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 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 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 complex numbers.

Front

How do you find the magnitude of a complex number?

Back

The magnitude of a complex number a + bi is given by the formula |a + bi| = √(a² + b²).

Front

What is the result of (1 + 2i) + (3 + 4i)?

Back

The result is (1 + 3) + (2 + 4)i = 4 + 6i.

Front

What is the result of (5 + 6i) - (2 + 3i)?

Back

The result is (5 - 2) + (6 - 3)i = 3 + 3i.

Front

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

Back

The result is (2*4 + 2*(-i) + 3i*4 + 3i*(-i)) = 8 - 2i + 12i + 3 = 11 + 10i.