Menu

A complex number \( z \) is generally written in the form:

\[

z = a + bi

\]

Where \( a \) and \( b \) are real numbers, and \( i \) (the imaginary unit) has the property that \( i^2 = -1 \). The real numbers \( a \) and \( b \) are called the real and imaginary parts of \( z = a + bi \), respectively.

The complex conjugate of \( z \) is denoted by \( \bar{z} \), and it is defined by:

\[

\bar{z} = a – bi

\]

Thus, \( a + bi \) and \( a – bi \) are conjugates of each other.

Two complex numbers are equal if and only if their corresponding real and imaginary parts are equal:

\[

a + bi = c + di \quad \text{if and only if} \quad a = c \text{ and } b = d

\]

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 + di)(c – di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc – ad}{c^2 + d^2}i

\]

Real numbers can be represented by points on a line (the real line), and similarly, complex numbers can be represented by points in the plane (the Argand plane or Gaussian plane). The point \( (a, b) \) in the plane represents the complex number \( z = a + bi \).

The absolute value (or modulus) of a complex number \( z = a + bi \), written \( |z| \), is defined as:

\[

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

\]

A complex number \( z = a + bi \) can also be represented in polar form. Let \( r = |z| \) be the modulus of \( z \) and \( \theta \) be the argument (angle), then:

\[

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

\]

This is often written as:

\[

z = r e^{i \theta}

\]

Where \( r = \sqrt{a^2 + b^2} \) and \( \theta = \text{arg}(z) \).

For any real number \( n \), De Moivre’s theorem states that:

\[

(r e^{i \theta})^n = r^n e^{i n \theta}

\]

This can also be written as:

\[

(\cos \theta + i \sin \theta)^n = \cos (n \theta) + i \sin (n \theta)

\]

The \( n \)-th roots of a complex number \( z = r e^{i \theta} \) are given by:

\[

z^{1/n} = r^{1/n} \left[ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right]

\]

Where \( k = 0, 1, 2, \dots, n-1 \).

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,

– \(b\) is the imaginary part, and

– \(i\) is the imaginary unit, where \(i^2 = -1\).

The set of complex numbers includes both real numbers (when \(b = 0\)) and purely imaginary numbers (when \(a = 0\)).

De Moivre’s Theorem is a fundamental result in complex number theory that relates complex numbers and trigonometry. It states that for any complex number in polar form, \[ z = r (\cos \theta + i \sin \theta), \] and any integer \(n\), the power of the complex number is given by:

\[

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

\]

Where:

– \(r\) is the magnitude (or modulus) of the complex number, \(r = \sqrt{a^2 + b^2}\),

– \(\theta\) is the argument (or angle) of the complex number, \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

**Applications of De Moivre’s Theorem:**

1. Powers of Complex Numbers: De Moivreâ€™s Theorem provides a straightforward way to compute powers of complex numbers in polar form.

2. Roots of Complex Numbers: The theorem is also used to find the roots of complex numbers by solving for \(z^{1/n}\).

3. Trigonometry: It aids in deriving multiple angle identities for sine and cosine functions.

Example:

To find \( (1 + i)^4 \), we first express \( 1 + i \) in polar form:

– \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \),

– \( \theta = \tan^{-1}(1/1) = \frac{\pi}{4} \).

**Using De Moivreâ€™s Theorem:**

\[

(1 + i)^4 = \left(\sqrt{2}\right)^4 \left[\cos(4 \cdot \frac{\pi}{4}) + i\sin(4 \cdot \frac{\pi}{4})\right]

= 4(\cos \pi + i\sin \pi)

= 4(-1 + 0i) = -4.

\]