Numbers Systems

Rational Numbers:
A rational number is a number which can be put in the form p/q where p, q ∈ Z ^ q ≠ 0. The numbers √16, 3.7, 4 etc., are rational numbers.
Irrational Numbers:
Irrational numbers are those numbers which cannot be put into the form p/q where p, q ∈ Z and q ≠ 0.
The numbers √2, √3, 7/√5, √(5/16) are irrational numbers.
Decimal Representation Of Rational and Irrational Numbers
Terminating Decimals:
A decimal which has only a finite number of digits in its decimal part, is called a terminating decimal.
Thus 202.04, 0.0000415, 100000.41237895 are examples of terminating decimals.
Since a terminating decimal can be converted into a common fraction, so every terminating decimal represents a rational number. 
Recurring Decimals:
This is another type of rational numbers. In general, a recurring or periodic decimal is a decimal in which one or more digits repeat indefinitely.
A recurring decimal can be converted into a common fraction. So every recurring decimal represents a rational number. 
Vice Versa:
A non-terminating , non-recurring decimal is a decimal which neither terminates nor it is recurring. It is not possible to convert such a decimal into a common fraction. Thus a non-terminating, non-recurring decimal represents an irrational number.
Pi(π):
Pi(π) is defined as a constant ratio of the circumference of any circle to the length of its diameter i.e.,
π=Circumference Of Any Circle/Length of Its Diameter
An approximate value of π is 22/7, a better approximation is 355/113 and a still better approximation is 3.14159.
Properties Of Real Numbers:
1. Addition Laws:

• Closure law of addition
• Associative law of addition
• Additive identity
• Additive inverse
• Communicative law of addition

2. Multiplication Laws:

• Closure law of multiplication
• Associative law of multiplication
• Multiplicative identity
• Multiplicative Inverse
• Communicative law of multiplication

3. Multiplication - Addition Laws:

• Distributive law of multiplication over addition
• Distributive law of addition over multiplication

4. Properties Of Equality:

• Reflexive Property
• Symmetric Property
• Transitive property
• Additive Property
• Multiplicative Property
• Cancellation property w.r.t addition
• Cancellation property w.r.t multiplication

5. Properties Of Inequality (Order Properties):

• Trichotomy property
• Transitive property
• Additive property
• Multiplicative property

Complex Numbers:
The numbers of the form x + iy, where x, y ∈ ℝ and i=√-1, are called complex numbers, here x is the real part and y is called imaginary part of the complex number. For example, 3 + 4i, 2-(5/7)i etc, are complex numbers.
Important Note:
Every real number is a complex number with 0 as its imaginary part.
Conjugate Complex Numbers:
Complex numbers of the form (a + bi) and (a - bi) which have the same real parts and whose
imaginary parts differ in sign only, are called conjugates of each other. Thus 5 + 4i and 5 - 4i ,
 -2 + 3i and -2 - 3i , -√5i and √5i are three pairs of conjugate numbers.
Important Note: 
A real number is self-conjugate.
Coordinates:
If a point A of the coordinate plane corresponds to the ordered pair (a, b) then a, b are called the coordinates of A, a is called the x-coordinate or abscissa and b is called the y-coordinate or ordinate.
Modulus Of The Complex Number:
The modulus of a complex number is the distance from the origin of the point representing the number. The modulus of a complex number is generally denoted as: |x + iy| or |(x,y)|. For convenience, a complex number is denoted by z.
If z = x + iy = (x,y), then |z| = √(x² + y²).
Polar Form Of a Complex Number:
As we know x = rcosθ and y = rsinθ where r = |z| and θ is called argument of z.
Hence, x + iy = rcosθ + irsinθ     .........(1)
Where, r = √(x² + y²) and θ = tan^-1(y/x).
Equation (1) is called the polar form of the complex number z.
De Moivre's Theorem:
(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ).