Imaginary Numbers

Front

What are imaginary numbers?

Back

Imaginary numbers are numbers 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 express imaginary numbers.

Front

What is the standard form of a complex number?

Back

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part.

Front

How do you add complex numbers?

Back

To add complex numbers, add the real parts and the imaginary parts separately. For example, (a + bi) + (c + di) = (a+c) + (b+d)i.

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract the real parts and the imaginary parts separately. For example, (a + bi) - (c + di) = (a-c) + (b-d)i.

Front

What is the product of two imaginary numbers?

Back

The product of two imaginary numbers, such as (bi)(di), is given by (bd)(i^2) = -bd, since i^2 = -1.

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 division of complex numbers.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property: (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 √(a² + b²). It represents the distance from the origin in the complex plane.

Front

How do you find the roots of a quadratic equation with imaginary solutions?

Back

To find the roots of a quadratic equation ax² + bx + c = 0 with imaginary solutions, use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). If the discriminant (b² - 4ac) is negative, the roots will be imaginary.