Flashcard 4.4 Operations on Complex Numbers

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What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i = √(-1).

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CCSS.HSN.CN.A.1

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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, (-9 + 5i) + (3 - 2i) = (-9 + 3) + (5i - 2i) = -6 + 3i.

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CCSS.HSN.CN.A.2

Front

What is the simplified form of (-9 + 5i) + (3 - 2i)?

Back

The simplified form is -6 + 3i.

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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 apply the fact that i^2 = -1. For example, 3i(4 - i) = 12i - 3i^2 = 12i + 3 = 3 + 12i.

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CCSS.HSN.CN.A.2

Front

What is the result of 3i(4 - i)?

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The result is 3 + 12i.

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CCSS.HSN.CN.A.2

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What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used in division and to simplify expressions involving 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 to eliminate the imaginary part in the denominator.

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CCSS.HSN.CN.A.3

Front

What is the simplified form of \( \frac{4}{5i} \)?

Back

The simplified form is -\( \frac{4i}{5} \).

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CCSS.HSN.CN.A.3

Front

What is the square root of a negative number?

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The square root of a negative number can be expressed using the imaginary unit 'i'. For example, \( \sqrt{-18} = 3i\sqrt{2} \).