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# NUMBERS

Kusal Thapa

The simple digits or numbers (0,1,2,3,4 …) that we learn in our childhood influence our daily life than we could ever imagine. Have you ever wondered how our lives would be without these numbers? Our Birthdays, telephone numbers, wake up time, Bank accounts everything would not have arranged in this way as they are nowadays. Now let us define the term “Number”

Number is an arithmetical value and is defined as a Mathematical object which is used to count, measure and label. This can be expressed by word, symbol or figure.

For example: Five (word), 5 (symbol)

Types of Numbers:

Over the time, many of the infinite arrays of numbers derivable from basic ten digits have been classified into a variety of number types according to their properties that they follow. And they are

1. Natural Number:

The numbers {1, 2, 3, 4 …} are commonly called natural numbers the natural numbers are also referred as counting numbers. However, some other definitions also include Zero. For example from Peanos first axiom 0 is also a natural number and also from Axiom of infinity 0 is a natural number. It is denoted by script N.

2. Whole Number:

Whole numbers are simply the natural number or counting number along with zero. That is W= {0, 1, 2,3,4…}.           Whole number does not consist of negative numbers. Every Natural number is a whole number.

3. Integers:

Integer is a Latin word which gives the meaning whole and Integers are the numbers which can be written without a fractional component, so called as whole. Simply Integer is the set of positive Whole numbers, Negative whole numbers and Zero. Z= { … -3, -2, -1, 0, 1, 2, 3 …} 4. Rational numbers:

Rational numbers are those numbers which can be expressed as ratio of any two integers i.e. p/q. Where p is numerator and q is a non-zero denominator.

For example:

1. 2/3 is a rational number.

Here, p=2 and q=3 (q ≠ 0)

1. √(3)/ 4 is also a rational number.

Here, p= √3 and q=4 (q ≠ 0)

1. 0 is also a Rational Number as 0 can be written in the ratio as: 0/5    Here, p=0 and q=5 (q ≠ 0) so it satisfies definition of rational number.

5. Real Numbers:

The Real numbers are every those numbers which lies in a real line. Generally, it is the union of rational and irrational number. That means it also contains whole numbers and natural number. In mathematics, unknown or unspecified real numbers are usually denoted by lower case italic letter u through z.

Or we can simply define real number as the numbers except infinity and complex numbers.  “Every integer is a Real number but, converse may not be true.”

For example:  a) π (pi) b) ¾ c) 13 d) 12.68 e) -0.8645

6.Irrational Number:

Irrational numbers are those numbers which cannot be expressed as exact fraction of any two integers (That is p/q where both p and q are integers.). Irrational numbers has decimal expansion that neither terminates nor become periodic. The most famous Irrational number is √2 , which is also known as Pythagoras number.

Other examples of irrational numbers are: a) π (pi) (3.14159265359…) b) e (exponential) (2.71828182846…) c) √2 (1.41421356237…) etc Here in every example the numbers after decimals are neither terminates nor repeats or become periodic. Or simply we can define Irrational numbers as all the real number which are not rational. “The irrational numbers π  and e are also known as transcendental numbers”

7. Imaginary Number:

An Imaginary number is defined as a Real number multiplied by imaginary unit “i”. Where i=  In simple words it can also be defined as a number which gives negative result when a number is squared. If we try every square of real number we always get positive numbers. Even the squares of negative number give positive result. So we can consider a letter i which on squaring gives negative result i.e.    i2=-1,   which implies that i= √-1 So i is defined as imaginary unit. Examples of imaginary numbers are 2i, 3i etc. Lastly..

8. Complex Number:

The numbers of the form a+bi is known as Complex number. Where, a and b are real numbers and i is imaginary unit. Simply we can say combination of real and imaginary number is defined as Complex number. Normally it is denoted by  z= a+bi

Few examples of complex numbers are; a) 2+3i    b) 3-5i    c) 45i