12-05 Operations of 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 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 two complex numbers?

Back

To add two complex numbers, add 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 given by (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.

Front

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

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify the division of complex numbers.

Front

How do you divide two complex numbers?

Back

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)].

Front

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

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²). It represents the distance from the origin to the point (a, b) in the complex plane.

Front

What is the argument of a complex number?

Back

The argument of a complex number a + bi is the angle θ formed with the positive real axis, calculated as θ = arctan(b/a). It is usually expressed in radians.

Front

What is the square of a complex number (a + bi)²?

Back

The square of a complex number is given by (a + bi)² = a² + 2abi + (bi)² = (a² - b²) + (2ab)i.