3.4a Imaginary Numbers

Front

What is an imaginary number?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -1.

Front

What is the value of i^2?

Back

i^2 = -1.

Front

How do you add imaginary numbers?

Back

To add imaginary numbers, combine the real parts and the imaginary parts separately. For example, (3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i.

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.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL). For example, (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

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 from the origin to the point (a, b) in the complex plane.

Front

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 x-axis in the complex plane, calculated as θ = arctan(b/a).

Front

How do you express a complex number in polar form?

Back

A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the modulus and θ is the argument.

Front

What is Euler's formula?

Back

Euler's formula states that e^(iθ) = cos θ + i sin θ, linking complex exponentials to trigonometric functions.

Front

What is the square root of a negative number?

Back

The square root of a negative number can be expressed using imaginary numbers. For example, √(-a) = i√a.