Complex Numbers

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What is a complex number?

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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 the square root of -1.

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CCSS.HSN.CN.A.1

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What is the imaginary unit 'i'?

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The imaginary unit 'i' is defined as the square root of -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

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How do you add complex numbers?

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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

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How do you subtract complex numbers?

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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)?

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The product is calculated using the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, it simplifies to (ac - bd) + (ad + bc)i.

Tags

CCSS.HSN.CN.A.2

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What is the conjugate of a complex number?

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The conjugate of a complex number a + bi is a - bi. It is used in division and to find the magnitude of complex numbers.

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CCSS.HSN.CN.A.3

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How do you find the magnitude of a complex number?

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The magnitude of a complex number a + bi is given by the formula |a + bi| = √(a² + b²).

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What is the result of (1 + 2i) + (3 + 4i)?

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The result is (1 + 3) + (2 + 4)i = 4 + 6i.

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CCSS.HSN.CN.A.2

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What is the result of (5 + 6i) - (2 + 3i)?

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The result is (5 - 2) + (6 - 3)i = 3 + 3i.

Tags

CCSS.HSN.CN.A.2

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What is the result of (2 + 3i)(4 - i)?

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The result is (2*4 + 2*(-i) + 3i*4 + 3i*(-i)) = 8 - 2i + 12i + 3 = 11 + 10i.

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CCSS.HSN.CN.A.2