Complex numbers

Complex Numbers

Complex numbers are an extension of the real numbers and are used in various fields of mathematics, engineering, and physics. They are essential in solving equations that do not have real solutions and in representing oscillations and waves.

Definition

A complex number is a number of the form:

<math> z = a + bi </math>

where:

• <math>a</math> and <math>b</math> are real numbers,
• <math>i</math> is the imaginary unit, defined by the property <math>i^2 = -1</math>.

In this expression, <math>a</math> is called the real part of <math>z</math>, and <math>b</math> is called the imaginary part of <math>z</math>.

Arithmetic of Complex Numbers

Addition and Subtraction

To add or subtract two complex numbers, simply add or subtract their real and imaginary parts separately:

<math> (a + bi) + (c + di) = (a + c) + (b + d)i </math>

<math> (a + bi) - (c + di) = (a - c) + (b - d)i </math>

Multiplication

The product of two complex numbers is given by:

<math> (a + bi)(c + di) = (ac - bd) + (ad + bc)i </math>

This formula is derived using the distributive property and the fact that <math>i^2 = -1</math>.

Division

To divide one complex number by another, multiply the numerator and the denominator by the conjugate of the denominator:

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

Complex Conjugate

The complex conjugate of a complex number <math>z = a + bi</math> is denoted by <math>\overline{z}</math> and is defined as:

<math> \overline{z} = a - bi </math>

The complex conjugate has the property that <math>z \overline{z} = a^2 + b^2</math>, which is always a non-negative real number.

Modulus and Argument

The modulus of a complex number <math>z = a + bi</math> is denoted by <math>|z|</math> and is defined as:

<math> |z| = \sqrt{a^2 + b^2} </math>

The argument of a complex number is the angle <math>\theta</math> in the complex plane, measured from the positive real axis to the line segment that represents the complex number. It is often denoted as <math>\arg(z)</math>.

Polar Form

A complex number can also be represented in polar form:

<math> z = r(\cos \theta + i \sin \theta) </math>

where <math>r = |z|</math> is the modulus and <math>\theta = \arg(z)</math> is the argument. This can also be written using Euler's formula as:

<math> z = re^{i\theta} </math>

Applications

Complex numbers are used in many areas of science and engineering. They are crucial in the study of electrical engineering, quantum mechanics, and signal processing. In mathematics, they are used in complex analysis, which studies functions of complex variables.

See Also

• Imaginary number
• Real number
• Euler's formula
• Complex plane