Operations with Complex Numbers

Front

Define complex numbers.

Back

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as i = √(-1).

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CCSS.HSN.CN.A.1

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √(-1). It is used to extend the real number system to include solutions to equations that do not have real solutions.

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CCSS.HSN.CN.A.1

Front

How do you add complex numbers?

Back

To add complex numbers, combine their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

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CCSS.HSN.CN.A.2

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

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CCSS.HSN.CN.A.2

Front

What is the product of two complex numbers (a + bi)(c + di)?

Back

The product is given by (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

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CCSS.HSN.CN.A.2

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method) and remember that i^2 = -1. For example, (a + bi)(c + di) = ac + adi + bci + bd(-1).

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CCSS.HSN.CN.A.2

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

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CCSS.HSN.CN.A.3

Front

How do you divide complex numbers?

Back

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide (a + bi) by (c + di), multiply by (c - di)/(c - di).

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CCSS.HSN.CN.A.3

Front

Simplify the expression: (8 + 9i) + (4 - 6i)

Back

12 + 3i.

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CCSS.HSN.CN.A.2

Front

Simplify the expression: (5 + 4i) - (-1 - 2i)

Back

6 + 6i.

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CCSS.HSN.CN.A.2