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 i² = -1.

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CCSS.HSN.CN.A.1

Front

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

Front

How do you multiply two complex numbers?

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To multiply two complex numbers (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi². Remember that i² = -1.

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CCSS.HSN.CN.A.2

Front

What is the result of (2i)(3i)?

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The result is -6.

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CCSS.HSN.CN.A.2

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How do you simplify (7 + 2i)(9 - 6i)?

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The simplified form is 75 - 24i.

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CCSS.HSN.CN.C.8

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What is the process for subtracting complex numbers?

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To subtract complex numbers (a + bi) - (c + di), subtract the real parts and the imaginary parts separately: (a - c) + (b - d)i.

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CCSS.HSN.CN.A.2

Front

What is the result of (10 + 15i) - (48 - 30i)?

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The result is -38 + 45i.

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CCSS.HSN.CN.A.2

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How do you simplify the expression 2i(4 - 3i)?

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The simplified form is 6 + 8i.

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CCSS.HSN.CN.A.2

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What is the definition of 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 find the magnitude of complex numbers.

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CCSS.HSN.CN.A.3

Front

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).