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 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

What is the relationship between imaginary numbers and complex numbers?

Back

A complex number is a number that has both a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the imaginary part.

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

How do you multiply two complex numbers?

Back

To multiply two complex numbers, use the distributive property (FOIL method). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, simplify accordingly.

Front

What is the conjugate of a complex number?

Back

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

Front

How do you divide complex numbers?

Back

To divide 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)].

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as |a + bi| = √(a² + b²).

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 real axis, calculated using θ = arctan(b/a).

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.