Complex Numbers

Front

What is a complex number?

Back

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.

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.

Front

What is the complex conjugate of a complex number a + bi?

Back

The complex conjugate of a complex number a + bi is a - bi.

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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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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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How do you multiply complex numbers?

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

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What is the modulus of a complex number a + bi?

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The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²). It represents the distance from the origin to the point (a, b) 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 as θ = arctan(b/a). It is usually expressed in radians.

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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.

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How do you convert a complex number from rectangular to polar form?

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To convert a complex number a + bi to polar form, calculate r = √(a² + b²) and θ = arctan(b/a), then express it as r(cos θ + i sin θ).