Complex Numbers: Simplifying, Adding, and Subtracting

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

How do you add complex numbers?

Back

To add complex numbers, combine 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 imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, which means i² = -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 in division and simplifying expressions.

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 + bdi² = (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| = √(a² + b²). It represents the distance from the origin in the complex plane.

Front

How do you simplify √-n?

Back

To simplify √-n, express it as √n * √-1 = √n * i. For example, √-4 = 2i.

Front

What is i raised to the power of 4?

Back

i raised to the power of 4 is equal to 1, since i² = -1 and (i²)² = (-1)² = 1.

Front

What is the result of i raised to the power of 5?

Back

i raised to the power of 5 is equal to i, since i⁴ = 1 and i⁵ = i⁴ * i = 1 * i = i.