Test: Operations with Complex Numbers

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).

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part.

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

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

The product of two complex numbers (a + bi)(c + di) is calculated using the distributive property: ac + adi + bci + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

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).

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

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

The result of multiplying two imaginary numbers, such as (ai)(bi), is -ab, since i^2 = -1.