5.1.1: Resources and Key Concepts

Resources

Videos

• Complex numbers
  • Arithmetic - Sets - Complex Numbers - Definitions, Theorem, and Example
• Complex plane
  • Arithmetic - Sets - Complex Numbers - The Complex Plane and Plotting Complex Numbers
• Adding and Subtracting complex numbers
  • Arithmetic - Sets - Complex Numbers - Adding and Subtracting Complex Numbers
• Multiplying complex numbers
  • Arithmetic - Sets - Complex Numbers - Multiplying Complex Numbers
• Dividing complex numbers
  • Arithmetic - Sets - Complex Numbers - Dividing Complex Numbers
• Powers of \(i\)
  • Arithmetic - Sets - Complex Numbers - Simplifying Powers of i

Key Concepts

Definitions

• Unit Imaginary Number: The number, denoted as \(i\), which is defined as the square root of negative one (\(\sqrt{-1} = i\)).
• Complex Number: A number that can be expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers.
  • Real Part: For a complex number \(a+bi\), \(a\) is the real part.
  • Imaginary Part: For a complex number \(a+bi\), \(b\) is the imaginary part.
  • Imaginary Number: A complex number \(a+bi\) where \(a=0\) and \(b \neq 0\).
  • Standard Form: The representation of a complex number as \(a+bi\).
• Complex Number System: The entire set of numbers of the form \(a+bi\), symbolically denoted as \(\mathbb{C}\).
• Complex Plane: A coordinate system used for graphing complex numbers, where the horizontal axis (real axis) represents the real part and the vertical axis (complex axis) represents the imaginary part.
  • Real Axis: The horizontal axis in the complex plane.
  • Complex Axis: The vertical axis in the complex plane.
• Addition of Complex Numbers: If \(z_1 = a+bi\) and \(z_2 = c+di\), their sum is \(z_1+z_2 = (a+c) + (b+d)i\).
• Subtraction of Complex Numbers: If \(z_1 = a+bi\) and \(z_2 = c+di\), their difference is \(z_1-z_2 = (a-c) + (b-d)i\).
• Multiplication of Complex Numbers: If \(z_1 = a+bi\) and \(z_2 = c+di\), their product is \(z_1 \cdot z_2 = (ac-bd) + (ad+bc)i\). (Treat this like binomial multiplication rather than memorizing the formula.)
• Complex Conjugate: For a complex number \(z = a+bi\), its complex conjugate is \(\bar{z} = a-bi\). Similarly, the complex conjugate of \(z = a-bi\) is \(\bar{z} = a+bi\).
• Division of Complex Numbers: For two complex numbers \(z_1 = a+bi\) and \(z_2 = c+di\) (where \(z_2 \neq 0\)), their quotient \(\frac{z_1}{z_2}\) is found by multiplying the numerator and denominator by the conjugate of the denominator, resulting in \(\frac{(ac+bd) + (bc-ad)i}{c^2+d^2}\). (Perform the multiplication by the conjugate rather than memorizing this formula.)

Theorems

• The Square of the Imaginary Unit: \(i^2 = -1\).
• Products of Complex Conjugates: Let \(z = a+bi\), so that its complex conjugate is \(\bar{z} = a-bi\). Then \(z \cdot \bar{z} = a^2+b^2\).
• Powers of the Imaginary Unit: Let \(p \in \mathbb{W}\) (the Whole Number system) and \(r\) be the remainder when \(p\) is divided by 4. Then \(i^p = i^r\). (This implies \(i^0=1, i^1=i, i^2=-1, i^3=-i\)).

Common Mistakes

• Memorizing the formula for the product of two complex numbers instead of treating the multiplication like that of two binomials. The text advises: "you should never memorize the result of this definition. Instead, treat multiplying complex numbers like multiplying binomials."
• Leaving a complex number in the denominator of a fraction when expressing a complex number. The text states: "Any fraction must have a real-valued denominator."
• Memorizing the formula for the division of two complex numbers instead of understanding and applying the process of multiplying the numerator and denominator by the complex conjugate of the denominator. The text advises: "it is best not to memorize this resulting formula. You should be able to just 'do the math.'"