N.CN: The Complex Number System

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

N.CN.A.1: Know there is a complex number i such that i^2 = -1, and show that every complex number has the form a + bi where a and b real.

Points in the Complex Plane

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

Points in the Complex Plane

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

Points in the Complex Plane

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.

Points in the Complex Plane

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.

Points in the Complex Plane

N.CN.B.6: Calculate the distance between numbers in the complex plane as the absolute value of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Points in the Complex Plane

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.

Roots of a Quadratic

N.VM: Vectors and Matrix Quantities

N.VM.A: Represent and model with vector quantities.

N.VM.A.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Adding Vectors
Vectors

N.VM.A.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Adding Vectors

N.VM.A.3: Solve problems involving velocity and other quantities that can be represented by vectors.

2D Collisions
Adding Vectors
Golf Range

N.VM.B: Perform operations on vectors.

N.VM.B.4: Add and subtract vectors.

N.VM.B.4.a: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Adding Vectors
Vectors

N.VM.C: Perform operations on matrices and use matrices in applications.

N.VM.C.7: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Dilations

N.VM.C.8: Add, subtract, and multiply matrices of appropriate dimensions.

Translations

N.VM.C.10: Demonstrate understanding that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Solving Linear Systems (Matrices and Special Solutions)

N.VM.C.12: Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Dilations
Translations