Complex Number Practice

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 i = √(-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

What is the complex conjugate of a complex number?

Back

The complex conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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 + bdi^2, and since i^2 = -1, combine like terms.

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a^2 + b^2). It represents the distance from the origin to the point (a, b) in the complex plane.

Front

How do you simplify the square of a complex number?

Back

To simplify (a + bi)^2, use the formula: (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi.

Front

What is the result of (5 - 2i) + (-7 + 8i)?

Back

The result is -2 + 6i.

Front

What is the simplified form of (8 - 3i)^2?

Back

The simplified form is 55 - 48i.

Front

How do you simplify √(-28)?

Back

√(-28) can be simplified to 2i√7.