Pre-Calculus Complex Numbers

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 complex numbers?

Back

To add complex numbers, combine their real parts and their imaginary parts separately. For example, (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i.

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract their real parts and their imaginary parts separately. For example, (5 + 3i) - (2 + 1i) = (5 - 2) + (3i - 1i) = 3 + 2i.

Front

What is the product of two complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, it simplifies to (ac - bd) + (ad + bc)i.

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, to divide (3 + 2i) by (1 + i), multiply by (1 - i)/(1 - i).

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²). 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

What is Euler's formula?

Back

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