Kalkulus 1 ITS : Bilangan Kompleks #MathBySovia

Introduction

This tutorial provides a comprehensive guide to understanding complex numbers as covered in the Kalkulus 1 course at Institut Teknologi Sepuluh Nopember (ITS). We will explore definitions, operations, and key concepts related to complex numbers, which are essential in calculus and advanced mathematics.

Step 1: Understanding Complex Numbers

• Definition: Complex numbers consist of a real part and an imaginary part, typically expressed in the form ( a + bi ), where ( a ) is the real part, ( b ) is the imaginary part, and ( i ) is the imaginary unit (( i^2 = -1 )).
• Examples:
  • ( 3 + 4i ) (real part: 3, imaginary part: 4)
  • ( -2 - 5i ) (real part: -2, imaginary part: -5)

Step 2: Operations on Complex Numbers

• Addition:

  • To add complex numbers, combine their real and imaginary parts:
  • ( (a + bi) + (c + di) = (a + c) + (b + d)i )

• Subtraction:

  • Similar to addition, subtract the real and imaginary parts:
  • ( (a + bi) - (c + di) = (a - c) + (b - d)i )

• Multiplication:

  • Use the distributive property (FOIL method):
  • ( (a + bi)(c + di) = ac + adi + bci + bdi^2 )
  • Remember that ( i^2 = -1 ), so simplify accordingly.

• Division:

  • To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator:
  • ( \frac{a + bi}{c + di} \times \frac{c - di}{c - di} )

Step 3: Conjugate and Modulus of Complex Numbers

• Conjugate:

  • The conjugate of a complex number ( a + bi ) is ( a - bi ).
  • Useful for simplifying division of complex numbers.

• Modulus:

  • The modulus (or absolute value) of a complex number ( z = a + bi ) is given by:
  • ( |z| = \sqrt{a^2 + b^2} )

Step 4: Complex Plane and Argand Diagram

• Complex Plane:

  • A graphical representation where the x-axis represents the real part and the y-axis represents the imaginary part.

• Plotting:

  • To plot a complex number ( a + bi ), locate the point ( (a, b) ) on the plane.

Step 5: Polar Form of Complex Numbers

• Converting to Polar Form:
  • The polar form is expressed as ( r(\cos \theta + i \sin \theta) ) or ( re^{i\theta} ),
  • Where ( r = |z| ) (modulus) and ( \theta = \tan^{-1}(\frac{b}{a}) ) (argument).

Step 6: De Moivre's Theorem

• Theorem Statement:
  • For a complex number in polar form, ( (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) ).
• Application:
  • Useful for finding powers and roots of complex numbers.

Step 7: Finding Roots of Complex Numbers

• Using De Moivre's Theorem:
  • To find the nth roots of a complex number, use:
  • ( r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) ) for ( k = 0, 1, 2, \ldots, n-1 ).

Conclusion

In this tutorial, we covered the essential concepts of complex numbers, including definitions, operations, conjugates, modulus, and their representation in the complex plane. Understanding these principles is crucial for further studies in calculus and complex analysis. For additional resources, consider exploring the playlists provided in the video description for a deeper dive into calculus topics.