Understanding Definition Based Problems in Complex Numbers

Complex numbers are an extension of the real numbers and are used in many fields of science and engineering. They are particularly useful in solving equations that have no real solutions. In this article, we will explore the definition of complex numbers and how to solve definition-based problems.

What is a Complex Number?

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$. The real part of the complex number is $a$, and the imaginary part is $b$.

Standard Form of a Complex Number

The standard form of a complex number is:

$$ z = a + bi $$

where:

• $z$ is the complex number,
• $a$ is the real part,
• $b$ is the imaginary part,
• $i$ is the imaginary unit.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using the following rules:

Addition

$$ (a + bi) + (c + di) = (a + c) + (b + d)i $$

Subtraction

$$ (a + bi) - (c + di) = (a - c) + (b - d)i $$

Multiplication

$$ (a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i $$

Division

$$ \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} $$

Conjugate of a Complex Number

The conjugate of a complex number $z = a + bi$ is denoted by $\bar{z}$ and is defined as:

$$ \bar{z} = a - bi $$

The conjugate has the property that $z \bar{z} = a^2 + b^2$, which is a real number.

Modulus of a Complex Number

The modulus (or absolute value) of a complex number $z = a + bi$ is denoted by $|z|$ and is defined as:

$$ |z| = \sqrt{a^2 + b^2} $$

Argument of a Complex Number

The argument of a complex number $z = a + bi$ (denoted by $\arg(z)$) is the angle $\theta$ formed by the real axis and the line representing the complex number in the complex plane. It is measured in radians and can be calculated using:

$$ \theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) $$

Polar Form of a Complex Number

A complex number can also be expressed in polar form as:

$$ z = r(\cos \theta + i\sin \theta) $$

where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument.

Euler's Formula

Euler's formula provides a powerful connection between complex numbers and trigonometry:

$$ e^{i\theta} = \cos \theta + i\sin \theta $$

Using Euler's formula, a complex number in polar form can be written as:

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

Solving Definition Based Problems

When solving definition-based problems in complex numbers, it is essential to understand the properties and operations mentioned above. Let's look at some examples to illustrate these concepts.

Example 1: Addition of Complex Numbers

Add the complex numbers $3 + 4i$ and $1 - 2i$.

Solution:

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

Example 2: Multiplication of Complex Numbers

Multiply the complex numbers $2 + 3i$ and $4 - i$.

Solution:

$$ (2 + 3i) \cdot (4 - i) = (2 \cdot 4 - 3 \cdot -1) + (2 \cdot -1 + 3 \cdot 4)i = 8 + 3 + (-2 + 12)i = 11 + 10i $$

Example 3: Modulus and Argument

Find the modulus and argument of the complex number $-1 + \sqrt{3}i$.

Solution:

$$ |-1 + \sqrt{3}i| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 $$

$$ \arg(-1 + \sqrt{3}i) = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} $$

Example 4: Euler's Formula

Express the complex number $z = 5(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})$ using Euler's formula.

Solution:

$$ z = 5e^{i\frac{\pi}{4}} $$

Summary Table

Property/Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(3 + 4i) + (1 - 2i) = 4 + 2i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(5 + 2i) - (3 + i) = 2 + i$
Multiplication	$(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$	$(2 + 3i) \cdot (4 - i) = 11 + 10i$
Division	$\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$	$\frac{1 + i}{1 - i} = \frac{(1 + 1) + (1 - 1)i}{1 + 1} = 1$
Conjugate	$\bar{z} = a - bi$	$\overline{3 + 4i} = 3 - 4i$
Modulus	$	z
Argument	$\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$	$\arg(-1 + \sqrt{3}i) = -\frac{\pi}{3}$
Polar Form	$z = r(\cos \theta + i\sin \theta)$	$5(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})$
Euler's Formula	$e^{i\theta} = \cos \theta + i\sin \theta$	$5e^{i\frac{\pi}{4}}$

By understanding these fundamental concepts and properties of complex numbers, you can solve a wide range of definition-based problems in mathematics and related fields.