The Complex Number System

N.CN.1(11) 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.N.CN.10(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.N.CN.2(11) Use the relation i^2 = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.N.CN.3(11) Find the conjugate of a complex number.N.CN.4(+) Use conjugates to find moduli and quotients of complex numbers.N.CN.5(+) 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.N.CN.6(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 ⋅ i)^3 = 8 because (-1 + √3 ⋅ i) has modulus 2 and argument 120°.N.CN.7(+) 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.N.CN.8(11) Solve quadratic equations with real coefficients that have complex solutions.N.CN.9(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x-2i).
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