Key Concepts of Complex Numbers

Complex numbers combine real and imaginary parts, expressed as ( a + bi ). They expand our understanding of numbers beyond the real line, allowing for two-dimensional representation. This concept connects algebra and trigonometry, enhancing problem-solving techniques.

1. Definition of a complex number

   • A complex number is expressed in the form ( a + bi ), where ( a ) and ( b ) are real numbers.
   • The number ( a ) is called the real part, and ( b ) is called the imaginary part.
   • Complex numbers extend the concept of one-dimensional number lines to two dimensions.

2. Real and imaginary parts

   • The real part of a complex number ( z = a + bi ) is ( a ).
   • The imaginary part is ( b ), which is multiplied by the imaginary unit ( i ).
   • Real and imaginary parts can be extracted using ( \text{Re}(z) = a ) and ( \text{Im}(z) = b ).

3. The imaginary unit i

   • The imaginary unit ( i ) is defined as ( i = \sqrt{-1} ).
   • It allows for the representation of numbers that are not on the real number line.
   • Powers of ( i ) cycle through four values: ( i, -1, -i, 1 ).

4. Complex conjugates

   • The complex conjugate of a complex number ( z = a + bi ) is ( \overline{z} = a - bi ).
   • Conjugates are useful for simplifying division of complex numbers.
   • The product of a complex number and its conjugate yields a real number: ( z \cdot \overline{z} = a^2 + b^2 ).

5. Arithmetic 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: To divide, multiply the numerator and denominator by the conjugate of the denominator.

6. Graphing complex numbers on the complex plane

   • The complex plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part.
   • Each complex number corresponds to a point in this plane, denoted as ( (a, b) ).
   • The distance from the origin to the point represents the modulus of the complex number.

7. Polar form of complex numbers

   • A complex number can be expressed in polar form as ( r(\cos \theta + i \sin \theta) ), where ( r ) is the modulus and ( \theta ) is the argument.
   • The modulus ( r ) is calculated as ( r = \sqrt{a^2 + b^2} ).
   • The argument ( \theta ) is the angle formed with the positive x-axis, found using ( \theta = \tan^{-1}(\frac{b}{a}) ).

8. Euler's formula

   • Euler's formula states that ( e^{i\theta} = \cos \theta + i \sin \theta ).
   • This connects complex exponentials with trigonometric functions.
   • It provides a powerful way to represent complex numbers in polar form: ( z = re^{i\theta} ).

9. De Moivre's theorem

   • De Moivre's theorem states that ( (r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i \sin(n\theta)) ).
   • It is useful for finding powers and roots of complex numbers.
   • The theorem simplifies calculations involving complex numbers raised to integer powers.

10. Roots of complex numbers

    • The ( n )-th roots of a complex number can be found using the formula: ( z_k = 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, \ldots, n-1 ).
    • There are ( n ) distinct roots for any non-zero complex number.
    • The roots are evenly spaced around a circle in the complex plane.