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 i² = -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., 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, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

How do you multiply complex numbers?

Back

To multiply 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 in division of complex numbers to eliminate the imaginary part from the denominator.

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 of the point (a, b) from the origin 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 real axis, calculated as θ = arctan(b/a). It is often expressed in radians.

Front

What is the result of (2 + 3i)²?

Back

(2 + 3i)² = 4 + 12i - 9 = -5 + 12i.

Front

What is the result of (3 + 8i)(-2 - i)?

Back

(3 + 8i)(-2 - i) = -6 - 3i - 16i - 8i² = -6 - 19i + 8 = 2 - 19i.

Front

How do you solve the equation 3x² - 5 = -446?

Back

3x² = -446 + 5 = -441; x² = -441/3; x = ±√(-147) = ±7i√3.