Operations with 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 i = √(-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, 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

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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

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What is the product of two complex numbers?

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The product of two complex numbers (a + bi)(c + di) is given by ac + adi + bci + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

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CCSS.HSN.CN.A.2

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How do you multiply complex numbers?

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To multiply complex numbers, use the distributive property (FOIL method) and remember that i^2 = -1. For example, (a + bi)(c + di) = ac + adi + bci + bd(-1).

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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 to simplify the division of complex numbers.

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CCSS.HSN.CN.A.3

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How do you divide complex numbers?

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To divide 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)/(c - di).

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CCSS.HSN.CN.A.3

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What is the modulus of a complex number?

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The modulus of a complex number a + bi is given by |a + bi| = √(a^2 + b^2). It represents the distance from the origin to the point (a, b) in the complex plane.

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How do you find 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 as θ = arctan(b/a).