KNOW THERE IS A COMPLEX NUMBER I SUCH THAT I^2 = –1, AND EVERY COMPLEX NUMBER HAS THE FORM A + BI WITH A AND B REAL.
USE THE RELATION I^2 = –1 AND THE COMMUTATIVE, ASSOCIATIVE, AND DISTRIBUTIVE PROPERTIES TO ADD, SUBTRACT, AND MULTIPLY COMPLEX NUMBERS.
(+) FIND THE CONJUGATE OF A COMPLEX NUMBER; USE CONJUGATES TO FIND MODULI AND QUOTIENTS 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 – ROOT 3I)3 = 8 BECAUSE (1 – ROOT 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.
(+) EXTEND POLYNOMIAL IDENTITIES TO THE COMPLEX NUMBERS. FOR EXAMPLE, REWRITE X^2 + 4 AS (X + 2I)(X – 2I).
(+) KNOW THE FUNDAMENTAL THEOREM OF ALGEBRA; SHOW THAT IT IS TRUE FOR QUADRATIC POLYNOMIALS.
Alaska: The Complex Number System Math Standards