The Complex Number System

Understand there is a complex number i such that i2 = ?1, and every complex number has the form a + bi where a and b are real numbers.

Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Find the conjugate of a complex number; use the conjugate to find the absolute value (modulus) and quotient of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + ?3i)3 = 8 because (-1 + ?3i) has modulus 2 and argument 120.

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square roots, completing the square, and the quadratic formula.

Extend polynomial identities to include factoring with complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i).

Use the Fundamental Theorem of Algebra to find all roots of a polynomial equation