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 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.

Front

How do you add complex numbers?

Back

To add 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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How do you subtract complex numbers?

Back

To subtract complex numbers, you subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

Front

What is the product of two complex numbers (a + bi)(c + di)?

Back

The product of two complex numbers is given by (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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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).

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by |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?

Back

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

Front

What is the polar form of a complex number?

Back

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.