Imaginary and Complex Numbers

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as $$i = \frac{\text{sqrt}(-1)}{1}$$ .

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.

The product is given by: (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

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

The powers of i cycle every four: $$i^1 = i$$ , $$i^2 = -1$$ , $$i^3 = -i$$ , $$i^4 = 1$$ . For any integer n, find n mod 4 to determine the equivalent power.

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

To add two complex numbers, simply add their real parts and their imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.

The result is (a - c) + (b - d)i.

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

To multiply a complex number by a real number, multiply the real part and the imaginary part by that real number: k(a + bi) = ka + kbi.