Complex numbers extend the real number system by including imaginary numbers (multiples of *i*, where *i*² = -1). Theyre crucial in H2 Math for solving polynomial equations, understanding oscillations, and have applications in physics and engineering.
Addition and subtraction: Combine real and imaginary parts separately. Multiplication: Use the distributive property and remember *i*² = -1. Division: Multiply both numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.
The conjugate of *a + bi* is *a - bi*. Its used in division to rationalize the denominator and find the modulus of a complex number.
The Argand diagram represents a complex number *a + bi* as a point (*a, b*) on a Cartesian plane, where the x-axis is the real axis and the y-axis is the imaginary axis.
The modulus is the distance from the origin to the point representing the complex number on the Argand diagram (√(a² + b²)). The argument is the angle between the positive real axis and the line connecting the origin to the point (arctan(b/a), considering the quadrant).
Cartesian form: *a + bi*. Polar form: *r(cos θ + i sin θ)*, where *r* is the modulus and *θ* is the argument. Use *a = r cos θ* and *b = r sin θ* for conversion.
De Moivres Theorem states that *(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)*. Use it to find powers of complex numbers in polar form. For roots, find all *n* solutions by adding multiples of 2π/n to the argument.
Forgetting that *i*² = -1, not considering the correct quadrant when finding the argument, making errors in algebraic manipulation during division, and not expressing answers in the required form (Cartesian or polar).