Operations on 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 the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -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 two complex numbers?

Back

To add two 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 two complex numbers?

Back

To subtract two 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 product of two complex numbers (a + bi)(c + di)?

Back

The product is calculated using the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, it simplifies to (ac - bd) + (ad + bc)i.

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 to find the magnitude of the complex number.

Front

How do you multiply a complex number by its conjugate?

Back

Multiplying a complex number by its conjugate results in a real number: (a + bi)(a - bi) = a^2 + b^2.

Front

What is the magnitude of a complex number a + bi?

Back

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

Front

What is the simplified form of (a + bi)^2?

Back

The simplified form is (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi.

Front

What is the result of (a + bi) + (c + di)?

Back

The result is (a + c) + (b + d)i, where you add the real parts and the imaginary parts separately.