NUMBER THEORY

NUMBER THEORY:

Natural Numbers:

The numbers 1 , 2 , 3 , 4 … are called natural numbers, their set is denoted by N. Thus N = {1 , 2 , 3 , 4 , 5 …}

Integers:

The numbers … −3 , −2 , −1 , 0 , 1 , 2 , 3 … are called integers and the set is denoted by I or Z.
Thus I (or Z) = {… −3 , −2 , −1, 0,1, 2, 3 …}.

Remarks:

⋄ Integers 1 , 2 , 3 , … are called positive integers or natural number and denoted by I+ or N.

⋄ Integers …, −3, −2, −1 are called negative integers and denoted by I

⋄ Integers 0 , 1 , 2 , 3 , … are called whole numbers or non-negative integers.

⋄ Integers …, −3, −2, −1, 0 are called non-positive integers

Rational Numbers:

The numbers which can be expressed in the form p/q where p and q are integers, H.C.F. of p and q is 1 and q ≠ 0, are called rational numbers and their set is denoted by Q.

Thus Q = { p/q : p , q ∈I and q ≠0 and HCF of p , q is 1 }.

It may be noted that every integer is a rational number since it can be written as p/1.

It may also be noted that all recurring decimals are rational numbers. e.g., p = = 0.33333…

Then, 10 p − p = 3

⇒  p = 1/3 , which is a rational number.

Irrational Numbers:

There are numbers, which can not be expressed in p/q form. These numbers are called irrational numbers and their set is denoted by Qc (i.e. complementary set of Q)

e.g. √2 , 1 +, √3 , π , e , √5 etc.

Irrational numbers can not be expressed as terminating decimals or recurring decimals.

Real Numbers:

The complete set of rational and irrational numbers is the set of real numbers and is denoted by R. Thus R = Q ∪ QC

It may be noted that N⊂ I⊂ Q⊂ R. The real numbers can also be expressed in terms of the positions of a point on the straight line.

The straight line is defined as the real number line wherein the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa.

All the numbers defined so far follow the order property i.e. if there are two numbers a and b then either a < b or a = b or a > b.
There is another set of numbers called Complex numbers (Universal set of numbers), denoted by C.
Thus C = {x + iy : x ∈ R, y ∈ R} where i (iota) is a symbol defined as i2 = -1
It may be noted that R ⊂ C. The study of complex numbers is not the part of this phase therefore we will deal only with the set of Real numbers.

Number Chart:

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