3.4a Imaginary Numbers

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What is an imaginary number?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -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 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

What is the relationship between imaginary numbers and complex numbers?

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A complex number is a number that has both a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the imaginary part.

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CCSS.HSN.CN.A.1

Front

How do you add two complex numbers?

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To add two complex numbers, add 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 multiply two complex numbers?

Back

To multiply two complex numbers, use the distributive property (FOIL method). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, simplify accordingly.

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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 the 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, (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)].

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CCSS.HSN.CN.A.3

Front

What is the modulus of a complex number?

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The modulus of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as |a + bi| = √(a² + b²).

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What is the argument of a complex number?

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The argument of a complex number a + bi is the angle θ formed with the positive real axis, calculated using θ = arctan(b/a).

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What is Euler's formula?

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Euler's formula states that e^(iθ) = cos(θ) + i sin(θ), linking complex exponentials to trigonometric functions.

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CCSS.HSN.CN.C.8