## The Real Number System

### Operations and Algebraic Thinking

Code
Standard
HSN-RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
HSN-RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
HSN-RN.B.3
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

### Quantities

Code
Standard
HSN-Q.A.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
HSN-Q.A.2
Define appropriate quantities for the purpose of descriptive modeling.
HSN-Q.A.3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

### The Complex Number System

Code
Standard
HSN-CN.A.1
Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
HSN-CN.A.2
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN-CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
HSN-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.
HSN-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 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
HSN-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.
HSN-CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.
HSN-CN.C.8
(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
HSN-CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

### Vector and Matrix Quantities

Code
Standard
HSN-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).
HSN-VM.A.2
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
HSN-VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
HSN-VM.B.4
HSN-VM.B.4a
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.
HSN-VM.B.4b
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
HSN-VM.B.4c
Understand vector subtraction v – w as v + (–w), where –w is the additive inverse ofw, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
HSN-VM.B.5
(+) Multiply a vector by a scalar.
HSN-VM.B.5a
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction;
HSN-VM.B.5b
perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c> 0) or against v (for c < 0).
HSN-VM.C.6
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
HSN-VM.C.7
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
HSN-VM.C.8
(+) Add, subtract, and multiply matrices of appropriate dimensions.
HSN-VM.C.9
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
HSN-VM.C.10
(+) Understand 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.
HSN-VM.C.11
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
HSN-VM.C.12
(+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.