Summary of Complex Numbers: Conjugate

TOPICS

Keywords:

• Complex Numbers
• Conjugate
• Real Part
• Imaginary Part
• Conjugate Symbol
• Operations with Conjugates
• Properties of Conjugates

Key Questions:

• What defines the conjugate of a complex number?
• How is the conjugate of a complex number calculated?
• What are the properties involving the conjugate of a complex number in operations?
• How does the conjugate affect the graphical representation of a complex number?

Crucial Topics:

• Definition of a complex number: a + bi, where a is the real part and bi is the imaginary part.
• Conjugate of a complex number: what changes and what remains the same?
• Importance of the conjugate in divisions and in the polar form of complex numbers.

Formulas:

• Conjugate of a complex number: If z = a + bi, then the conjugate of z is \bar{z} = a - bi.
• Product of a complex number by its conjugate: z * \bar{z} = (a + bi)(a - bi) = a^2 + b^2.
• Properties of the conjugate:
  1. \overline{z + w} = \bar{z} + \bar{w}
  2. \overline{z * w} = \bar{z} * \bar{w}
  3. \overline{z/w} = \bar{z} / \bar{w} (when w ≠ 0)
  4. \overline{\overline{z}} = z

NOTES

• Key Terms:

  • Complex Numbers: Form a + bi, where a is the real part and b the imaginary part. The imaginary unit i is such that i^2 = -1.
  • Conjugate: The mirror of a complex number over the real axis. Keeps the real part intact, reverses the sign of the imaginary part.
  • Real and Imaginary Part: In a + bi, a is the real part and bi is the imaginary part of the complex number.

• Main ideas and information:

  • Conjugation in division: The conjugate is vital for performing division between complex numbers, eliminating the imaginary part of the denominator.
  • Graphical representation: The conjugate of a complex number has the same distance to the real axis, but in the opposite semiplane.
  • Conjugate in polar form: In r(cosθ + isenθ), the conjugate is r(cosθ - isenθ).

• Topic Contents:

  • Conjugate of a complex number: Involves changing the sign of the imaginary part. If z = 3 + 4i, then \bar{z} = 3 - 4i.
  • Product of a number by its conjugate: Always results in a real number. The expression (a + bi)(a - bi) will be equal to a^2 + b^2.
  • Properties of the conjugate:
    • Conjugation is distributive with respect to addition and multiplication.
    • The conjugation of a fraction is the fraction of the conjugates.
    • Conjugating twice returns to the original number.

• Examples and Cases:

  • Calculating the conjugate: If z = 5 - 3i, then the conjugate is \bar{z} = 5 + 3i.
  • Using the conjugate in divisions: To divide z = 1 + i by w = 1 - i, multiply by the conjugate of w: (1 + i)/(1 - i) * (1 + i)/(1 + i) = (1 + 2i + i^2)/(1 - i^2) = 2i/2 = i.
  • Verifying the properties:
    • Distributivity: \overline{(1 + i) + (2 - 3i)} = \overline{3 - 2i} = 3 + 2i.
    • Conjugate of the product: \overline{(1 + i)(2 - 3i)} = \overline{2 + i - 3i^2} = 2 + i + 3 = 5 + i.
    • Conjugate of the conjugate: \overline{\overline{1 + i}} = \overline{1 - i} = 1 + i.

SUMMARY

• Summary of the most relevant points:

  • A complex number is represented by a + bi, where a is the real part and bi is the imaginary part.
  • The conjugate of a complex number z = a + bi is \bar{z} = a - bi; it changes the sign of the imaginary part.
  • The product of a complex number by its conjugate always results in a real number, z * \bar{z} = a^2 + b^2.
  • The conjugate is essential in the operations of division of complex numbers to rationalize the denominator.
  • Conjugates have the same radial representation on a graph, but are in opposite semiplanes relative to the real axis.

• Conclusions:

  • The concept of conjugate is fundamental to simplify expressions and perform calculations with complex numbers.
  • Knowing the properties of conjugates allows for easier manipulation of complex expressions.
  • The ability to calculate the conjugate of a complex number is crucial for understanding the geometric structure of complex numbers and their applications.
  • Repeated conjugation of a complex number confirms the reversible nature of the operation, returning to the original number.