Complex Numbers Revision Notes BTER Polytechnic 1st Semester

 

Complex Numbers: Detailed Notes

Complex numbers are a fundamental concept in mathematics and engineering. They allow us to solve equations that do not have real solutions, and they are crucial in areas such as electrical engineering, control systems, signal processing, and quantum mechanics.

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3.1 Definition, Real and Imaginary Parts of a Complex Number

A complex number is a number that can be expressed in the form:

$$z = a + bi$$

Where:

• $a$ is the real part of the complex number, denoted as $\text{Re}(z)$.
• $b$ is the imaginary part of the complex number, denoted as $\text{Im}(z)$.
• $i$ is the imaginary unit, which satisfies $i^2 = -1$.

Examples:

• $5 + 3i$ is a complex number where the real part is $5$ and the imaginary part is $3$.
• $-2 - 4i$ is a complex number where the real part is $-2$ and the imaginary part is $-4$.
• A real number is a complex number with $b = 0$, for example, $7 + 0i = 7$.
• An imaginary number is a complex number with $a = 0$, for example, $0 + 5i$ is purely imaginary.

Notation:

• The set of all complex numbers is denoted as $\mathbb{C}$.
• A complex number $z = a + bi$ can also be represented as $z = a + b i$ in polar or exponential form.

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3.2 Addition, Subtraction, Multiplication, and Division of Complex Numbers

Addition of Complex Numbers:

To add two complex numbers, simply add their corresponding real and imaginary parts. If $z_1 = a + bi$ and $z_2 = c + di$, then:

$$z_1 + z_2 = (a + c) + (b + d)i$$

Example:

$$(3 + 2i) + (1 + 4i) = (3 + 1) + (2 + 4)i = 4 + 6i$$

Subtraction of Complex Numbers:

To subtract two complex numbers, subtract their corresponding real and imaginary parts. If $z_1 = a + bi$ and $z_2 = c + di$, then:

$$z_1 - z_2 = (a - c) + (b - d)i$$

Example:

$$(5 + 3i) - (2 + 4i) = (5 - 2) + (3 - 4)i = 3 - i$$

Multiplication of Complex Numbers:

To multiply two complex numbers, use the distributive property (expand the product) and simplify by using the fact that $i^2 = -1$. If $z_1 = a + bi$ and $z_2 = c + di$, then:

$$z_1 \cdot z_2 = (a + bi)(c + di)$$

Expanding:

$$z_1 \cdot z_2 = ac + adi + bci + bdi^2 = ac + adi + bci - bd$$ $$z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$$

Example:

$$(3 + 2i) \cdot (1 + 4i) = (3 \cdot 1 - 2 \cdot 4) + (3 \cdot 4 + 2 \cdot 1)i = (3 - 8) + (12 + 2)i = -5 + 14i$$

Division of Complex Numbers:

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. If $z_1 = a + bi$ and $z_2 = c + di$, then:

$$\frac{z_1}{z_2} = \frac{a + bi}{c + di}$$

Multiply both the numerator and denominator by the conjugate of $z_2$, which is $c - di$:

$$\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$

The denominator becomes $(c + di)(c - di) = c^2 + d^2$. Now, expand the numerator:

$$(a + bi)(c - di) = ac - adi + bci - bdi^2 = ac + bdi + (bc - ad)i$$

Thus:

$$\frac{z_1}{z_2} = \frac{ac + bd + (bc - ad)i}{c^2 + d^2}$$

Example:

$$\frac{3 + 2i}{1 + 4i} \quad \text{Multiply numerator and denominator by} \quad 1 - 4i$$ $$\frac{(3 + 2i)(1 - 4i)}{(1 + 4i)(1 - 4i)} = \frac{3 - 12i + 2i - 8}{1^2 + 4^2} = \frac{-5 - 10i}{17}$$

Thus, the result is:

$$\frac{-5}{17} - \frac{10}{17}i$$

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3.3 Introduction to De Moivre’s Theorem

De Moivre's theorem provides a formula for raising a complex number to a power when the complex number is in polar form. It is especially useful for simplifying powers of complex numbers and for finding the roots of complex numbers.

Statement of De Moivre’s Theorem:

If $z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus (or absolute value) of the complex number and $\theta$ is the argument (or angle), then for any integer $n$:

$$z^n = r^n \left[ \cos(n\theta) + i \sin(n\theta) \right]$$

This theorem can also be written in exponential form as:

$$z = re^{i\theta}$$

Then:

$$z^n = r^n e^{in\theta} = r^n \left( \cos(n\theta) + i \sin(n\theta) \right)$$

Example:

If $z = 2(\cos 30^\circ + i \sin 30^\circ)$, find $z^3$:

$$z^3 = 2^3 \left[ \cos(3 \times 30^\circ) + i \sin(3 \times 30^\circ) \right]$$ $$z^3 = 8 \left[ \cos 90^\circ + i \sin 90^\circ \right]$$ $$z^3 = 8(0 + i) = 8i$$

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3.4 Application of De Moivre’s Theorem

De Moivre’s theorem has numerous applications, including calculating powers and roots of complex numbers, and solving problems in engineering, especially in signal processing and electrical engineering.

Finding Powers of Complex Numbers:

Using De Moivre’s theorem, we can easily find the power of a complex number in polar form. This is particularly useful when dealing with oscillations or alternating current (AC) circuits.

Finding Roots of Complex Numbers:

De Moivre’s theorem can also be used to find the $n$-th roots of a complex number. If $z = r(\cos \theta + i \sin \theta)$, the $n$-th roots of $z$ are given by:

$$z_k = r^{1/n} \left[ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right], \quad k = 0, 1, 2, \dots, n-1$$

Where $k$ represents the different roots.

Example: Finding the Cube Roots of $8$

To find the cube roots of $8$, express $8$ in polar form:

$$8 = 8 \left( \cos 0^\circ + i \sin 0^\circ \right)$$

Now, applying De Moivre’s theorem to find the cube roots:

$$z_k = 8^{1/3} \left[ \cos \left( \frac{0^\circ + 2k\pi}{3} \right) + i \sin \left( \frac{0^\circ + 2k\pi}{3} \right) \right], \quad k = 0, 1, 2$$

Thus, the three cube roots are:

• $z_0 = 2 \left( \cos 0^\circ + i \sin 0^\circ \right) = 2$
• $z_1 = 2 \left( \cos 120^\circ + i \sin 120^\circ \right) = -1 + \sqrt{3}i$
• $z_2 = 2 \left( \cos 240^\circ + i \sin 240^\circ \right) = -1 - \sqrt{3}i$

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Summary of Key Formulas:

1. Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
2. Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$
3. Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
4. Division: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$
5. De Moivre's Theorem: $z^n = r^n (\cos(n\theta) + i \sin(n\theta))$
6. Roots of Complex Numbers: $z_k = r^{1/n} \left[ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right], \quad k = 0, 1, \dots, n-1$

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These concepts form the foundation for working with complex numbers and are used widely in various fields of engineering and science.