Examples for

Common Core Math: High School Number & Quantity: Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is defined as the imaginary unit and is equal to the principal square root of -1. High school students learn to perform arithmetic with complex numbers and to plot them in the complex plane. At advanced levels, students may also learn to represent complex numbers in terms of trigonometric functions and polar coordinates. Students will find this domain relevant to their previous studies of polynomials, as complex numbers provide solutions to polynomials with roots that cannot be found by considering only the real numbers. With knowledge of complex numbers, students can learn and apply the fundamental theorem of algebra, which states that a polynomial of degree n has exactly n complex roots counted with multiplicity.
Common Core Standards

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CCSS.Math.Content.HSN-CN.C.7
hs numbers cn.a.2 common core standard

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common core standards about complex numbers
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Arithmetic with Complex Numbers

Add, subtract, multiply and divide complex numbers.

Define complex numbers (CCSS.Math.Content.HSN-CN.A.1):

what are complex numbers?
what is i?
3 + 8i
i^2
2(cos(pi/3) + i*sin(pi/3))
5 (cos(tan^(-1)(4/3)) + i sin(tan^(-1)(4/3)))

Add, subtract and multiply complex numbers (CCSS.Math.Content.HSN-CN.A.2):

4i + 5i
(8i + 4) - (3i + 2)
4 + 7 + 3i - i
6i * (-i)
3i (5i + 7)
(11i + 4) (2i - 3)
(2+3i)(5-i)
-i*4i + 3 (2i - 8)
(1 + 6i) * (1 - 6i)

Find conjugates, moduli and quotients of complex numbers (CCSS.Math.Content.HSN-CN.A.3):

conjugate of (5 - 2i)
(8 + 4i)^*
1/(-3i)
(-4 + i)/(6i)
(3 + 9i)/(9 + 18i)
1/(12+4i)
(2 - 4i)/(6 - 12i)
modulus of (4 + 2i)
magnitude of (2i - 1)
|-6i + 2|

Evaluate powers of complex numbers (CCSS.Math.Content.HSN-CN.A.2):

(6i + 5)^2
3(1 + 2i)^3
(4 (cos(pi/4) + i*sin(pi/4)))^3
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Visualize Complex Numbers

Define and plot complex numbers using rectangular or polar coordinates.

Convert between rectangular and polar coordinates (CCSS.Math.Content.HSN-CN.B.4):

convert 3 + 7i to polar coordinates
2 - 9i to polar coordinates
(3, pi/4) to rectangular coordinates
express (-1.5, 200 degrees) in Cartesian coordinates

Find the distance between two complex numbers (CCSS.Math.Content.HSN-CN.B.6):

distance between (3 + 2i) and (4 - 3i)
|(5 + i) - (3 + 2i)|
modulus of ((2 - i) minus (9 + 4i))
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Complex Numbers in Context

Generalize knowledge of polynomials to include complex numbers.

Find complex solutions to quadratic equations (CCSS.Math.Content.HSN-CN.C.7):

solve 4x^2 - 3x + 2 = 0
zeros of f(x) = 3x^2 - 4x + 3
solve for x if 8x^2 + 6x + 5 = 0

Use polynomial identities with complex numbers (CCSS.Math.Content.HSN-CN.C.8):

factor x^2 + 9 over the complex numbers
a^8 + b^4 factored over complexes
solve x^4 + 1 = 0

Reason with the fundamental theorem of algebra (CCSS.Math.Content.HSN-CN.C.9):

solve 0 = x^2 + 5x + 8 over the complex numbers
zeros of x^5 + 3x^3 + 5x^2 - 10
solutions of ax^2 + bx + c = 0
Fundamental Theorem of Algebra
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