NUMBER SYSTEM | BEST CONCEPTS OF DEFINITIONS ,TYPES&TRICKS

 SOHEL SAHOO: Hello Everyone,

What is Number System in Maths?

A number system is defined as a system by which we represent  any sort of numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a continuous  way.

It provides a unique manner of writing  and expressing  the arithmetic and algebraic structures  i.e. to operate arithmetic operations like addition, subtraction,Multiplication and division.

Some Important Definitions:-

Number:-

A number is a mathematical object  used to count, measure, and label or mark ;  moreover, really an idea in our minds. The original or genuine examples are the counting numbers 1,2,3,4…and so on.

We write or talk about numbers using numerals such as "5" or "five".

But we could also hold up our 5 fingers, or show some figure indicating sense 5 or tap the ground 5 times.

These are all different ways of referring to the same number.

There are also special numbers (like s π) that can't be written  exactly as it is non-terminating ,but are still numbers because we know the idea behind them. 

Numerals:-

But, numbers are represented by certain symbols or names called numerals which also can be a letter or combinations of letters from any alphabet (like roman numerals).

Examples:-  4,50, five all are numerals.

NOTE:-  The number is an idea whereas the numeral is how we write it.                                  Digits -> Numerals -> Numbers

So digits make up numerals, and numerals stand for an idea of a number.

Digits:-

A digit is a basic single symbol used to make numerals. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numerals i.e. decimal system.

   In hexadecimal system we use other symbols too like letter A,B,C,D,E,F.

Types of Number System:-

There are various types of number system in mathematics & actually we can make infinite systems. Till now discovered & used; the four most common number systems are:

1. Decimal number system (Base- 10) [Digits: 0,1,2,3,4,5,6,7,8,9]
2. Binary number system (Base- 2) [Digits: 0 and 1]
3. Octal number system (Base-8) [Digits: 0,1,2,3,4,5,6,7]
4. Hexadecimal number system (Base- 16) [Digits: 0,1,2,3,4,5,6,7,8,9,(A=10),(B=11),(C=12),(D=13),(E=14),(F=15)]

Now , we shall discuss about general  numbers i.e. Decimal Number system.

Classification Of Numbers:-

Natural Numbers:-

Counting numbers, viz. {1, 2, 3, 4, ……} . Denoted by N

0 is NOT a Natural number .                                                     

Whole Numbers:-

Invention of 0 gave rise to set of Whole Numbers.

Set of Natural numbers and also 0 (Zero) viz. {0, 1, 2, 3, 4, ……} . Denoted by W 

Integers:-

Invention of positives and negatives gave rise to set of Integers.

Negative and positive Natural numbers and 0 viz. {……, –3, –2, –1, 0, 1, 2, 3, ……}

Remember:-

1.  Zero is neither positive nor negative.
2. Thus, set of positive integers will not include 0 and will be {1, 2, 3, ……} i.e. set of Natural numbers.
3. Similarly, set of non-negative integers will include 0 (as it is not negative) and will be {0, 1, 2, 3, ……} i.e. set of Whole numbers.

Fractions:-

 In addition to an integer part, fractions also have ‘part of 1’.

.Example:-  one-half of 1 i.e.`\frac{1}{2}` ; one-tenth of 1 i.e. `\frac{1}{10}` ; two-third of 1 i.e. `\frac{2}{3}`

Fractions could also have an Integral part in addition to a ‘part of 1’. 

Fraction like `\frac{15}{4}` = 3 + `\frac{3}{4}` i.e. 3 integer and three-fourth of 1.

And can be written in the mixed-form as `3\frac{3}{4}` _.

_NOTE:-

• _{Proper fraction is a fractional number less than 1 i.e. it does not have any integral part and is just a ‘part of 1’.} In a proper fraction the numerator is less than the denominator. 

E.g.  `\frac{2}{3}`,`\frac{7}{10}`, etc.

• Improper fraction is a fractional number more than 1 i.e. it has an integral part and also a ‘part of 1’. In an improper fraction, the numerator is more than the denominator. 

E.g. `\frac{7}{2}`,`\frac{5}{4}` etc.

All these improper fractional numbers can be written in the mixed-form e.g.  `3\frac{1}{2}`, `1\frac{1}{4}`.

Rational Numbers:-

All Integers and Fractions together form the set of Rational Numbers and the set of Rational numbers is denoted by Q

By definition, it is the set of all numbers that can be expressed in the form, `\frac{p}{q}` i.e. all fractions.    where p and q are Integers and obviously, q ≠ 0.

• All integers can be expressed in the required form,`\frac{p}{q}`, with q = 1. E.g. 3, 0, –10, are

same as `\frac{3}{1}`,`\frac{0}{1}`,`\frac{-10}{1}`.

Hence, all integers are rational numbers.

Irrational number:-

Numbers  that cannot be converted in the form `\frac{p}{q}`  i.e.

which are not rationals called irrational numbers.

Such numbers are `\sqrt{2}`,`\sqrt{3}` .

Real Numbers:-

The set of Rational and Irrational numbers together

is called as Real Numbers and is denoted by R.

• Real Numbers are numbers that can be

plotted on a number line.
Thus even 0.3333... is a unique point on the number

and so is `\sqrt{2}` 

and other Irrational numbers like `\pi `.

Every real number can be uniquely expressed in

decimal representation.

All terminating fractions can be also be expressed in

the form, `\frac{p}{q}`,

 with q being a power of 10 such that the decimal

point is eliminated

. E.g. 0.3, 4.57, –7.3333 are same as `\frac{3}{10}`,`\frac{457}{100}`&`\frac{-73333}{10000}`.

All non-terminating but recurring decimals

can also be expressed in the form,

`\frac{p}{q}`

. E.g. 0.333…=`\frac{1}{3}` ; 1.4545…=`\frac{16}{11}` ; -3.222…= `\frac{-29}{9}`

Just do the actual division to check that the equality

holds. Later we shall see

How to find the `\frac{p}{q}` form of recurring decimals.

As saw in  above diagram, decimals that are not

rational are those that are

non-terminating and non-recurring. This is the

set of Irrational Numbers.

Such numbers are `\pi` ,e, `\phi`(Golden ratio).

We found, the numbers do not terminate  nor

does form any recurring(repeating) pattern.

Even and Odd Numbers:-
Even and Odd are properties of Integers only. For entrance exam purposes, we would limit ourselves to non-negative Integers i.e. 0, 1, 2, 3,... otherwise whole integer set would taken into consideration.

 Even Number:-
A number which is divisible by 2 is called even number. viz. 0, 2, 4, 6, 8, …….
Even numbers are represented by 2n, where n = 0, 1, 2, ……
Zero is also an Even number.

 Odd Number:-
A number that is not divisible by 2 is called odd number. viz. 1, 3, 5, 7, 9, ……
Odd numbers are represented by 2n + 1, where n  = 0, 1, 2, …

Prime and Composite Numbers:-

Prime and Composite are properties of positive integer or Natural numbers ONLY.

 Prime numbers:-

Natural numbers (>1) that have exactly two distinct factors viz. 1 and itself

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, .......}

 Composite numbers:-

Natural numbers that have more than two distinct factors.

Prime numbers are important to us because we can consider all natural numbers to be made up of product of prime numbers. 

12 = 2 × 2 × 3 = 22 × 3,        60 = 2 × 2 × 3 × 5  = 22 × 3 × 5

400 = 2 × 2 × 2 × 2 × 5 × 5 = 24 × 52    540 = 2 × 2 × 3 × 3 × 3 × 5 = 22 × 33 × 5

This process of writing any number as a product of prime numbers is called Factorisation .

Points to note:-

• 1 is neither Prime nor Composite.
• 2 is the only even Prime number. 
• There are 25 prime numbers less than 100.15 from 1 to 50 &10 from 51 to 100.
•  Prime numbers greater than 3 are of the form 6n ± 1 i.e. one less than a multiple of 6 or one more than a multiple of 6. Not all numbers of the form 6n ± 1 are Prime BUT Prime numbers (> 3) have to be of the form 6n ± 1.

Relatively Prime:-

Two numbers are said to be relatively-prime to each other if they do not have any common factor, other than 1. .Or Set of two number having HCF/GCD=1.

example: {1,3},{9,25},{8,21} etc.

Co-Prime:-

It is the set of two prime numbers only and no other

condition.

Obviously HCF=1 so every coprime is relatively

prime but converse is not true.

Example: {2,3}, {7,19} etc.

Twin Prime:-

Set of two prime numbers having difference 2 is

called twin prime numbers.

example: {5,7},{11,13} etc.

Perfect Number:-

If sum of all factors(+ve divisors) of a number

(excluding number) is equal to

original number ,then no. is called a perfect number.

Or, If sum of all factors of a number

(including number) is equal to twice of original

number

,then it is called a perfect number.

example: 6 has divisors 1,2&3 (exluding 6)

1+2+3=6. So, 6 is a perfect number.

28 is also a perfect number. 28=1+2+4+7+14

NOTE:-

The general rule of finding rational numbers between a&b is given by

;where k is a natural number.

The general rule of finding irrational number

between a&b is given by

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