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 the square root of -1.

Front

What does i^2 equal?

Back

i^2 = -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 product of two complex numbers (a + bi)(c + di)?

Back

The product is given by (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.

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 modulus of a complex number a + bi?

Back

The modulus is given by |a + bi| = √(a^2 + b^2).

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 x-axis, calculated as θ = arctan(b/a).

Front

How do you divide complex numbers?

Back

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, (a + bi) / (c + di) = [(a + bi)(c - di)] / (c^2 + d^2).