Complex Numbers Basic

A complex number is a number that can be expressed in the form $$a + bi$$ , where $$a$$ is the real part, $$b$$ is the imaginary part, and $$i$$ is the imaginary unit defined as $$\text{sqrt}(-1)$$ .

The imaginary unit 'i' represents the square root of -1, i.e., $$i = \text{sqrt}(-1)$$ .

To add complex numbers, combine their real parts and their imaginary parts separately. For example, $$(a + bi) + (c + di) = (a + c) + (b + d)i$$ .

To subtract complex numbers, subtract their real parts and their imaginary parts separately. For example, $$(a + bi) - (c + di) = (a - c) + (b - d)i$$ .

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

To multiply complex numbers, use the distributive property: $$(a + bi)(c + di) = ac + adi + bci + bdi^2$$ , and remember that $$i^2 = -1$$ .

The modulus of a complex number $$a + bi$$ is given by $$|a + bi| = \text{sqrt}(a^2 + b^2)$$ .

The argument of a complex number $$a + bi$$ is the angle $$\theta$$ formed with the positive real axis, calculated as $$\theta = \text{tan}^{-1}(\frac{b}{a})$$ .

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

The result of $$i^2$$ is -1.