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

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

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How do you add two complex numbers?

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To add two complex numbers, you 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

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How do you subtract two complex numbers?

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To subtract two complex numbers, you 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.

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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 of complex numbers and has the property that the product of a complex number and its conjugate is a real number.

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CCSS.HSN.CN.A.3

Front

How do you divide two complex numbers?

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To divide two 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) to eliminate the imaginary part in the denominator.

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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 given by the formula |a + bi| = √(a^2 + b^2). It represents the distance of the complex number from the origin in the complex plane.

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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). It represents the direction of the complex number in the complex plane.

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What is the polar form of a complex number?

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The polar form of a complex number is expressed as r(cos θ + i sin θ), where r is the modulus and θ is the argument. It can also be written as re^(iθ) using Euler's formula.

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CCSS.HSN.CN.B.4