Complex Numbers

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 $$i = \frac{\text{sqrt}(-1)}{1}$$ .

The imaginary unit $$i$$ is defined as $$i = \text{sqrt}(-1)$$ . It is used to extend the real number system to include solutions to equations that do not have real solutions.

$$\text{sqrt}(-1) = i$$ .

$$i * i = i^2 = -1$$ .

$$\text{sqrt}(-x) = i\text{sqrt}(x)$$ .

The standard form of a complex number is $$a + bi$$ , where $$a$$ is the real part and $$b$$ is the imaginary part.

To add complex numbers, combine the real parts and the imaginary parts: (3 + 5) + (4i + 2i) = 8 + 6i.

To subtract complex numbers, subtract the real parts and the imaginary parts: (7 - 2) + (3i - 5i) = 5 - 2i.

Using the distributive property: 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i.

The conjugate of $$a + bi$$ is $$a - bi$$ .