[ \dfrac{4 + 3i}{2 - i} \times \dfrac{2 + i}{2 + i} = \dfrac{8 + 4i + 6i + 3i^2}{4 - i^2} \]

Substitute \(i^2 = -1\):

\[ \dfrac{8 + 10i + 3(-1)}{4 - (-1)} = \dfrac{5 + 10i}{5} = 1 + 2i \]

Answer: \(1 + 2i\)

Complex Number Raised to Exponent n

De Moivre’s Theorem states that raising a complex number in polar form to a power \(n\) raises its modulus to power \(n\) and multiplies its argument by \(n\).

De Moivre’s Theorem for Exponents

\[ (r \angle \theta)^n = r^n \angle (n\theta) \]

\[ [r(\cos \theta + j \sin \theta)]^n = r^n (\cos n\theta + j \sin n\theta) \]

Example. Find the value of \((1 + i)^5\).

Solution. Convert \(1 + i\) to polar form:

\[ r = \sqrt{1^2 + 1^2} = \sqrt{2}, \qquad\qquad \theta = \arctan(1) = 45^\circ \implies 1 + i = \sqrt{2} \angle 45^\circ \]

Apply De Moivre’s Theorem for \(n = 5\):

\[ (1 + i)^5 = (\sqrt{2})^5 \angle (5 \times 45^\circ) = 4\sqrt{2} \angle 225^\circ \]

Convert back to rectangular coordinates:

\[ 4\sqrt{2}(\cos 225^\circ + i \sin 225^\circ) = 4\sqrt{2}\left(-\dfrac{\sqrt{2}}{2} - i \dfrac{\sqrt{2}}{2}\right) = -4 - 4i = -4(1 + i) \]

Answer: \(-4(1 + i)\)

nth Root of a Complex Number

The \(n\text{th}\) roots of a complex number are \(n\) distinct values having equal modulus \(r^{1/n}\) and arguments symmetrically spaced by \(\dfrac{360^\circ}{n}\).

nth Root Formula

\[ (r \angle \theta)^{1/n} = r^{1/n} \angle \left( \dfrac{\theta + k \cdot 360^\circ}{n} \right), \qquad\qquad k = 0, 1, 2, \dots, n-1 \]

Example. Find the principal \(5\text{th}\) root of \(50(\cos 150^\circ + j \sin 150^\circ)\).

Solution. For the principal root, set \(k = 0\) in the root formula:

\[ [50 \angle 150^\circ]^{1/5} = (50)^{1/5} \angle \left(\dfrac{150^\circ}{5}\right) = 2.1867 \angle 30^\circ \]

Convert to rectangular form:

\[ 2.1867(\cos 30^\circ + j \sin 30^\circ) = 2.1867(0.8660 + j 0.5000) = 1.893 + j 1.093 \approx 1.9 + j 1.1 \]

Answer: \(1.9 + j 1.1\)

Logarithm of a Complex Number

The natural logarithm of a complex number is evaluated by writing the number in exponential form \(z = r e^{j\theta}\) (with \(\theta\) in radians) and applying logarithmic laws.

Complex Logarithm Formula

\[ \ln z = \ln(r e^{j\theta}) = \ln r + j\theta \]

Example. Evaluate \(\ln(3 + j4)\).

Solution. Convert \(3 + j4\) to exponential form:

\[ r = \sqrt{3^2 + 4^2} = 5 \]

\[ \theta = \arctan\left(\dfrac{4}{3}\right) = 53.13^\circ = 53.13^\circ \times \dfrac{\pi}{180^\circ} = 0.9273\text{ rad} \]

Apply the complex natural logarithm formula:

\[ \ln(3 + j4) = \ln(5 e^{j0.9273}) = \ln 5 + j 0.9273 = 1.609 + j 0.927 \]

Answer: \(1.609 + j 0.927\)

Euler’s Identities

Euler’s formula establishes the relationship between trigonometric functions and complex exponential functions.

Euler’s Formulas

\[ e^{j\theta} = \cos \theta + j \sin \theta, \qquad\qquad e^{-j\theta} = \cos \theta - j \sin \theta \]

\[ \cos \theta = \dfrac{e^{j\theta} + e^{-j\theta}}{2}, \qquad\qquad \sin \theta = \dfrac{e^{j\theta} - e^{-j\theta}}{j2} \]

Example. Simplify \(x = 2 e^{j10\pi t} + 2 e^{-j10\pi t}\) at \(t = 1\).

Solution. Substitute \(t = 1\) and factor out \(2\):

\[ x = 2\left(e^{j10\pi} + e^{-j10\pi}\right) \]

Use Euler’s identity \(e^{j\theta} + e^{-j\theta} = 2 \cos \theta\):

\[ x = 2\left[2 \cos\left(10\pi \times \dfrac{180^\circ}{\pi}\right)\right] = 4 \cos(1800^\circ) = 4(1) = 4 \]

Answer: \(4\)

Hyperbolic Functions of Complex Numbers

Hyperbolic functions are defined using exponential functions and relate directly to circular functions of imaginary arguments.

Hyperbolic Function Formulas

\[ \sinh x = \dfrac{e^x - e^{-x}}{2}, \qquad\qquad \cosh x = \dfrac{e^x + e^{-x}}{2}, \qquad\qquad \tanh x = \dfrac{e^x - e^{-x}}{e^x + e^{-x}} \]

\[ \cosh(j\theta) = \cos \theta, \qquad\qquad \sinh(j\theta) = j \sin \theta, \qquad\qquad \tanh(j\theta) = j \tan \theta \]

Example. Evaluate \(\cosh\left(j \dfrac{\pi}{4}\right)\).

Solution. Apply the identity \(\cosh(j\theta) = \cos \theta\):

\[ \cosh\left(j \dfrac{\pi}{4}\right) = \cos\left(\dfrac{\pi}{4}\right) = \cos(45^\circ) = \dfrac{\sqrt{2}}{2} \approx 0.707 \]

Answer: \(0.707\)

Inverse Trigonometric Functions of Complex Numbers

Inverse trigonometric functions for complex variables are expressed in terms of complex logarithmic functions.

Inverse Trigonometric Formulas

\[ \arcsin z = -j \ln\left(jz \pm \sqrt{1 - z^2}\right) \]

\[ \arccos z = -j \ln\left(z \pm \sqrt{z^2 - 1}\right) \]

\[ \arctan z = -\dfrac{j}{2} \ln\left(\dfrac{1 + jz}{1 - jz}\right) \]

Example. Express \(\arcsin(\cos x)\) in simplified form.

Solution. Use the trigonometric cofunction identity \(\cos x = \sin\left(\dfrac{\pi}{2} - x\right)\):

\[ \arcsin(\cos x) = \arcsin\left(\sin\left(\dfrac{\pi}{2} - x\right)\right) = \dfrac{\pi}{2} - x \]

Answer: \(\dfrac{\pi}{2} - x\)

Inverse Hyperbolic Functions of Complex Numbers

Inverse hyperbolic functions are logarithmic functions that solve for arguments in hyperbolic relationships.

Inverse Hyperbolic Formulas

\[ \operatorname{arcsinh} z = \ln\left(z \pm \sqrt{z^2 + 1}\right) \]

\[ \operatorname{arccosh} z = \ln\left(z \pm \sqrt{z^2 - 1}\right) \]

\[ \operatorname{arctanh} z = \dfrac{1}{2} \ln\left(\dfrac{1 + z}{1 - z}\right) \]

Example. Evaluate \(\operatorname{arctanh}(0.5)\).

Solution. Substitute \(z = 0.5\) into the inverse hyperbolic tangent formula:

\[ \operatorname{arctanh}(0.5) = \dfrac{1}{2} \ln\left(\dfrac{1 + 0.5}{1 - 0.5}\right) = \dfrac{1}{2} \ln\left(\dfrac{1.5}{0.5}\right) = \dfrac{1}{2} \ln(3) \approx 0.549 \]

Answer: \(0.549\)

Formula Summary

Formula Name	Formula
Imaginary Unit Powers	\(i=\sqrt{-1}, \quad i^2=-1, \quad i^3=-i, \quad i^4=1\)
Complex Number Forms	\(z=a+jb=r(\cos \theta+j \sin \theta)=r \angle \theta=r e^{j\theta}\)
Modulus and Argument	\(r=\sqrt{a^2+b^2}, \quad \theta=\arctan\left(\dfrac{b}{a}\right)\)
Addition and Subtraction	\((a_1+jb_1) \pm (a_2+jb_2)=(a_1 \pm a_2)+j(b_1 \pm b_2)\)
Polar Multiplication	\((r_1 \angle \theta_1)(r_2 \angle \theta_2)=r_1 r_2 \angle (\theta_1+\theta_2)\)
Polar Division	\(\dfrac{r_1 \angle \theta_1}{r_2 \angle \theta_2}=\dfrac{r_1}{r_2} \angle (\theta_1-\theta_2)\)
De Moivre’s Theorem	\((r \angle \theta)^n=r^n \angle (n\theta)\)
nth Root of Complex Number	\((r \angle \theta)^{1/n}=r^{1/n} \angle \left(\dfrac{\theta+k \cdot 360^\circ}{n}\right)\)
Complex Natural Logarithm	\(\ln z=\ln(r e^{j\theta})=\ln r+j\theta\)
Euler’s Formula	\(e^{j\theta}=\cos \theta+j \sin \theta, \quad e^{-j\theta}=\cos \theta-j \sin \theta\)
Exponential Form of Trig Functions	\(\cos \theta=\dfrac{e^{j\theta}+e^{-j\theta}}{2}, \quad \sin \theta=\dfrac{e^{j\theta}-e^{-j\theta}}{j2}\)
Hyperbolic Imaginary Arguments	\(\cosh(j\theta)=\cos \theta, \quad \sinh(j\theta)=j \sin \theta, \quad \tanh(j\theta)=j \tan \theta\)
Inverse Circular Functions	\(\arcsin z=-j \ln\left(jz \pm \sqrt{1-z^2}\right)\)
Inverse Hyperbolic Tangent	\(\operatorname{arctanh} z=\dfrac{1}{2}\ln\left(\dfrac{1+z}{1-z}\right)\)