Complex numbers are an extension of the real number system. They introduce the imaginary unit 'i', where i squared equals negative one. A complex number is written as a plus bi, where a is the real part and b is the imaginary part. We can visualize complex numbers on a coordinate plane called the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
Complex numbers follow specific rules for basic operations. Addition and subtraction work component-wise: we add or subtract the real parts and imaginary parts separately. Multiplication uses the distributive property, remembering that i squared equals negative one. Division requires multiplying by the conjugate of the denominator. Geometrically, addition of complex numbers follows the parallelogram rule, just like vector addition.
Complex numbers can also be represented in polar form using their modulus and argument. The modulus is the distance from the origin to the point, calculated as the square root of a squared plus b squared. The argument is the angle the line makes with the positive real axis. This gives us the polar form r times cosine theta plus i sine theta, which can also be written using Euler's formula as r e to the i theta.
De Moivre's theorem provides a powerful way to compute powers of complex numbers in polar form. When we raise a complex number to the nth power, we raise the modulus to the nth power and multiply the argument by n. This theorem makes calculations much simpler than using the rectangular form. Watch how the complex number rotates and scales as we increase its power.
Finding roots of complex numbers reveals beautiful geometric patterns. The nth roots of a complex number are equally spaced around a circle, forming a regular polygon. For example, the fourth roots of unity form a square on the unit circle. Each root can be found using the formula involving the nth root of the modulus and dividing the argument plus 2 pi k by n, where k goes from 0 to n minus 1. This completes our journey through complex numbers, from basic definitions to advanced operations and geometric interpretations.