Complex numbers - polar form

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 polar form of a complex number?

Back

The polar form of a complex number is expressed as r(cos θ + i sin θ) or re^(iθ), where r is the modulus (magnitude) and θ is the argument (angle) of the complex number.

Front

How do you calculate the modulus of a complex number z = a + bi?

Back

The modulus of a complex number z = a + bi is calculated as |z| = √(a² + b²).

Front

What is the argument of a complex number?

Back

The argument of a complex number is the angle θ formed with the positive x-axis in the complex plane, typically measured in radians or degrees.

Front

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

Back

To convert from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)), calculate r = √(a² + b²) and θ = arctan(b/a).

Front

What is the significance of the angle in the polar form of a complex number?

Back

The angle (argument) indicates the direction of the complex number in the complex plane, while the modulus indicates its distance from the origin.

Front

If z = -7 - 8i, what is the modulus of z?

Back

The modulus of z = -7 - 8i is |z| = √((-7)² + (-8)²) = √(49 + 64) = √113.

Front

What quadrant does the complex number -7 - 8i lie in?

Back

The complex number -7 - 8i lies in the third quadrant of the complex plane.

Front

What is the value of the argument for the complex number -7 - 8i?

Back

The argument for the complex number -7 - 8i is between -90° and -180°.

Front

What is the relationship between the modulus and the argument of a complex number?

Back

The modulus represents the distance from the origin, while the argument represents the angle of the complex number in the complex plane.