Complex Numbers Unit Review

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

How do you add two complex numbers?

Back

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

Front

What is the distance between two complex numbers z1 and z2?

Back

The distance between two complex numbers z1 = a + bi and z2 = c + di is given by the formula |z1 - z2| = √((a - c)² + (b - d)²).

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers, use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi². Remember that i² = -1.

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 (distance from the origin) and θ is the argument (angle with the positive real axis).

Front

How do you find the modulus of a complex number?

Back

The modulus of a complex number z = a + bi is given by |z| = √(a² + b²). It represents the distance from the origin in the complex plane.

Front

What is the argument of a complex number?

Back

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

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number z = a + bi is denoted as z̅ = a - bi. It reflects the complex number across the real axis.

Front

How do you divide two complex numbers?

Back

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)].