Complex Numbers
Introduction to Complex Numbers
- Complex numbers are an extension of the real numbers and contain all the numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property that i² = -1.
- Any complex number can be represented as a point in the complex plane, with the horizontal (real) axis representing the real part and the vertical (imaginary) axis representing the imaginary part.
Arithmetic with Complex Numbers
- The standard rules of algebra apply to complex numbers, including commutativity, associativity, and distributivity.
- Adding and subtracting complex numbers works component-wise, i.e., real parts with real parts and imaginary parts with imaginary parts.
- When multiplying complex numbers, apply the FOIL method (first, outside, inside, last), remembering to substitute i² with -1 where necessary.
Modulus and Argument of Complex Numbers
- The modulus of a complex number, represented as r, is the distance from the number to the origin in the complex plane. It is calculated using Pythagoras’ theorem: r = sqrt(a² + b²).
- The argument of a complex number, represented as θ, is the angle that the line joining the number and the origin makes with the positive real axis, measured anti-clockwise. Applying trigonometry, tan(θ) = b/a.
Conjugates and Division of Complex Numbers
- The conjugate of a complex number is the number with the same real part but with an imaginary part of the opposite sign. If a complex number is a + bi, its conjugate will be a - bi.
- Dividing complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator, simplifying where possible.
Polar Form of Complex Numbers
- A complex number can also be represented in polar form: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.
- Euler’s formula provides a shortcut for transforming from Cartesian to polar form. According to this formula, e^(iθ) = cos θ + i sin θ.
De Moivre’s Theorem
- De Moivre's theorem is powerful for making computations with complex numbers easier. It states that for any integer n, (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).
Complex Roots
- Polynomials can have complex roots, thanks to the Fundamental Theorem of Algebra. If a polynomial's real roots do not account for all its roots, the remaining roots will be complex.
- To find these roots, set the polynomial equal to zero and solve for the variable. If you encounter a square root of a negative number, remember that this can be simplified using i.