Lecture Notes: Complex Numbers (Section 3.1)

Key Topics Covered

• Plotting complex numbers
• Basic operations: addition, subtraction, multiplication, division
• Finding the conjugate of a complex number
• Expressing square roots of negative numbers as multiples of i

Imaginary Unit and Complex Numbers

• Definition of i: ( i ) is defined as ( i^2 = -1 ). Therefore, ( i = \sqrt{-1} ).
• Standard Form: A complex number is in the form ( a + bi ), where:
  • ( a ) is the real part
  • ( b ) is the imaginary part
• Real Numbers: When ( b = 0 ), the complex number becomes a real number ( a + 0i = a ).
• Imaginary Numbers: When ( a = 0 ), the complex number is purely imaginary ( 0 + bi = bi ).

Graphing Complex Numbers

• Graphed on the complex plane:
  • X-axis represents the real part.
  • Y-axis represents the imaginary part.
  • Example: (-2 + 3i) is plotted at ((-2, 3)).

Operations on Complex Numbers

Addition and Subtraction

• Combine like terms (real with real, imaginary with imaginary).
• Example:
  • ( (4 - 7i) + (-5 - 2i) = -1 - 9i )
  • ( (-4 + 2i) - (6 - 7i) = -10 + 9i )

Multiplication

• Use distributive property, similar to multiplying binomials.
• Remember ( i \times i = i^2 = -1 ).
• Example:
  • ( (-6i) \times (2i) = -12 \times i^2 = 12 )
  • ( (3i) \times (4 - 5i) = 12i + 15 = 15 + 12i )

Division

• Multiply the numerator and the denominator by the conjugate of the denominator.
• Simplify using the property that a complex number times its conjugate is a real number.
  • Example: ( \frac{3}{4+2i} ) becomes ( \frac{3(4+2i)}{4^2+2^2} = \frac{12+6i}{20} = \frac{6+3i}{10} )

Complex Conjugates

• The conjugate of ( a + bi ) is ( a - bi ).
• Multiplying a complex number by its conjugate results in a real number: ( a^2 + b^2 ).
  • Example: ( (5+3i)(5-3i) = 25 + 9 = 34 )

Square Roots of Negative Numbers

• Express ( \sqrt{-b} ) as ( i\sqrt{b} ).
• Examples:
  • ( \sqrt{-25} = 5i )
  • ( -\sqrt{-24} = -2\sqrt{6}i )
  • ( \sqrt{-7} = \sqrt{7}i )

Complex Numbers in Operations

• Combine real and imaginary components separately when multiplying or adding.
• Example of a complex multiplication:
  • ( (5 + \sqrt{-18})(-2 - \sqrt{-50}) = -10 - 25\sqrt{2}i - 6i + 30 )
  • Final form: ( 20 - 31\sqrt{2}i )

Key Takeaways

• Complex numbers have both a real and imaginary component.
• Operations on them involve combining or manipulating these components separately based on algebraic rules.
• The conjugate is a very useful tool for simplifying expressions involving complex numbers, especially for rationalizing denominators in division.