Number system

 

Number

    A number is a arithmetic value used for representing quantity and used for calculations.

Example : 1,2,5 7,-1,3,0....

Types of numbers

There are many types of numbers such as

1. Natura numbers
2. Whole numbers
3. Integers
4. Fractions
5. Rational numbers
6. Irrational numbers
7. Decimal numbers
8. Real numbers

Natural Numbers 

Natural numbers are the countable numbers.Natural numbers are the positive countable numbers,start from 1.

Example:1,2,5,100,10000........

Natural numbers can be one digit number such as 1 to 9.two digits number 10 to 99.three digit numbers 100 to 999 and so on.

In Indian system number can be read as unit for ones digit,ten for tens digit hundred for hundred digit thousand for thousands digit ten thousand ,then lakh ,ten lakh and so on.

In international number system one for ones digit ten for tens digit hundred for hundreds digits thousand for thousands digits ten thousand for ten thousand digit hundred thousand for lakh digit and million for ten lakh digits ten million and hundred million and billion so on.

The smallest Natural number is one.

The smallest odd Natural number is 1.

The smallest even Natural numbers is 2.

The smallest prime number is 2.

The smallest composite Natural number is 4.

The Natural numbers are denoted by N.

Whole numbers 

The whole numbers are all Natural numbers including 0.the Whole numbers starts from 0.

Example:any Natural numbers,0 ,24578....

The smallest Whole number is 0.

The smallest odd whole number is 1.

The smallest even whole numbers is 0.

The smallest prime number is 2.

The smallest composite Natural number is 4.

The whole numbers are denoted by W.

Integers

All the whole numbers including negative numbers are called Integers.

Here all positive negative numbers are Integers including zero also.

Example: 0,97,-35,76,-43,-09,7 ,all natural numbers are integers and all whole numbers are integers.....

• All Natural numbers are integers but all the integers not Natura numbers numbers.
• All the whole numbers are integers but all the integers are not whole numbers.
• Integers are denoted by English alphabets Z.
• Z is for zhelen means negative numbers.

Fractions

       The numbers which can written in the form of `\frac (P)Q`,where P and Q are whole numbers called Fractions.fractions are the part of a whole.

The fractions indicate the whole thing is divided into how many parts,and how many parts being selected. 

Example: `\frac{3}{4}`.

In above example  3 parts have been selected in 4 parts of given whole.

There 3 types of Fractions 

1. Proper fractions 
2. Improper fractions 
3. Mixed fractions 

Proper fractions 

The fraction number which has numerator less than the denominator called Proper fraction. 

Example: `\frac{5}{8}`

Here denominator shows the total number of parts of a whole.and numerator indicates number of parts selected from total number of parts of a whole.

Improper fractions 

The fraction with greater numerator and lessor denominator called Improper fractions.

The numerator of the given fraction will be more than that if denominator in Improper fractions. 

In Improper fractions The number of parts which is to be selected are more than the total number of parts of a whole.

Example `\frac{5}3`

Here numerator is 5 which is greater than the denominator that is 3.

Mixed fractions 

The fraction which has a whole with part of a whole called mixed fractions.

In mixed fractions The whole part and it's part will given.

The mixed fraction can written in form as `8\frac 5/7`.here it is saying that there are 8 whole are there and among 7 parts if a while 5 are selected.

Example `8\frac {5}{7}`.

Rational numbers 

Rational numbers the numbers which can written in the form of `\frac{P}{Q}`. Where P and Q are integers.

The main difference between Rational numbers and fractions is ,Rational numbers can be negative numbers also but fractions are never negative.

All integers are Rational numbers.

Example:-3,2 -23,`\frac {-4}{5}`.......

Rational numbers can have terminating Decimal expansion or non terminating repeating Decimal expansion.

Example:- `\frac{4}{8}`=0.5 which have terminating Decimal expansion.

       `\frac{2}{3}`=0.6.......... have non terminating Decimal expressions. 

In both cased the numbers are Rational numbers. 

Irrational numbers

Numbers other than rational numbers are irrational numbers.irrational numbers have non terminating and non repeating Decimal expansion. 

Example:`\sqrt3`,`\sqrt2 `.

Here the root of any non perfect square is irrational number.

Irrational numbers have non terminating non repeating Decimal expansion.

Example:`\frac{4}{7}` =0.5714...........

For finding whether the given devision is terming or non terminating, we can easily find by checking the prime Factorisation of denominator is in the from of `2^m`or `5^n`or both where m and n are natural numbers.then the Decimal expansion is terminating. 

If the prime Factorisation of denominator of given devision is of the form `7^m`or `13^n`then the Decimal expressions ate non terminating non repeating.hence the given numbers are irrational numbers.

Decimals

The numbers with Decimal point called Decimal numbers.

Example : 10.6

Read as ten point six

Here the numbers left to Decimal points are unit tens hundred so on.but the numbersright to the Decimal point are divisible of give place as tenth,hundredth,thousandth.the number how much away from the given Decimal the by that place value we should decide it.

Example: 1245.789

Here it will written as `1×10^4+2×10^3+4×10^2+5×10^2+\frac{7}{10}+\frac{8}{100}+\frac{9}{100}`.

It is the expanded form of 1245.789.