Mathematics Common Core Standards: The Complex Number System

By Jared Myracle

High school students will need to know about the complex number system for Common Core Standards. The complex number system includes both real and imaginary numbers. An imaginary number, represented as i, is the square root of –1; i is imaginary because no number multiplied by itself results in a negative value.

In Grade 11, students encounter imaginary numbers as a translation on the imaginary plane. Here’s what students need to know and be able to do when dealing with the complex number system:

  • Explain what a complex (imaginary) number is:

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    or

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    • a + bi = a complex number, with both a and b being real numbers

  • Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties:

    • Commutative enables you to add or multiply numbers in any order, for example 4 + 2 = 2 + 4

    • Associative means you can add or multiply numbers in any grouping, for example (3 × 5) × 4 = 3 × (5 × 4)

    • Distributive is commonly represented as

      a × (b + c) = (a × b) + (a × c)

    So, students should be able to solve equations such as:

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  • Find the conjugate of a complex number and use conjugates to find moduli and quotients of complex numbers. A conjugate is a binomial expression (representing the sum or difference of two terms) formed by negating the second term of a binomial; for example, the conjugate of a + b is a – b.

  • When an imaginary number is involved, you have a complex conjugate; for example, in the expression m = a + bi the complex conjugate represented is:

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    A sample problem may provide a given and ask you to use the conjugate to find the modulus and quotient; for example, given that y = 3 – 7i and z = 5 + 2i, find the modulus of y and the quotient of z and y:

    To find the modulus of y using its complex conjugate, students may solve an equation like the following:

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    or

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    As you can see, you use the distributive property to multiply the two binomials in the first step. You can use the FOIL method (first terms, outside terms, inside terms, last terms) to remember how to do this: first (3 × 3), outside (3 × 7i), inside (–7i × 3), and last (–7i × 7i). After multiplying these terms, you arrive at a polynomial with four terms.

    Then you combine like terms and complete any remaining operations. Because i is the square root of –1, you can change i2 into –1 and multiply it by –49, resulting in changing it to a positive number. After calculating 9 + 49 = 58, take the square root of 58 because you’re solving for y2 and want to find y instead.

    To find the quotient of z and y:

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  • Represent complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a complex number represent the same number. In the complex plane, the horizontal axis (X ) represents real numbers, and the vertical axis (Yi) represents imaginary numbers. Imaginary numbers can be represented on the complex plane in two forms:

    • Rectangular form: The intersection of the real and imaginary numbers is shown as the intersection of coordinates on the X and Yi axes.

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    • Polar form: The real number represents the vector length (how far the vector reaches into the imaginary plane), and θ represents the angle the vector forms with the real axis (the familiar axis represented by x and y). Polar form is derived from the Pythagorean theorem, r2 = a2 + b2.

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  • Represent addition, subtraction, multiplication, and conjugation on the complex plane.

  • Solve quadratic equations (equations in which the highest power of an unknown is a square) with real coefficients that have complex solutions. For example, students may be asked to solve x2 + 2x = 0 over complex numbers.

  • Extend polynomial identities to complex numbers. For example, x + 7 using complex numbers can be expressed as (x + 7i) × (x – 7i).

  • Grasp the Fundamental Theorem of Algebra, which states that any polynomial of n degree has n roots (places where the polynomial is equal to zero when graphed). For example, in a polynomial with one variable, such as 5×6 + 8x – 2, the n degree is 6, so the polynomial has 6 roots.

You can support your child at home by showing interest; monitoring progress; encouraging your child to seek help, if necessary; and expressing any concerns you have to your child’s math teacher. You can also track down websites that can help you understand these concepts, like Khan Academy or Illustrative Mathematics.