Any number that matches to a number from \(1, 2, 3, 4, ….\) is called a Natural Number or Counting number.

Example

\(4, 9 , 11, 35, 99, 500\) or \(9876\) are natural numbers.

Natural numbers are used for counting and ordering. On the basis of that they can be further divided into cardinal and ordinal numbers.

Cardinal numbers are used for counting of objects. For example, there are 7 apples in a box or there are 7 days in a week. In both examples, we are counting objects which are 7.

Ordinal numbers are used to order the positions of objects or in other words, we can tell the position of an object. For example, Ist position or 2nd position and so on. The example of ordinal numbers is, girls stood Ist in the race. Another example is, this is the second largest country in the world.

Note

Set of natural numbers is denoted by **N**.

Any number that matches to a number from \(0, 1, 2, 3, 4, ….\) is called a Whole Number.

Example

\(0, 2, 14 , 23, 48, 172, 623\) or \(1245\) are whole numbers.

Note

Set of whole numbers is denoted by **W**.

All natural numbers and negative of natural numbers including zero are called Integers. It includes 0 and all negative numbers and positive natural numbers.

Example

\(-65, -12, -5, 0, 56, 89\) or \(354\) are Integers.

Also, the negative numbers are **additive inverse** of positive corresponding numbers.

What are additive inverse? Additive inverse refers if sum of a positive number and corresponding negative number is equal to 0.

Example

2 is **additive inverse** of -2.
Because -2 + 2 = 0.

Note

Zero is neither a negative number nor positive number. It is a neutral number.

Set of integers is denoted by **Z**.

The numbers each of which can be expressed in the form of \(\frac{p}{q}\), where p and q are integers and q is not equal to zero, are called Rational Numbers. Besides, the rational numbers are also non terminating and repeating numbers.

Example

\(\frac{2}{5}\), \(\frac{1}{3}\), \(\frac{8}{9}\), \(\frac{3}{10}\) etc.

Note

Set of rational numbers is denoted by **Q**.

Rational number is also a **real number.**

The numbers which are not rational are called irrational numbers. Besides, irrational numbers are also non terminating and non repeating numbers.

Example

\(\sqrt 2\), \(\sqrt 5\) etc.

Set of rational numbers and irrational numbers is called Real Numbers. In other words, real numbers include all rational and irrational numbers.

Example

\(\sqrt 7\), \(\frac{1}{8}\), \(\frac{3}{10}\) etc.

All real numbers can be represented on a number line which is also called a real line.

Note

Set of real numbers is denoted by **R**.

Any Natural number greater than \(1\), which is divisible by \(1\) and only by itself is called Prime Number.

Example

\(2\), \(3\), \(5\), \(7\), \(11\) etc.

Note

\(2\) is the smallest Prime Number.

Prime number cannot be represented by a rectangle or square.

Representation of prime numbers

Any Natural number, which is greater than \(1\) and is not a Prime Number is called a Composite Number.

Example

\(4\), \(6\), \(8\), \(9\), \(10\) etc.

Note

\(1\) is neither a Prime Number nor Composite Number.

Composite number can be represented by a rectangle or square.

Representation of composite numbers

Two numbers are called to be coprime or co-prime if they have only \(1\) as common factor.

In other way, we can say if greatest common divisor of two numbers is 1, then the two numbers are coprimes.

Example

\((2,3)\), \((14,15)\), \((8,13)\) etc.

Two numbers are called Twin Primes if they differ by 2 only or we can say that there is only one composite number between them. One prime number can be 2 more or 2 less than another prime number.

Example

\((3,5)\), \((29,31)\), \((71,73)\) etc.

A set of three consecutive Prime Numbers if the smallest and the largest differ by 6 and must be in the form of (p, p + 2, p + 6) is called a Prime Triplet.

Example

\((5,7,11)\)

Why?

Because, 5, 7 and 11 is in the form of p, p + 2, p + 6

where p = 5

p + 2 = 5 + 2 = 7

and p + 6 = 5 + 6 = 11