Complex Numbers: A Brief Overview
What are Complex Numbers?
Complex numbers are a mathematical system that extends the real numbers by including the imaginary unit, denoted by 'i'. This imaginary unit is defined by the property that i² = -1.
Form of a Complex Number
A complex number is typically represented in the form:
- a + bi
where:
- a and b are real numbers.
- a is called the real part.
- b is called the imaginary part.
Operations with Complex Numbers
Addition:
- (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction:
- (a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication:
- (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
Division:
- (a + bi) / (c + di) = [(ac + bd) / (c² + d²)] + [(bc - ad) / (c² + d²)]i
Conjugate of a Complex Number
The conjugate of a complex number a + bi is a - bi. It is denoted by the bar over the number: a̅ + bi̅ = a - bi.
Magnitude or Modulus of a Complex Number
The magnitude or modulus of a complex number a + bi is the distance of the point representing the number from the origin in the complex plane. It is denoted by |a + bi| and is calculated as:
- |a + bi| = √(a² + b²)
Polar Form of a Complex Number
A complex number can also be represented in polar form:
- r(cosθ + i sinθ)
where:
- r is the magnitude of the complex number.
- θ is the angle between the positive x-axis and the line connecting the origin to the point representing the number in the complex plane.
Euler's Formula
Euler's formula provides a connection between exponential functions and trigonometric functions:
- e^(iθ) = cosθ + i sinθ
Applications of Complex Numbers
Complex numbers have numerous applications in various fields, including:
- Electrical engineering (AC circuits)
- Quantum mechanics
- Signal processing
- Control systems
Key Points to Remember
- The imaginary unit i is defined by i² = -1.
- Complex numbers can be represented in rectangular or polar form.
- Operations with complex numbers involve combining real and imaginary parts.
- The conjugate of a complex number is found by changing the sign of the imaginary part.
- The magnitude of a complex number is its distance from the origin in the complex plane.
- Euler's formula provides a connection between exponential and trigonometric functions.