# Number System

A number system is defined as a writing system as it expresses numbers. We can say it is a mathematical notation for representing numbers of a provided set by utilizing digits or other symbols in a consistent manner. It gives a unique representation of each number and represents the arithmetic and algebraic structure of the figures. On the other hand, It also permits us to use arithmetic operations like addition, subtraction and division.

Natural Number-: Counting numbers are known as natural numbers.

Thus, 1,2,3,4,5,6,….,etc., are all natural numbers

Whole Number-: All natural numbers together with 0 form the collection of all whole numbers.

Thus, 0,1,2,3,4,5,6,….,etc., are all whole number.

Remarks-:     (i) Every natural number is a whole number.

(ii) 0 is a whole number which is not a natural number.

Integers-: All natural numbers, 0 and negatives of natural numbers form the collection of all integers.

Thus, …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …,etc., are all integers.

Remarks-:   (i) Every natural number is an integer.

(ii) Every whole number is an integer.

## Representation of Integers on Number Line

Draw a line XY which extends endlessly in both the directions, as indicated by the arrowhead in the diagram below.

Take any point O on this line. Let this point represent the integer 0 (zero). Now, taking a fixed length, called unit length, set off equal distances to the right as well to the left of O.

On the right-hand side of O, the points at distance of i unit, 2 units, 3 units, 4 units, etc., from O denote respectively the integers 1,2,3,4, etc.

Similarly, on the left-hand side of O, the points at distance of i unit, 2 units, 3 units, 4 units, etc., from O denote respectively the integers -1 -,2, -3, -4, etc.

Since the line XY can be extended endlessly on both sides of O, it follows that we can represent each and every integer by some point on this line.

For instance, starting from O and moving to its right, after 836 units, we get a pont, which represents the integer 836.

Similarly, starting from O and moving to its left, a point after 750 units, represents the integer -750.

Thus, each and every integer can be represented by some point on this line.

This line is known as number line. Number system

We shall show further that there are some more types of numbers lying on this line.

## Rational Numbers

The numbers of the form \frac { p }{ q } , where p and q are integers and q \neq 0, are known as Rational Numbers.

Remarks-:       (i) 0 is a rational number, since we can write, 0=\frac { 0 }{ 1 }.

(ii) Every natural number is a rational number, since we can write, 1=\frac { 1 }{ 1 }, 2=\frac { 2 }{ 1 }, 3=\frac { 3 }{ 1 }, etc.

(iii) Every integer is a rational number, since an integer a can be written as \frac { a }{ 1 }. (Number system)

## IRRATIONAL NUMBERS

A number which can] neither be expressed as a terminating decimal nor as a repeating decimal is called an irrational number.

Thus, nonterminating, non repeating decimals are irrational numbers.

## REAL NUMBERS

A number whose square is non- negative, is called a real number.

In fact, all rational and all irrational numbers form the collection of all real numbers.

Every real number is either rational or irrational.

Consider a real number. (Number System)

(i) If it is an integer or it has a terminating or repeating decimal representation then it is rational.

(ii) If it has a nonterminating and nonrepeating decimal representation then it is irrational.

The totality of rational and irrationals form the collection of all real number.

## SURD

If the root of a number cannot be exactly obtained, the root is called a surd or an irrational number. (Number system)

## Types of Number System

### Decimal Number System

The decimal system of 10 numerals or symbols (Deca means 10, that is why this system is called decimal system). These 10 symbols are 0,1,2,3,4,5,6,7,8,9 ; using these symbols as digit of number, we can express any quantity. The decimal system, also called the base-10 system because it has 10 digits, has evolved naturally as a result of the fact that man has 10 fingers.

### Binary Number System

In the binary system there are only two symbols or possible digit values, 0 and 1.

Even so, this base-2 system can be used to represent any quantity that can be represented in decimal or other number systems.

The binary system is also a positional-value system, wherein each binary digit has its own value or weight expressed as a power of 2.

### Octal Number System

The octal number system is very important in digital computer work. The octal number system has a base of eight, meaning that it has eight unique symbols: 0,1,23,4,5,6 and 7. Thus, each digit of an octal number can have any value from 0 to 7. (Number System)

The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It uses the digits 0 through 9 plus the letters A, B, C, D, E, and F as the 16 digit symbols.

## Representation of Numbers on The Number Line by Means of Magnifying Glass-:

The process of visualization of numbers on the number line through a magnifying glass is known as successive magnification.

Sometimes, we are unable to check the numbers line 3.765 and 4.\overline { 26 } on the number line, we seek the help of magnifying glass by dividing the part into subparts and subparts into again equal subparts to ensure the accuracy of the given number. (Number System)

### Method to find such numbers on the number line-:

1. Choose the two consecutive integral number in which the given number lies.
2. Choose the two consecutive decimal points in which the given decimal part lies by dividing the two given decimal part into required equal parts.
3. Visualise the required number through magnifying glass.

## Fundamental Theorem of Arithmetic, Unique Factorization Theorem

Statement-: Every positive integer a>1 can be expressed as the unique product of positive primes except for the order of the factors.

Proof of the unique factorization theorem

1.  Existence-: Let a> be factored

If a is a prime number, the theorem is proved.

If a be composite then, there exists a prime { p }_{ 1 } such that

a= { p }_{ 1 }b

## G.C.D and L.C.M. of Numbers with the help of prime Factorization

The students are well versed in resolving numbers into prime factors and finding the g.c.d. And l.c.m. with the help of prime factors. This method of finding g.c.d and l.c.m by prime factorisation is popular as it requires division by small numbers. We now take one example to explain the procedure. (Number System)

Example- Find the g.c.d and l.c.m of 144, 112 and 418

Solution- 144=2 x 2 x 2 x 2 x 3 x 3

112=2 x 2 x 2 x 2 x 7

418=2 x 11 x 19

\therefore  g.c.d= 2, l.c.m= { 2 }^{ 4 }\times { 3 }^{ 2 }\times 7\times 11\times 19=210672

## Working rule for Determining H.C.F.

1. Factorise the numbers into product of primes expressed in exponential form.
2. Select the lowest of the powers of common primes.
3. The product of primes with lowest powers in H.C.F.

## Working rule for Determining L.C.M.

1. Factorise into product of primes expressed in exponential form.
2. Select the highest power of a prime present in all or some of the numbers.
3. The product of prime with highest powers is L.C.M.

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Source-: Ts aggarwal & R.L Arora References

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