Operations with 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 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

How do you subtract complex numbers?

Back

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

Front

What is the conjugate of a complex number?

Back

The 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, simplify accordingly.

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 of the point (a, b) from the origin in the complex plane.

Front

How do you simplify the expression (a + bi)^2?

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 (1 + i)(1 - i)?

Back

The result of (1 + i)(1 - i) is 1 - i^2 = 1 - (-1) = 2.

Front

What is the simplified form of (3 + 4i) + (2 - 3i)?

Back

The simplified form is (3 + 2) + (4 - 3)i = 5 + 1i = 5 + i.