Mathematics » High School: Number & Quantity » The Complex Number System

Perform arithmetic operations with complex numbers.

• 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.

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• 2. Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

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• 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

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Represent complex numbers and their operations on the complex plane.

• 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.

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• 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°.

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• 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.

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Use complex numbers in polynomial identities and equations.

• 7. Solve quadratic equations with real coefficients that have complex solutions.

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• 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).

  Suggested tag: cc-9-12m-n-cn-8

• 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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