Imaginary & Complex Numbers

Imaginary numbers are defined by the unit imaginary number i, where i^2 = -1. Imaginary numbers involve the square root of negative numbers.
A complex number is a number of the form a + bi, where a and b are real numbers and i represents the square root of -1.
The ‘a’ part is referred to as the real part of the complex number, and ‘bi’ as the imaginary part.
When a=0, the complex number is purely imaginary, and when b=0, it is a real number.
The conjugate of a complex number is obtained by changing the sign of its imaginary part. So, the conjugate of a + bi is a - bi.
Adding or subtracting complex numbers is carried out by separately adding or subtracting the real and imaginary parts.
In contrast, multiplying complex numbers involves applying the distributive law and recalling that i^2 = -1.
The absolute value, or modulus, of a complex number a + bi is the distance from the origin to the point (a,b) in the complex plane and is given by sqrt(a^2 + b^2).
Complex numbers can also be represented in polar form, r(cosθ + isinθ), where r is the modulus of the complex number and θ is the argument or angle.
Euler’s Formula provides a useful way to handle complex numbers in polar form: e^(iθ) = cosθ + i sinθ.
De Moivre’s theorem, another key concept, allows powers of complex numbers to be calculated easily. It states that (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ).
Roots of complex numbers can be obtained using De Moivre’s theorem.
It’s valuable to know how to plot complex numbers in the Argand Diagram, which is similar to plotting (a,b) in the Cartesian coordinate system.
Finally, remember how to use the principle of mathematical induction when grappling with proofs involving complex numbers for multiple cases.