N-CN. Complex Numbers

N-CN.A. Perform arithmetic operations with complex numbers.

• Complex number patterns
• Computations with Complex Numbers

N-CN.A.1. 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.

• Complex number patterns
• Vertex of a parabola with complex roots

N-CN.A.2. Use the relation $i^2 = -1$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

• Complex Cube and Fourth Roots of 1
• Complex Square Roots
• Powers of a complex number

N-CN.A.3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

• No tasks yet illustrate this standard.

N-CN.B. Represent complex numbers and their operations on the complex plane.

N-CN.B.4. 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.

• No tasks yet illustrate this standard.

N-CN.B.5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, $(-1 + \sqrt{3} i)^3 = 8$ because $(-1 + \sqrt3 i)$ has modulus $2$ and argument $120^\circ$.

• Complex Cube and Fourth Roots of 1

N-CN.B.6. 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.

• Complex Distance

N-CN.C. Use complex numbers in polynomial identities and equations.

N-CN.C.7. Solve quadratic equations with real coefficients that have complex solutions.

• Completing the square

N-CN.C.8. Extend polynomial identities to the complex numbers. For example, rewrite $x^2 + 4$ as $(x + 2i)(x - 2i)$.

• Complex Cube and Fourth Roots of 1

N-CN.C.9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

• No tasks yet illustrate this standard.