Powers of i and 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 = √(-1).

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √(-1). It is used to represent imaginary numbers.

The distance from the origin to a point (a, b) in the complex plane is calculated using the formula: distance = $$\text{sqrt}(a^2 + b^2)$$ .

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

The polar form of a complex number is expressed as r(cos θ + i sin θ), where r is the modulus and θ is the argument of the complex number.

To convert a complex number a + bi to polar form, calculate r = $$\text{sqrt}(a^2 + b^2)$$ and θ = $$\tan^{-1}(\frac{b}{a})$$ .

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

To add two complex numbers (a + bi) and (c + di), combine the real parts and the imaginary parts: (a + c) + (b + d)i.

To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (ac - bd) + (ad + bc)i.

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